The Collected Papers of

STEPHEN SMALE Volume 1

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The Collected Papers of

STEPHEN SMALE Volume 1

•iMorMfi :..Ki"- KM& J*m& fewMMLtM-

Edited by

F. Cucker R. Wong

City University of Hong Kong

SINGAPORE UNIVERSITY PRESS NATIONAL UNIVERSITY OF SINGAPORE

^ | f e World Scientific WM Singapore *New Jersey • London • Hong Kong

Published by

World Scientific Publishing Co. Pte Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NI0766! UK office: 57 Shelton Street, Covent Garden. London WC2H 9HE

Library of Congreu Cataloglng-in-PnUication Data Smale, Stephen. 1930-

[Works. 2000] The collected papers of Stephen Smale / edited by F. dicker, R. Wong.

p. cm. ISBN 9810243073 (set) - ISBN 9810249918 (v. 1) -- ISBN 9810249926 (v. 2) - ISBN

9810249934 (v. 3) 1. Mathematics. 2. Computer science. 3. Economics. I. Cucker, Felipe, 1958- II.

Wong, R. (Roderick), 1944- III. Tide.

QA3 .S62525 2000 510~dc21 00-031992

British Library Cataloguirjg-bi-PubHcatk>n Data A catalogue record for this book is available from the British Library.

Copyright © 2000 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center. Inc.. 222 Rosewood Drive, Danvers. MA 01923, USA. In this case permission to photocopy is not required from the publisher.

Printed in Singapore.

Roderick Wong, Stephen Smale and Felipe Cucker

V

Digging a well at Hilberry's, 1953.

Our first ascent (in 1953) of the Grand Teton, New Route, with Jack Hilberry and Anne Blackwell.

vi

A young Stephen Smale.

Nat and Stephen Smale, Thanksgiving, Chicago, 1957.

vii

Tiger Leaping Gorge of the Yangtze River, 1999.

Clara and Stephen Smaie, Shenzhen, 1996.

viii

In the office, Hong Kong, 1998.

Mike Shub, Lenore Blum, Felipe Cucker and Stephen Smale, Dagstuhl, 1995.

ix

Clara Smale, Stephen Smale, Dick Karp and Bill Clinton.

X

xi

Foreword

On July 15, 2000 Steve Smale will turn seventy. To celebrate his birthday, an international conference will be held at City University of Hong Kong. Some months ago, we had the idoa of taking this opportunity to publish Steve's collected works so that the Bnal volumes could be ready by July 15, 2000. It was probably a timely decision since Steve, who had turned down previous suggestions to publish his collected work, accepted our proposal right from the start.

The reasons for publishing Steve's collected papers are only too clear. Steve Smale is one of the great mathematicians of this century, a fact acknowledged not only by the different distinctions he has been granted, but also by the breadth and the depth of his work. Its breadth is witnessed by his fundamental contributions to so many diverse areas of mathematics, ranging from Dynamical Systems to Theory of Computation, from Differential Topology to Mathematical Economics and from Calculus of Variations to Mechanics. Its depth can be seen from the fact that he has an amazing ability to lay foundations, invent techniques, and create new concepts and ideas, as well as being able to crack hard problems such as the Pomcare" conjecture for dimensions greater than or equal to five.

The structure of these volumes of collected papers reflects the breadth of Steve's contributions. We have avoided a purely chronological ordering of the papers. Each volume is divided into several parts, and each of these parts contains the papers Steve wrote on a specific subject. Within each part, the papers are sorted chronologically. In some sections, we have included a paper written by a world leader commenting on Steve's contributions to that particular subject. Volume I also contains papers by close friends and colleagues of Steve that describe different aspects of his work, and a paper written by Steve himself, specifically for these volumes, with retrospective remarks on his own work.

Many people and institutions have helped make this project possible. These include the different publishing houses that have kindly given us the permission to reprint Steve's papers, and World Scientific for its constant willingness to cooperate with us. Ms Colette Lam, Executive Assistant of the Liu Bie Ju Centre for Mathe­matical Sciences here at the City University of Hong Kong, has been of invaluable help in putting the volumes together. Last but not least, we want to thank Steve Smale for his continuous availability and his warm disposition towards our project.

As a final thought, we would like to point out that Steve Smale is a man of many accomplishments, a good number of which are not related to mathematics. For instance, he is a very kc en hiker, has been an accomplished sailor for years, and has managed to put together one of the finest collections of crystals in the world. He has produced excellent photographs of his crystals, and some of these photos grace the covers of these three volumes.

Felipe Cucker Roderick Wong

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XU1

Contents

VOLUME I

Research Themes 1

Luncheon Talk and Nomination for Stephen Smale (R. Bott) 8

Some Recollections of the Early Work of Steve Smale (M. M. Peixoto) 14

Luncheon Talk (R. Thorn) 17

Banquet Address at the Smalefest (E. C. Zeeman) 20

Some Retrospective Remarks 22

Part I. Topology

The Work of Stephen Smale in Differential Topology (M. Hirsch) 29

A Note on Open Maps 53

A Vietoris Mapping Theorem for Homotopy 56

Regular Curves on Riemannian Manifolds 63

On the Immersion of Manifolds in Euclidean Space (with R. K. Lashof) 84

Self-Intersections of Immersed Manifolds (with R. K. Lashof) 106

A Classification of Immersions of the Two-Sphere 121

The Classification of Immersions of Spheres in Euclidean Spaces 131

Diffeomorphisms of the 2-Sphere 149

On Involutions of the 3-Sphere (with M. Hirsch) 155

The Generalized Poincare Conjecture in Higher Dimensions 163

On Gradient Dynamical Systems 166

Generalized Poincar6's Conjecture in Dimensions Greater Than Four 174

Differentiable and Combinatorial Structures on Manifolds 190

On the Structure of 5-Manifolds 195

On the Structure of Manifolds 204

A Survey of Some Recent Developments in Differential Topology 217

The Story of the Higher Dimensional Poincard Conjecture (What actually happened on the beaches of Rio) 232

Part II. Economics

Stephen Smale and the Economic Theory of General Equilibrium (G. Debreu) 243

Global Analysis and Economics, I: Pareto optimum and a generalization of Morse theory 259

Global Analysis and Economics, IIA: Extension of a theorem

of Debreu 271

Global Analysis and Economics, III: Pareto optima and price equilibria 285

Global Analysis and Economics, IV: Finiteness and stability of

equilibria with general consumption sets and production 296

Global Analysis and Economics, V: Pareto theory with constraints 305

Dynamics in general equilibrium theory 314

Global Analysis and Economics, VI: Geometric analysis of Pareto

Optima and price equilibria under classical hypotheses 321

A Convergent Process of Price Adjustment and Global Newton Methods 335

Exchange Processes with Price Adjustment 349

Some Dynamical Questions in Mathematical Economics 365

An Approach to the Analysis of Dynamic Processes in Economic Systems 368

On Comparative Statics and Bifurcation in Economic Equilibrium Theory 373

The Prisoner's Dilemma and Dynamical Systems Associated to

Non-Cooperative Games 380

Global Analysis and Economics 398

Gerard Debreu Wins the Nobel Prize 438

Global Analysis in Economic Theory 440

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XV

Part ID. Miscellaneous

Scientists and the Arms Race 445

On the Steps of Moscow University 454

Some Autobiographical Notes 461

Mathematical Problems for the Next Century 480

VOLUME II

Part IV. Calculus of Variations (Global Analysis) and PDE's

Smale and Nonlinear Analysis: A personal perspective (A. J. Tromba) 491

A Generalized Morse Theory (with R. Palais) 503

Morse Theory and a Non-Linear Generalization of the Dirichlet Problem 511

On the Calculus of Variations 526

An Infinite Dimensional Version of Sard's Theorem 529

On the Morse Index Theorem 535

A correction to "On the Morse Index Theorem" 542

What is Global Analysis? 544

Book Review on "Global Variational Analysis: Weierstrass Integrals on a Riemannian Manifold" by Marston Morse 550

Smooth Solutions of the Heat and Wave Equations 561

Part V. Dynamics

On the Contribution of Smale to Dynamical Systems (J. Palis) 575

Discussion (S. Newhouse, R. F. Williams and others) 589

Morse Inequalities for a Dynamical System 596

On Dynamical Systems 603

Dynamical Systems and the Topological Conjugacy Problem for Diffeomorphisms 607

Stable Manifolds for Differential Equations and Diffeomorphisms 614

XVI

A Structurally Stable Differentiable Homeomorphism with an Infinite

Number of Periodic Points 634

Diffeomorphisms with Many Periodic Points 636

Structurally Stable Systems Are Not Dense 654

Dynamical Systems on /j-Dimensional Manifolds 660

Differentiable Dynamical Systems 664

Nongenericity of ^-Stability (with R. Abraham) 735

Structural Stability Theorems (with J. Palis) 739

Notes on Differential Dynamical Systems 748

The Q-Stability Theorem 759

Stability and Genericity in Dynamical Systems 768

Beyond Hyperbolicity (with M. Shub) 776

Stability and Isotopy in Discrete Dynamical Systems 781

Differential Equations 785 Dynamical Systems and Turbulence 791

Review of "Catastrophe Theory: Selected Papers, 1972-1977" by E. C. Zeeman 814

On the Problem of Reviving the Ergodic Hypothesis of Boltzmann

and Birkhoff 823

On How I Got Started in Dynamical Systems 831

Dynamics Retrospective: Great problems, attempts that failed 836

What is Chaos? 843

Finding a Horseshoe on the Beaches of Rio 859

The Work of Curtis T. McMullen 865

Part VI. Mechanics

Steve Smale and Geometric Mechanics (J. E. Marsden) 871

Topology and Mechanics, I. 889

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Topology and Mechanics, II. 916

Problems on the Nature of Relative Equilibria in Celestial Mechanics 936

Personal Perspectives on Mathematics and Mechanics 941

Part VII. Biology, Electric Circuits, Mathematical Programming

On the Mathematical Foundations of Electrical Circuit Theory 951

A Mathematical Model of Two Cells via Turing's Equation 969

Optimizing Several Functions 979

Sufficient Conditions for an Optimum 986

The Qualitative Analysis of a Difference Equation of Population Growth

(with R. F. Williams) 993

On the Differential Equations of Species in Competition 997

The Problem of the Average Speed of the Simplex Method 1000

On the Average Number of Steps of the Simplex Method of Linear Programming 1010

VOLUME ID

PartVIII. Theory of Computation

On the Work of Steve Smale on the Theory of Computation (M. Shub) 1035

The Work of Steve Smale on the Theory of Computation: 1990-1999

(L. Blum and F. Cucker) 1056

On Algorithms for Solving/(jc) = 0 (with M. Hirsch) 1076

The Fundamental Theorem of Algebra and Complexity Theory 1108

Computational Complexity: On the geometry of polynomials and

a theory of cost, Part I (with M. Shub) 1144

On the Efficiency of Algorithms of Analysis 1180

Computational Complexity: On the geometry of polynomials and a theory of cost, Part II (with M. Shub) 1215 On the Existence of Generally Convergent Algorithms (with M. Shub) 1232

XVU1

Newton's Method Estimates from Data at One Point 1242

On the Topology of Algorithms, I. 1254

Algorithms for Solving Equations 1263

The Newtonian Contribution to Our Understanding of the Computer 1287

On a Theory of Computation and Complexity over the Real Numbers: NP-completeness, recursive functions and universal machines

(with L. Blum and M. Shub) 1293

Some Remarks on the Foundations of Numerical Analysis 1339

Theory of Computation 1349

Complexity of Bezout's Theorem I: Geometric aspects (with M. Shub) 1359

Complexity of Bezout's Theorem II: Volumes and probabilities (with M. Shub) 1402 Complexity of Bezout's Theorem III: Condition number and packing (with M. Shub) 1421

Complexity of Bezout's Theorem IV: Probability of success;

Extensions (with M. Shub) 1432

Complexity of Bezout's Theorem V: Polynomial time (with M. Shub) 1453

The Godel Incompleteness Theorem and Decidability over a Ring (with L. Blum) 1477 Separation of Complexity Classes in Koiran's Weak Model (with F. Cucker and M. Shub) 1496

On the Intractability of Hilbert's Nullstellensatz and an Algebraic Version of ' W/V/>?" (with M. Shub) 1508

Complexity and Real Computation: A Manifesto (with L. Blum, F. Cucker and M. Shub) 1516

Algebraic Settings for the Problem "P*/VP?"

(with L. Blum, F. Cucker and M. Shub) 1540

Complexity Theory and Numerical Analysis 1560

Some Lower Bounds for the Complexity of Continuation Methods (with J.-P. Dedieu) 1589

A Polynomial Time Algorithm for Diophantine Equations in One Variable (with F. Cucker and P. Koiran) 1601

Complexity Estimates Depending on Condition and Round-off Error (with F. Cucker) 1610

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Permissions

Research themes Reprinted with permission from From Topology to Computation: Proceedings of the SMALEFEST, edited by M. W. Hirsch, J. E. Marsden and M. Shub, Copyright © 1993 Springer-Verlag, pp. xix-xxv.

Luncheon talk and nomination for Stephen Smale (by R. Bott) Reprinted with permission from From Topology to Computation: Proceedings of the SMALEFEST, edited by M. W. Hirsch, J. E. Marsden and M. Shub, Copyright © 1993 Springer-Verlag, pp. 67-72.

Some recollections of the early work of Steve Smale (by M. M. Peixoto) Reprinted with permission from From Topology to Computation: Proceedings of the SMALEFEST, edited by M. W. Hirsch, J. E. Marsden and M. Shub, Copyright © 1993 Springer-Verlag, pp. 73-75.

Luncheon talk (by R. Thorn) Reprinted with permission from From Topology to Computation: Proceedings of the SMALEFEST, edited by M. W. Hirsch, J. E. Marsden and M. Shub, Copyright © 1993 Springer-Verlag, pp. 76-78.

Banquet address at the Smalefest (by E. C. Zeeman) Reprinted with permission from From Topology to Computation: Proceedings of the SMALEFEST, edited by M. W. Hirsch, J. E. Marsden and M. Shub, Copyright © 1993 Springer-Verlag, pp. 79-80.

The work of Stephen Smale in differential topology (by M. Hirsch) Reprinted with permission from From Topology to Computation: Proceedings of the SMALEFEST, edited by M. W. Hirsch, J. E. Marsden and M. Shub, Copyright © 1993 Springer-Verlag, pp. 83-106.

A note on open maps Reprinted with permission from Proceedings of the AMS, Copyright © 1957 American Mathematical Society, Vol. 8, pp. 391-393.

A Vletoris mapping theorem for homotopy Reprinted with permission from Proceedings of the AMS, Copyright © 1957 American Mathematical Society, Vol. 8, pp. 604-610.

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Regular curves on Riemannian manifolds Reprinted with permission from Transactions of the AMS, Copyright © 1958 American Mathematical Society, Vol. 87, pp. 492-512.

On the immersion of manifolds in Euclidean space (with R. K. Lashof) Reprinted with permission from Annals of Mathematics, Copyright © 1953 Annals of Mathematics, Vol. 68, pp. 562-583.

Self-intersections of immersed manifolds (with R. K. Lashof) Reprinted with permission from Journal of Mathematics and Mechanics, Copyright © 1959 Indiana University Mathematics Journal, Vol. 8, pp. 143-157.

A classification of immersions of the two-sphere Reprinted with permission from Transactions of the AMS, Copyright © 1959 American Mathematical Society, Vol. 90, pp. 281-290.

The classification of immersions of spheres in Euclidean spaces Reprinted with permission from Annals of Mathematics, Copyright © 1959 Annals of Mathematics, Vol. 69, pp. 327-344.

Diffeomorphisms of the 2-sphere Reprinted with permission from Proceedings of the AMS, Copyright © 1959 American Mathematical Society, Vol. 10, pp. 621-626.

On involutions of the 3-sphere (with M. Hirsch) Reprinted with permission from American Journal of Mathematics, Copyright © 1959 The Johns Hopkins Press, Vol. 81, pp. 893-900.

The generalized Poincare conjecture in higher dimensions Reprinted with permission from Bulletin of the AMS, Copyright © 1960 American Mathematical Society, Vol. 66, pp. 373-375.

On gradient dynamical systems Reprinted with permission from Annals of Mathematics, Copyright © 1961 Annals of Mathematics, Vol. 74, pp. 199-206.

Generalized Poincare's conjecture in dimensions greater than four Reprinted with permission from Annals of Mathematics, Copyright © 1961 Annals of Mathematics, Vol. 74, pp. 391-406.

Differentiable and combinatorial structures on manifolds Reprinted with permission from Annals of Mathematics, Copyright © 1961 Annals of Mathematics, Vol. 74, pp. 498-502.

On the structure of 5-manifolds Reprinted with permission from Annals of Mathematics, Copyright © 1962 Annals of Mathematics, Vol. 75, pp. 38-46.

On the structure of manifolds Reprinted with permission from American Journal of Mathematics, Copyright © 1962 The Johns Hopkins Press, Vol. 84, pp. 387-399.

A survey of some recent developments in differential topology Reprinted with permission from Bulletin of the AMS, Copyright © 1963 American Mathematical Society, Vol. 69, pp. 131-145.

The story of the higher dimensional Poincare conjecture (What actually happened on the beaches of Rio) Reprinted with permission from The Mathematical Intelligence, Copyright © 1990 Springer-Verlag, Vol. 12, pp. 44-51.

Stephen Smale and the economic theory of general equilibrium Reprinted with permission from From Topology to Computation: Proceedings of the SMALEFEST, edited by M. W. Hirsch, J. E. Marsden and M. Shub, Copyright © 1993 Springer-Verlag, pp. 131-146.

Global analysis and economics, I: Pareto Optimum and a generalization of Morse theory, Reprinted with permission from Dynamical Systems, edited by M. M. Peixoto, Copyright © 1973 Academic Press, pp. 531-542.

Global analysis and economics, IIA: Extension of a theorem ofDebreu Reprinted with permission from Journal of Mathematical Economics, Copyright © 1974 Elsevier Science, Vol. 1, pp. 1-14.

Global analysis and economics. III: Pareto Optima and price equilibria Reprinted with permission from Journal of Mathematical Economics, Copyright © 1974 Elsevier Science, Vol. 1, pp. 107-117.

Global analysis and economics, IV: Finiteness and stability of equilibria with general consumption sets and production Reprinted with permission from Journal of Mathematical Economics, Copyright © 1974 Elsevier Science, Vol. 1, pp. 119-127.

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Global analysis and economics, V: Pareto theory with constraints Reprinted with permission from Journal of Mathematical Economics, Copyright © 1974 Elsevier Science, Vol. 1, pp. 213-221.

Dynamics in general equilibrium theory Reprinted with permission from American Economic Review, Copyright © 1976 The American Economic Association, pp. 288-294.

Global analysis and economics, VI: Geometric analysis of Pareto Optima price and equilibria under classical hypotheses Reprinted with permission from Journal of Mathematical Economics, Copyright © 1976 Elsevier Science, Vol. 3, pp. 1-14.

A convergent process of price adjustment and global Newton methods Reprinted with permission from Journal of Mathematical Economics Copyright © 1976 Elsevier Science, Vol. 3, pp. 107-120.

Exchange processes with price adjustment Reprinted with permission from Journal of Mathematical Economics, Copyright © 1976 Elsevier Science, Vol. 3, pp. 211-226.

Some dynamical questions in mathematical economics Reprinted with permission from Colloques International du Centre National de la Recherche Scientifique, No. 259: Systemes Dynamiques et Modeles Economiques, Copyright © 1976 Centre National de la Recherche Scientifique, pp. 95-97.

An approach to the analysis of dynamic processes in economic systems Reprinted with permission from Equilibrium and disequilibrium in Economic Theory edited by G. Schwodiauer, Copyright © 1977 D. Reidel Publishing Company, pp.363-367.

On comparative statics and bifurcation in economic equilibrium theory Republished with permission, originally in Annals of the New York Academy of Sciences, Copyright © 1979, New York Academy of Science, Vol. 316, pp.545-548.

The prisoner's dilemma and dynamical systems associated to non-cooperative games Reprinted with permission from Econometrica, Copyright © 1980, Blackwell publisher, Vol. 48, pp. 1617-1634.

Global analysis and economics Reprinted with permission from Handbook of Mathematical Economics, edited by K. J. Arrow and M. D. Intrilligator, Copyright © 1981 Elsevier Science, Vol. 1, pp. 331-370.

Gerard Debreu wins the Nobel Prize Reprinted with permission from Mathematical Intelligencer, Copyright © 1984 Springer-Verlag, Vol. 6, pp. 61-62.

Global analysis in economic theory Reprinted with permission from The New Palgrave: A Dictionary of Economics, 2, edited by John Eatwell, Murray Milgrate and Peter Newman, Copyright © 1987 Macmillan Press, pp. 532-534.

Scientists and the arms race Republished with permission, originally in (in German translation) Natur-Wissenschafter Gegen Atomrustung, edited by Hans-Peter Durr, Hans-Peter Harjes, Matthias Krech and Peter Starlinge, Copyright © 1983 Rowohlt Taschenbuch Verlag GmbH, pp. 327-334.

On the steps of Moscow University Reprinted with permission from Mathematical Intelligencer, Copyright © 1984 Springer-Verlag, Vol. 6, pp. 21-27.

Some autobiographical notes Reprinted with permission from From Topology to Computation: Proceedings of the Smalefest edited by M. Hirsch, J. Marsden and M. Snub, Copyright © 1993 Springer-Verlag, pp. 3-21.

Mathematical problems for the next century Reprinted with permission from Mathematical Intelligencer, Copyright © 1998 Springer-Verlag, Vol. 20, pp. 7-15.

Smale and nonlinear analysis: A personal perspective (by A. Tromba) Reprinted with permission from From Topology to Computation: Proceedings of the SMALEFEST, edited by M. W. Hirsch, J. E. Marsden and M. Shub, Copyright © 1993 Springer-Verlag, pp. 481-492.

A generalized Morse theory (with R. Palais) Reprinted with permission from Bulletin of the AMS, Copyright © 1964 American Mathematical Society, Vol. 70, pp. 165 172.

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Morse theory and a non-linear generalization of the Dirichlet problem Reprinted with permission from Annals of Mathematics, Copyright © 1964 the Annals of Mathematics.Vol. 80, pp. 382-396.

On the calculus of variations Reprinted with permission from Differential Analysis: papers presented at the Bombay colloquin, edited by Atiyah, Copyright © 1964 Oxford University Press and Tata Institute of Fundamental Research, pp. 187-189.

An infinite dimensional version of Sard's theorem Reprinted with permission from American Journal of Mathematics, 87 (1965) pp. 861-866.

On the Morse index theorem Reprinted with permission from Journal of Mathematics and Mechanics, Copyright © 1965 Indiana University Vol. 14, pp. 1049-1056.

What is global analysis? Reprinted with permission from American Math. Monthly, Copyright © 1969 Mathematical Association of America, Vol. 76, pp. 4-9.

Book review on Global Variational Analysis: Weierstress Integrals on a Riemannian Manifold by Marston Morse Reprinted with permission from Bulletin of the AMS, Copyright © 1977 American Mathematical Society Vol. 83, pp. 683-693.

Smooth solutions of the heat and wave equations Reprinted with permission from Commentarii Mathematici Helvetici, Copyright © 1980 Birkhauser Verlag, Vol. 55, pp. 1-12.

The work of Steve Smale in dynamics (by J. Palis) Reprinted with permission from From Topology to Computation: Proceedings of the SMALEFEST, edited by M. W. Hirsch, J. E. Marsden and M. Shub, Copyright © 1993 Springer-Verlag, pp. 165-178.

Discussion (by S. Newhouse, R. F. Williams and others) Reprinted with permission from From Topology to Computation: Proceedings of the SMALEFEST, edited by M. W. Hirsch, J. E. Marsden and M. Shub, Copyright © 1993 Springer-Verlag, pp. 179-185.

Morse inequalities for a dynamical system Reprinted with permission from Bulletin of the AMS, Copyright © 1960 American Mathematical Society, Vol. 66, pp. 43-49.

On dynamical systems Reprinted with permission from Boletin de la Sociedad Mathematics Mexicana, Copyright © 1960 Sociedad Mathematica Mexicana, pp. 195-198.

Dynamical systems and the topological conjugacy problem for diejfeomorphisms Reprinted with permission from Proceedings of the International Congress of Mathematicians 1962, Copyright © 1962 Mittag-Leffler Institute, pp. 490-496.

Stable manifolds for differential equations and dijfeomorphisms Reprinted with permission from Estratto dagli Annali della Scuola Normale Superiore di Pisa, Serie in, XVII (1963), pp. 97-116.

A structurally stable differentiable homeomorphism with an infinite number of periodic points Reprinted with permission from Report on the Symposium on Non Linear Oscillations, Kiev Mathematics Institute (1963), pp. 365-366.

Dijfeomorphisms with many periodic points Reprinted with permission from Differential and Combinatorial Topology (A symposium in honor ofMarston Morse), Copyright © 1965 Princeton University Press, pp. 63-80.

Structurally stable systems are not dense Reprinted with permission from American Journal of Mathematics, 88 (1966), pp. 491-496.

Dynamical systems on n-dimensional manifolds Reprinted with permission from Differential Equations and Dynamical Systems, Copyright © 1967 Academic Press, pp. 483-486.

Differentiable dynamical systems Reprinted with permission from Bulletin of the AMS, Copyright © 1967 American Mathematical Society, Vol. 73, pp. 747 SI7.

Nongenericity of Q-stability (with R. Abraham) Reprinted with permission from Global Analysis, Proceedings of Symposia in Pure Mathematics, Copyright © 1970 American Mathematical Society, Vol. 14, pp. 5-8.

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Structural stability theorems (with J. Palis) Reprinted with permission from Global Analysis, Proceedings of Symposia in Pure Mathematics, Copyright © 1970 American Mathematical Society, Vol. 14, pp. 223-231.

Notes on differential dynamical systems Reprinted with permission from Global Analysis, Proceedings of Symposia in Pure Mathematics, Copyright © 1970 American Mathematical Society, Vol. 14, pp. 277-287.

The Q-stability theorem Reprinted with permission from Global Analysis, Proceedings of Symposia in Pure Mathematics, Copyright © 1970 American Mathematical Society, Vol. 14, pp. 289-297.

Stability and genericity in dynamical systems Republished with permission, originally in Semincire Bourbaki, (1969-70), Copyright © Springer-Verlag, pp. 177-186.

Beyond hyperbolicity (with M. Shub) Reprinted with permission from Annals of Mathematics, Copyright © 1972 Annals of Mathematics, Vol. 96, pp. 587-591.

Stability and isotopy in discrete dynamical systems Reprinted with permission from Dynamical Systems, edited by M. M. Peixoto, Copyright © 1973 Academic Press, pp. 527-530.

Differential equations Reprinted with permission from the New Encyclopcedia Britannica, fifteenth Edition, Copyright © 1974 by Encyclopaedia Britannica Inc., pp. 762-767.

Dynamical systems and turbulence Reprinted with permission from Turbulence Seminar, Berkeley 1976/77, edited by P. Bernard and T. Ratiu, Lecture Notes in Mathematics 615, Copyright © 1977 Springer-Verlag, pp. 48-70.

Review of Catastrophe theory: Selected papers 1972-1977 by E. C. Zeeman Reprinted with permission from Bulletin oftheAMS, Copyright © 1978 American Mathematical Society, Vol. 84, pp. 1360-1368.

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On the problem of reviving the Ergodic Hypothesis ofBoltzmann and Birkhojf Reprinted with permission from International Conference on Non-linear Dynamics, Copyright © 1979 New York Academy of Sciences.

On how I got started in dynamical systems Reprinted with permission from Steve Smale, The Mathematics of Time, Copyright © 1980 Springer-Verlag, pp. 147-151.

Dynamics retrospective: great problems attempts that failed Reprinted with permission from Physica D, Copyright © 1991 Elsevier Science, Vol. 51, pp. 267-273.

What is chaos, originally published in Nobel Conference XXVI, Chaos the New Science, edited by J. Holte, Gustavus Adolphus College, University Press in American (1993), pp. 89-104.

Finding a horseshoe on the beaches of Rio Reprinted with permission of the publisher from Mathematical Intelligencer, Copyright © 1998, Springer-Verlag New York, Vol. 20, pp. 39-44.

The work of Curtis T. McMullen Republished with permission, originally published in Proceedings of the International Congress of Mathematicians 1998, Vol. 1, Documenta Mathematica, Bielefeld, Germany 1998, pp. 127-131.

Steve Smale and geometric mechanics (by J. E. Marsden) Reprinted with permission from From Topology to Computation: Proceedings of the SMALEFEST, edited by M. W. Hirsch, J. E. Marsden and M. Shub, Copyright © 1993 Springer-Verlag, pp. 499-516.

Topology and mechanics, I. Reprinted with permission from lnventiones Mathematicae, Copyright © 1970 Springer-Verlag, Vol. 10, pp. 305-331.

Topology and mechanics, II. Reprinted with permission from lnventiones Mathematicae, Copyright © 1970 Springer-Verlag, Vol. 11, pp. 45-64.

Problems on the nature of relative equilibria in celestial mechanics Reprinted with permission from Proceedings of Conference on Manifolds, Copyright © 1970 Springer-Verlag, pp. 194-198.

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Personal perspectives on mathematics and mechanics Republished with permission, originally in Statistical Mechanics: New Concepts, New Problems, New Applications, edited by Stuart A. Rice, Karl F. Freed and John C. Light, pp. 3-12, Copyright © 1972 by the University of Chicago. All rights reserved.

On the mathematical foundations of electrical circuit theory Reprinted with permission from Journal of Differential Geometry, Copyright © 1972, Journal of Differential Geometry, pp. 193-210.

A mathematical model of two cells via Turing's equation Reprinted with permission from Lectures on Mathematics in the Life Sciences, Copyright © 1974 American Mathematical Society, Vol. 6, pp. 17-26.

Optimizing several functions Reprinted with permission from Manifolds: Proceedings of the International Conference on Manifolds and Related Topics in Topology, Tokyo, 1973, pp. 69-74, Copyright © 1975 University of Tokyo Press.

Sufficient conditions for an optimum Republished with permission, originally in Warwick Dynamical Systems (1974), Lecture Notes in Mathematics, Copyright © 1975 Springer-Verlag, pp. 287-292.

The qualitative analysis of a difference equation of population growth (with R. F. Williams) Reprinted with permission from Journal of Mathematical Biology, Copyright © 1976 Springer-Verlag, Vol. 3, pp. 1-4.

On the differential equations of species in competition Reprinted with permission from Journal of Mathematical Biology, Copyright © 1976 Springer-Verlag, Vol. 3, pp. 5-7.

The problem of the average speed of the simplex method Reprinted with permission from Proceedings of the Xlth International Symposium on Mathematical Programming, edited by Bachem, Grotschel and Korte, Copyright © 1983 Springer-Verlag, pp. 530-539.

On the average number of steps of the simplex method of linear programming Reprinted from Mathematical Programming, 27 (1983), pp. 241-262, Copyright © 1983, with permission from Elsevier Science.

On the work of Steve Smale on the theory of computation (by M. Shub) Reprinted with permission from From Topology to Computation: Proceedings of the SMALEFEST, edited by M. W. Hirsch, J. E. Marsden and M. Shub, Copyright © 1993 Springer-Verlag, pp. 281-301.

On algorithms for solving fix) = 0 (with Morris W. Hirsch) Reprinted with permission from Communications on Pure and Applied Mathematics, 32 (1978) pp. 281-312. Copyright © 1978 John Wiley & Sons, Inc.

The fundamental theorem of algebra and complexity theory Reprinted with permission from Bulletin oftheAMS, 4 (1981), pp. 1-36. Copyright © 1981 American Mathematical Society.

Computational complexity: On the geometry of polynomials and a theory of cost, Part I (with M. Shub) Reprinted with permission from Annales Scientifiques de I'Ecole Normale Superieure, 18 (1985), pp. 107-142. Copyright © Gantheir-Villars.

On the efficiency of algorithms of analysis Reprinted with permission from Bulletin oftheAMS, Copyright © 1985 American Mathematical Society, Vol. 13, pp. 87-121.

Computational complexity: On the geometry of polynomials and a theory of cost. Part II (with M. Shub) Reprinted with permission from SI AM Journal of Computing, Copyright © 1986 Society for Industrial and Applied Mathematics, Vol. 15, pp. 145-161.

On the existence of generally convergent algorithms (with M. Shub) Reprinted with permission from Journal of Complexity, Copyright © 1986 by Academic Press Inc., Vol. 2, pp. 2-11.

Newton's method estimates from data at one point Reprinted with permission from The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics, edited by Richard E. Ewing, Kenneth I. Gross and Clyde F. Martin, Copyright © 1986 Springer-Verlag, pp. 185-196.

On the topology of algorithms, I. Reprinted with permission from Journal of Complexity, 3 (1987), pp. 81-89. Copyright © 1987 Academic Press.

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Algorithms for solving equations Reprinted with permission from Proceedings of the International Congress of Mathematicians 1986, American Mathematical Society Providence, (1987), pp. 172-195. Copyright © 1987 International Congress of Mathematicians 1986.

The Newtonian contribution to our understanding of the computer Republished with permission, originally in Queen's Quarterly, 95 (1988), pp. 90-95.

On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines (with L. Blum and M. Shub). Reprinted with permission from Bulletin (New Series) of the American Mathematical Society, 21(1) (1989), pp. 1-46. Copyright © 1989 American Mathematical Society.

Some remarks on the foundations of numerical analysis Reprinted with permission from SIAM Review, 32(2) (1990), pp. 211-220. Copyright © 1990, Society for Industrial and Applied Mathematics.

Theory of computation Reprinted with permission from Mathematical Research Today Tomorrow, edited by Casacuberta, 1992, pp. 60-69. Copyright © 1992 Springer-Verlag.

Complexity ofBezout's Theorem I: Geometric aspects (with M. Shub) Reprinted with permission from Journal of the AMS, 6 (1993), pp. 459-501. Copyright © 1993 American Mathematical Society.

Complexity ofBezout's Theorem II: Volumes and probabilities (with M. Shub) Reprinted with permission from Computational Algebraic Geometry edited by F. Eyssette and A. Galligo, Progress in Mathematics, 109 (1993), pp. 267-285. Copyright© 1993 Birkhauser.

Complexity ofBezout's Theorem III: Condition number and packing (with M. Shub). Reprinted with permission from Journal of Complexity, 9 (1993), pp. 4-14. Copyright © 1993 by Academic Press.

Complexity ofBezout's Theorem IV: Probability of success; Extensions (with M. Shub). Reprinted with permission from SIAM Journal of Numerical Analysis, 33 (1996), pp. 128-148. Copyright © 1996 Society for Industrial and Applied Mathematics.

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Complexity ofBezout's Theorem V: Polynomial Time (with M. Snub) Reprinted with permission from Theoretical Computer Science, 133 (1994), pp. 141-164. Copyright © 1994 Elsevier Science.

The Godel incompleteness theorem and decidability over a ring (with L. Blum) Reprinted with permission from From Topology to Computation: Proceedings of the Smalefest, edited by M. Hirsch, J. Marsden and M. Shub. Copyright © 1993 Springer-Verlag, pp. 321-339.

Separation of complexity classes in Koiran 's weak model (with F. Cucker and M. Shub). Reprinted with permission from Theoretical Computer Science, 133 (1994), pp. 3-14. Copyright © 1994 Elsevier Science.

On the intractability ofHilberts nullstellensatz and an algebraic version of 'W7VP?" (with M. Shub). Reprinted with permission from Duke Math. Journal 81 (1995), pp. 47-54 . Copyright © 1995 Duke University Press.

Complexity and real compulation: A manifesto (with L. Blum, F. Cucker and M. Shub). Reprinted with permission from International Journal of Bifurcation and Chaos 6 (1996) pp. 3-26. Copyright © 1996 World Scientific Publishing Company.

Algebraic settings for the problem "P*NP" (with L. Blum, F. Cucker and M. Shub) Reprinted with permission from Lectures in Applied Mathematics Vol. 32, edited by J. Renegar, M. Shub and S. Smale, pp. 125-144. Copyright © 1996 American Mathematical Society.

Complexity theory and numerical analysis Reprinted with permission from Acta Numerica (1997) pp. 523-551. Copyright © 1997 Cambridge University Press.

Some lower bounds for the complexity of continuation methods (with J-P Dedieu) Reprinted with permission from Journal of Complexity 14 (1998) 454-465. Copyright © 1998 Academic Press.

A polynomial time algorithm for diophantine equations in one variable (with F. Cucker and P. Koiran) Reprinted with permission from Journal of Symbolic Computation 27(1999)21-29. Copyright © 1999 Academic Press.

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Complexity estimates depending on condition and round-off error (with F. Cucker). Reprinted with permission from Journal of the Association of Computing Machinery 46, (1999) 113-184. Copyright © 1999 Association of Computing Machinery.

Research Themes

Introduction Many mathematicians have contributed powerful theorems in several fields. Smale is one of the very few whose work has opened up to research vast areas that were formerly inaccessible. From his early papers in differential topology to his current work in theory of computation, he has inspired and led the development of several fields of research: topology of nonlinear function spaces; structure of manifolds; structural stability and chaos in dynami­cal systems; applications of dynamical systems to mathematical biology, economics, electrical circuits; Hamiltonian mechanics; nonlinear functional analysis; complexity of real-variable computations. This rich and diverse body of work is outlined in the following subsection.

There are deep connections between Smale's work in apparently disparate fields, stemming from his unusual ability to use creatively ideas from one subject in other, seemingly distant areas. Thus, he used the homotopy theory of fibrations to study immersions of manifolds, and also the classification of differentiable structures. In another area, he applied handle body decomposi­tions of manifolds to structural stability of dynamical systems. Smale applied differential geometry and topology to the analysis of electrical circuits, and to several areas of classical mechanics. He showed how qualitative dynamical systems theory provides a natural framework for investigating complex phe­nomena in biology. A recent example is his application of algebraic topology to complexity of computation. In each case his innovative approach quickly became a standard research method. His ideas have been further developed by his more than 30 doctoral students, many of whom are now leading researchers in the fields he has pioneered.

This conference brought together mathematicians who are currently mak­ing important contributions to these fields. It had two purposes: First, to present recent developments in these fields; and second, to explore the con­nections between them. This was best done by examining the several areas of Smale's research in a single conference which crosses the traditional

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boundary lines between mathematical subjects. In this way, a stimulating environment encouraged a fruitful exchange of ideas between mathemati­cians working in topics that are formally separate, but which, as Smale's work demonstrates, have strong intellectual connections.

Through this conference proceedings we hope that important new insights may be achieved into the extraordinary diversity and unity of mathematics.

Topics

Differential Topology Smale's first work in differential topology, on the classification of immersions of spheres, led to the general classification of immersions of manifolds. But it also presented, for the first time, the use of fibrations of function spaces in what is now called geometric topology. Through fibrations, the power­ful tools of algebraic topology were applied in new ways to a host of geo­metrical problems. This became, in the hands of Smale and many others, a standard approach to many areas: embeddings, diffeomorphisms, differential structures, piecewise linear theory, submersions, and other fields. The classi­fication of differential structures on topological manifolds due to R. Kirby and L. Siebemmann, and many of the profound geometrical theories of M. Gromov, are based on Smale's technique of function space fibrations.

In 1960, Smale startled the mathematical world with his proofs of the Generalized Poincare Conjecture and what is now called the H-Cobordism Theorem. Up to that time, the topological classification of manifolds was stuck at dimension three. John Milnor's exciting discovery in 1956 of exotic differential structures on the 7-sphere had pointed to the need for a theory of differential structures, but beyond his examples nothing was known about sufficient conditions for diffeomorphism. Smale had the audacity to attack the problem in dimensions five and above. His results opened the flood­gates of research in geometric topology. His techniques of handle cancella­tion and his constructive use of Morse theory proved enormously fruitful in a host of problems and have become standard approaches to the structural analysis of manifolds. Michael Freedman's recent topological classification of 4-dimensional manifolds is a far-reaching generalization of Smale's handle-canceling methods. It is closely related to exciting developments in Yang-Mills theory by Donaldson, Uhlenbeck, Taubes, and others. This work in­volves other areas of nonlinear functional analysis and mechanics that will be discussed below.

Dynamical Systems In the early 1960s, Smale embarked on the study of dynamical systems. Like topology, this subject was founded by Poincare, who called it the qualitative

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Research Themes xxi

theory of differential equations. Intensively developed by G.D. Birkhoff, by 1960 it seemed played out as a source of new ideas. At this point, Smale introduced a new approach, based on geometrical assumptions about the dynamical process, rather than the standard method of examining specific equations coming from physics and engineering. The key notion was a hyper­bolic structure for the nonwandering set, a far-reaching generalization of the standard notion of hyperbolic fixed point. Under this hypothesis, Smale proved that the nonwandering set (of points that are recurrent in a certain sense) breaks up into a finite number of compact invariant sets in a unique way; these he called basic sets. Each basic set was either a single periodic orbit or contained infinitely many periodic orbits that were tangled in a way that today would be called "chaotic." Moreover, he proved the dynamics in a basic set to be structurally stable.

These new ideas led to a host of conjectures, proofs, examples, and counterexamples by Smale, his many students, and collaborators. Above all, they led to new ways of looking at dynamical systems. These led to precise constructions and rigorous proofs for phenomena that, until then, were only vaguely describable, or only known in very special cases.

For example, Smale's famous Horseshoe is an easily described transforma­tion of the two-dimensional sphere that he proved to be both chaotic (in a precise sense) and structurally stable, and completely describable in com­binatorial terms. Moreover, this construction and analysis generalized to all manifolds of all dimensions. But it was more than merely an artificial class of examples, for Smale showed that any system satisfying a simple hypothesis going back to Poincare (existence of a transverse homoclinic orbit) must have a horseshoe system embedded in it. Such a system is, therefore, not only chaotic, but the chaos is stable in the sense that it cannot be eliminated by arbitrarily small perturbations. In this way, many standard models of natural dynamical processes have been proved to be chaotic.

Smale's new dynamical ideas were quickly applied, by himself and many others, to a variety of dynamical systems in many branches of science.

Nonlinear Functional Analysis Smale has made fundamental contributions to nonlinear analysis. His ap­plication (with R.S. Palais) of Morse's critical point theory to infinite-dimensional Hilbert space has been extensively used for nonlinear problems in both ordinary and partial differential equations. The "Palais-Smale" con­dition, proving the existence of a critical point for many variational problems, has been used to prove the existence of many periodic solutions for nonlinear Hamiltonian systems. Another application has been to prove the existence of minimal spheres and other surfaces in Riemannian manifolds.

Smale also was a pioneer in the development of the theory of manifolds of maps. The well-known notes of his lectures by Abraham and the related work

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of Eells has undergone active development ever since. For example, mani­folds of maps were used by Arnol'd, Ebin, Marsden, and others in their work on the Lagrangian representation of ideal incompressible fluids, in which the basic configuration space is the group of volume-preserving difTeomorphisms, and for which the Poisson reduced equations are the standard Euler equa­tions of fluid mechanics.

In 1965, Smale proved a generalization of the famous Morse-Sard theo­rem on the existence of regular values to a wide class of nonlinear mappings in infinite-dimensional Banach spaces. This permitted the use of transver-sality methods, so useful in finite-dimensional dynamics and topology, for many questions in infinite-dimensional dynamics. An important example is A. Tromba's proof that, genetically (in a precise sense), a given simple closed curve in space bounds only a finite number of minimal surfaces of the topo­logical type of the disk. A similar result was proved by Foias and Temam for stationary solutions to the Navier-Stokes equations.

Physical and Biological Applications Smale's first papers in mechanics are the famous ones on Topology and Mechanics." These papers appeared in 1970 around the beginning of the geometric formulation of mechanics and its applications, when Mackey's book on the foundations of quantum mechanics and Abraham's book on the foundations of mechanics had just come out. Smale's work centered on the use of topological ideas, principally on the use of Morse theory and bifurca­tion theory to obtain new results in mechanics. Probably the best-known result in this work concerns relative equilibria in the planar n-body problem, which he obtained by exploiting the topological structure of the level sets of conserved quantities and the reduced phase space, so that Morse theory gave interesting results. For example, he showed that a result of Moulton in 1910, that there are in! collinear relative equilibria, is a consequence of critical point theory. Smale went on to determine the global topology and the bifur­cation of the level sets of the conserved angular momentum and the energy for the problem. These papers were a great influence: for example, they led to further work of his former student Palmore on relative equilibria in the planar n-body problem and in vortex dynamics, as well as a number of studies by others on the topology of simple mechanical systems such as the rigid body. This work also was the beginning of the rich symplectic theory of reduction of Hamiltonian systems with symmetry. Smale investigated the case of the tangent bundle with a metric invariant under a group action, which was later generalized and exploited by Marsden, Weinstein, Guillemin, Steinberg, and others for a variety of purposes, ranging from fluids and plasmas to representation theory. The international influence of these papers on a worldwide generation of young workers in the now burgeoning area of geometric mechanics was tremendous.

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Research Themes xxm

Smale's work on dynamical systems also had a great influence in mechan­ics. In particular, the Poincare-Birkhoff-Smale horseshoe construction has led to studies by many authors with great benefit. For example, it was used by Holmes and Marsden to prove that the PDE for a forced beam has chaotic solutions, by Kopell and Howard to find chaotic solutions in reac­tion diffusion equations, by Kopell, Varaiya, Marsden, and others in circuit theory, by Levi in forced oscillations, and by Wiggins and Leonard to estab­lish connections between dynamical chaos and Lagrangian or particle mixing rates in fluid mechanics. This construction is regarded as a fundamental one in dynamical systems, and it is also one that is finding the most applications.

In 1972, Smale published his paper on the foundations of electric cir­cuit theory. This paper, highly influenced by an interaction between Smale, Desoer, and Oster, examines the dynamical system defined by the equations for an electric circuit, and gives a study of the invariant sets defined by Kirchhoff's laws and the dynamical systems on these sets. Smale was the first to deal with the implications of the topological complexities that this in­variant set might have. In particular, he raised the question of how to deal with the hysteresis or jump phenomena due to singularities in the constraint sets of the form f(x,dx/dt) = 0, and he discussed various regularizing devices. This had an influence on the electrical engineering community, such as the 1981 paper of Sastry and Desoer, "The Jump Behavior of Circuits and Sys­tems," which provided the answers to some of the questions raised by Smale's work. (This paper actually originated with Sastry's Masters thesis written in the Department of Mathematics at Berkeley.) This work also motivated the studies of Takens on constrained differential equations.

The best known of Smale's several papers in mathematical biology is the first, in which he constructed an explicit nonlinear example to illustrate the idea of Turing that biological cells can interact via diffusion to create new spatial and/or temporal structure. His deep influence on mathematical biol­ogy came less from the papers that he wrote in this field than from the impetus that his pure mathematical work gave to the study of qualitative dynamical systems. Because of the difficulty in measuring all of the relevant variables and the need for clarifying simplification, qualitative dynamical systems provides a natural framework for investigating dynamically com­plex phenomena in biology. These include "dynamical diseases" (Glass and Mackey), oscillatory phenomena such as neural "central pattern generators" (Ermentrout and Kopell), complexity in ecological equations (May) and im­mune systems (Perelson), and problems involving spontaneous pattern for­mation (Howard and Kopell, Murray, Oster).

Another important paper constructed a class of systems of classical com­peting species equations in R" with the property that the simplex A"'1

spanned by the n coordinate unit vectors is invariant and the trajectories of the large system asymptotic to those of any dynamical system in A"-1; this demonstrated the possible complexity in systems of competing species. Hirsch has shown that arbitrary systems of competing species decompose

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into pieces which are virtually identical with Smale's construction. Thus, Smale's example, seemingly a very special case, turns out to be the basic building block for the general case.

Economics Smale's geometrical approach to dynamics proved fruitful not only in the physical and biological sciences, but also in economic theory. In 1973, he began a series of papers investigating the approach to equilibria in various economic models. In place of the then standard linear methods relying heav­ily on convexity, he used nonlinear differentiate dynamics with an emphasis on generic behavior. In a sense, this represented a return to an older tradition in mathematical economics, one that relied on calculus rather than algebra, but with intuitive arguments replaced by the rigorous and powerful methods of modern topology and dynamics.

Smale showed that under reasonable assumptions the number of equilibria in a large market economy is generically finite, generalizing work of Debreu. He gave a rigorous treatment of Pareto optimality. His interest in economic processes inspired his work on global Newton algorithms (see the section below). This led to an important paper on price-adjustment processes.

His interest in theoretical economics led Smale to work in the theory of games. In an original approach to the "Prisoner's Dilemma," which is closely related to economic competition, he showed that two players employing certain kinds of reasonable strategies will, in the long run, achieve optimal gains.

Smale's work in economics led to this research in the average stopping time for the simplex algorithm in linear programming, discussed below.

Theory of Computation Smale's work in economic equlibrium theory led him to consider questions about convergence of algorithms. The economists H. Scarf and C. Eaves had turned Sperner's classical existence proof for the Brouwer fixed-point theorem into a practical computational procedure for approximating a fixed point. Their methods were combinatorial; Smale transformed them into the realm of differentiate dynamics. His "global Newton" method was a simple-looking variant of the classical Newton-Raphson algorithm for solving f(x) = 0, where / is a nonlinear transformation of n-space. Unlike the classi­cal Newton's method, which guarantees convergence of the algorithm only if it is started near a solution, Smale proved that, under reasonable assump­tions, the global Newton algorithm will converge to a solution for almost every starting point which is sufficiently far from the origin. Because it is easy to find such starting points, this led to algorithms that are guaranteed to converge to a solution with probability one. The global Newton algorithm was the basis for an influential paper on the theory of price adjustment by Smale in mathematical economics.

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This geometrical approach to computation was developed further in a series of papers on Newton's method for polynomials, several of which were joint work with M. Shub. These broke new ground by applying to numer­ical analysis ideas from dynamical systems, differential topology and prob­ability, together with mathematical techniques from many fields: algebraic geometry, geometric measure theory, complex function theory, and differen­tial geometry.

Smale then turned to an algorithm that is of great practical importance, the simplex method for linear programming. It was known that this algo­rithm usually converged quickly, but that there are pathological examples requiring a number of steps that grows exponentially with the number of variables. Smale asked: What is the average number of steps required for m inequalities in n variables if the coefficients are bounded by 1 in absolute value? He translated this into a geometric problem which he then solved. The surprising answer is that the average number of steps is sublinear in n (or m) if m (or n) is kept fixed.

These new problems, methods, and results led to a great variety of papers by Smale and many others, attacking many questions of computational com­plexity. Smale always emphasized that he looks at algorithms as mathe­maticians do, in terms of real numbers, and not as computer scientists do, in terms of a finite number of bits of information. Most recently this led Smale, L. Blum, and M. Shub to a new algebraic approach to the general theory of computability and some surprizing connections with Godel's theorem.

7

6 Luncheon Talk and Nomination for Stephen Smale R. BOTT

Thank you, Moe. It is a great pleasure to be here, but I must say that I feel a bit like Karen who just remarked that she comes from a different world. I come from the beach! In fact, I'm hardly dry. On my beach, you don't usually wear clothes, but as you see, I put some on for this great occasion.

Now, I'm quite amazed to find what deep philosophical things we have becomes involved in here at lunch. But let me warn you about becoming too philosophical—it is a sure sign of age! In fact, this whole discussion reminds me of a happening at the Institute in Princeton circa 1950. Niels Bohr had come to lecture. He was grand, beautiful, but also very philosophical. After a while, von Neumann, who was sitting next to me, leaned over and whispered in my ear, "It's calcium, it's the calcium in the brain." Keep that in mind, youngsters!

In a sense, thinking back to those days at the Institute, the man who inspired me most was Carl Ludwig Siegel, and I seem to have inherited his feelings that it is not so much the theory that constitutes mathematics, but the theorem and, above all, the crucial example. He felt that mathematics had started to go down the road of abstraction with Riemann! He was skeptical of Hilbert's proofs; they were too abstract.

But I'm not going to become philosophical! In fact, tomorrow I'm heading straight back to my beach. You see, it is very easy for you youngsters to enjoy Steve's birthday, but after all, he was my student, and if he is 60, you can well imagine what that means about my age.

So instead of philosophy, let me reminisce, and tell you a little bit about how it all began. I came to Michigan in 1951 from the Institute having learned some topology there. In fact, I had a stellar cast: personal instruction from Specker and Reidemeister and Steenrod! Morse was there, of course, and I was great friends with Morse, but he was not really my mathematical mentor. He was interested in other things at that time—Hilbert spaces, etc. Anyway, I came to Michigan as a young instructor, a rather lowly position, and I cannot help telling you the anecdote that greeted me, so to speak, on arrival.

In the summer, there had been an International Congress at Harvard

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where I had met my Chairman-to-be, Professor Hildebrand. I had grown a full beard for the occasion as a lark and, indeed, the only other people whom I recall having beards at that time were the Italians. When I presented myself to Hildebrand, I had already shaved off my beard long ago. He looked at me and said, "You mean you don't have a beard?" And I said, "No, I don't usually wear one." Thereupon he said in his customary frank manner, "Well, young man. That beard at the Congress cost you $500. When I came home, I took $500 off your salary; I don't approve of beards." But there, I could entertain you all evening with Hildebrand stories.

Michigan had a great tradition in topology at that time. Samelson was there. Also Moise, Wilder, and Young; Steenrod and Sammy Eilenberg had been there. Samelson and I hit it off from the very start and we both started a very business-like seminar on Serre's thesis. This was algebraic topology with a capital A. In fact, that whole generation had lost faith in proving purely geometric theorems. That is why it was so amazing to my generation when later on Steve, and then progressively more and more other people, were able to go back to geometry and with their bare hands do things which at that time we thought were undoable.

When I offered my first course in topology, I had three enrolled students, but there were quite a few other people who attended. One was Steve, the other one was Jim Munkres, and the third, as I like to say, was a really bright guy. He was a universalist and just did mathematics on the side. I expect that those of you who know Steve well can imagine what he was like as a student. He sat in the back and it wasn't clear whether he was paying attention, but he always looked benign. I think it is fair to say that he wasn't really consid­ered a star at Michigan. In fact, there was some question whether he should be allowed to stay! Maybe that's why he picked me. But I think I gave you, Steve, a good question which in some ways still has unsolved parts to it today. My problem was this. If we take the space of regular curves, starting from a point and project each curve onto its end tangent, does this projection have the fiber homotopy property? This turned out to be very difficult. There was no clean way of doing it. But that didn't stop Steve! Actually I had only a small concrete application in mind related to a theorem of Whitney's about the winding numbers of curves in the plane. So that when Steve later elabo­rated his technique to immersion theory I was completely amazed. And in case you have not heard this story, when Steve wrote me that he could turn the sphere inside out, I immediately sent him a counterexample! That's true. I wrote him back, "Don't be foolish. It can't be done." So, you see, this shows how good it is to have a stupid thesis advisor. Once you overcome him, the rest will be easy.

Well, so many good things have been said about Steve that I do not want to add to them. I mean, he will become quite insufferable. Let me rather tell you about some of his other gifts. These include many techniques of nearly getting rid of people. Several times I've gone on excursions with Steve, and on returning I've often ended up on my knees: "Back home again and still

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6. Luncheon Talk and Nomination for Stephen Smale 69

alivel" There was a wonderful excursion organized by Steve on the beaches of the Olympic Peninsula. Maybe he did his best job there. This beach had many promontories which one could only traverse when the tide was low enough. This is what intrigued Steve and he had researched it carefully. Only he had done the trip the wrong way around, so that when we came along we were walking, so to speak, against the grain of his experience. On the last day, he told us that the end was in sight and that we could surely take a rather daring shortcut along the beach. The tide also did look very low. And so we were all walking along very nicely, relaxed, doing mathematics all over the place. My wife and one enterprising daughter were along. And then suddenly I remember turning a corner only to see a quite unexpected bay which we still had to traverse. And, indeed, that already my wife, who was quite a bit ahead of me, was being ushered through the ice-cold waters that were rising along the cliffs by one of our wonderful God-sent mountaineers who had joined our expedition. By the time I got to the spot I did not trust myself to carry my little daughter through the water, but rather put her on the back of another wonderful mountaineer, "Pham." Pham is about half my height, but he took her over to safety, and then ran back to alert all the people behind us, such as Steve, his son, the Shubs, etc. Then I got into the water with all my packs. By this time the waves were crashing in and between their onslaught I managed to get to the place where I was supposed to climb out again. I tried but couldn't quite make it. I tried once again, but of course was weaker. It never occurred to me to remove my pack! Instead, what went through my mind was, "Aha! This is how one drowns!" But at my third attempt that wonderful girl, Megan, another of our mountaineers, came back just in time to get me on my feet.

All right. This is enough. I think I have now lowered the level of discourse sufficiently! But in closing, I cannot resist one serious word. It was a great pleasure to have you as a student, Steve—even though you tried to drown me. I've always been very proud of you, and I am sure you will not cease to do great mathematics for many years to come. Happy birthday—you have now finally come of age!

Nomination for Stephen Smale* It is a pleasure and an honour for me to hereby place Stephen Smale's name in nomination for the Presidency of the American Mathematical Society.

Smale is one of the leading mathematicians of his generation, whose work has been foundational in differential topology, dynamical systems, and many

♦ Reprinted from Notices of the American Mathematical Society, 38, 7, 758-760, 1991.

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aspects of nonlinear analysis and geometry. It has been his genius also to bring these subjects to bear in a significant way on mechanics, economics, the theory of computation, and other brands of applied mathematics.

Smale's characteristic quality is courage, combined with great geometric insight, patience, and power. He is single-minded in his pursuit of under­standing a subject on his own terms and will follow it wherever it leads him. These qualities also emerge in many episodes from his personal life. Once his sails are set, it is literally impossible to stop him. Thus in a few short years he moved from Sunday sailor to skippering his own boat across the Pacific with a mathematical crew of hardy souls; and so his interest in minerals has, over the years, not only taken him to obscure and dangerous places all over the globe, but has culminated in one of the finest mineral collections anywhere.

Smale characteristically takes on one project at a time, thinks deeply about it and then turns to the next. He likes to share his thoughts with others, keeps his office door open, never seems in a hurry, and inspires his students with his own confidence. His willingness to run for this position therefore assures me that if he were elected he would grace our Society not only with his great mathematical distinction, ecumenical interests and quiet almost shy—manner, but that he would also do his homework thoroughly and give the serious problems that face our subject and our institutions his "prime time."

It was my good fortune to have Smale as one of three enrolled students in the first course on topology which I taught at Michigan (1952-1953). (Munkres was another one, but the third, whose name now escapes me, was—as I liked to say—"the really smart one." Indeed, he could play blind­folded chess, compose operas, etc.) Smale's manner in class was the same then as it is now. He preferably sits in the back; says little, and seems to let the mathematical waves wash over him, rather than confront them. However, Smale's courage surfaced soon at Michigan when he chose me, the greenhorn of topology—actually of mathematics altogether—as his thesis advisor. I proposed a problem concerning regular curves (i.e., curves with nowhere zero tangents) on manifolds; namely, that the projection of the space of such curves on its final tangent-direction satisfied the "covering homotopy prop­erty." This notion had just been invented in the late forties, and I had learned it from Steenrod in Princeton just the year before. The combination of anal­ysis and topology in this question appealed to Smale and he went to work. I was pleased and impressed by the geometric insight and technical power of his eventual solution, but completely amazed when he in the next few years extended these techniques to produce his famous "inside-out turning of the sphere" in R3 through regular deformations. In fact, when he wrote me about this theorem I replied curtly with a false argument which purported to prove the impossibility of such a construction!

More precisely, what Smale had proved was that the regular immersion classes of a it-sphere S" in R" correspond bijectively to nkVk „), the kth homo­topy group of the Stiefel-manifold of fc-frames in R". He had thus managed to

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reduce a difficult differential topology question to one in pure homotopy theory and so had set the stage for an obstruction theory of immersions. The final beautiful development of this train of thought came in the thesis of Morris Hirsch, directed by Spanier, but also with great interest and encour­agement from Smale.

In the 1960s, Smale produced his "Generalized Poincare Conjecture" and—what is still to this day the most basic tool in differential topolo­gy—the "//-cobordism theorem." All these were corollaries of his deep re­thinking of Morse theory, which he perfected to a powerful tool in all aspects of differential topology. Above all, Steve had the courage to look for concrete geometric results, where my generation was by and large taught to be content with algebraic ones.

Smale's rethinking of the Morse theory involved fitting it into the broader framework of dynamical systems. This enabled him not only to extend the Morse inequalities to certain dynamical systems, but also to use concepts from dynamical systems to understand the Morse theory more profoundly. By clearly formulating and using the transversality condition on the "de­scending" and "ascending" cells furnished by the gradient flow, he was able to control these cell-subdivisions much more accurately than Thorn had been able to do ten years earlier. I distinctly remember when he retaught me Morse theory in this new and exciting guise during a Conference in 1960 in Switzerland.

By that time he had actually also constructed his famous "horseshoe" map of the 2-sphere, and so was well on his way to laying the foundations of a subject we now call "chaos." Indeed, he showed that under the assumption of a hyperbolic structure on a "non-wandering" set, this set breaks up into a finite number of compact invariant sets in a unique way. Each of these was either a single periodic orbit or else was an infinite union of such orbits so inextricably tangled that we would call them "chaotic" today. Moreover, he showed that these basic sets, as he called them, were structurally stable (the chaos cannot be removed by a small perturbation) as well as ubiquitous!

In the years since these spectacular results—which earned him the Field Medal in 1966—Smale has not ceased to find new and exciting quests for his geometric and dynamical imagination.

In the later sixties, he and his students studied the Morse theory in infinite dimensions as a tool in non-linear differential equations. He proved a Sard type theorem in this framework and the Palais-Smale Axioms are now the foundation on which the modern school build their Morse theory beyond "Palais-Smale."

In mechanics, Smale was one of the initiators of the "geometric reduction" theory which occurs so prominently in the work of the symplectic school. In economics, Smale reintroduced differential techniques in the search for equi­librium with great success, and in computation theory his "probabilistic growth theory" applied to algorithms—in particular his estimates for a mod­ified form of Newton's algorithm is an exciting new development in that

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72 R. Bott

subject. And the last time I heard Smale talk, he was explaining Godel's theorem in dynamical system terms!

In fact, with an oeuvre of this magnitude and with more than thirty distin­guished doctoral students dispersed all over the world, one would have to invoke truly legendary names to best Smale's impact on today's mathe­matical world.

He is clearly a candidate of the first order whom we must not pass up.*

* The election was won by Ron Graham.

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7 Some Recollections of the Early Work of Steve Smale M.M. PEIXOTO

Ladies and gentlemen: It is for me a great honor and a great pleasure to speak at this Smalefest. I will focus here on the personal and mathematical contacts I had with

Steve during the period September 1958-June 1960. From September 1958 to June 1959,1 was living in Baltimore and Steve was living in Princeton at the Institute for Advanced Study. I visited him several times there. In June 1959,1 went back to IMPA (Instituto de Matematica Pura e Aplicada) in Rio de Janeiro. Steve spent the first six months of 1960 at IMPA and during that period I had daily contacts with him, mathematical and otherwise. This period of Steve's career is sometimes associated with the "Beaches of Rio" episode. But this episode—originated by an out-of-context reference to a letter of Steve's—is something that happened in the United States and sev­eral years later. It will be mentioned here only indirectly.

I was introduced to Steve on the initiative of Elon Lima in the late summer of 1958 in Princeton. After spending 1 year there, I was about to leave in a few weeks and Steve was coming for a 2-year stay at the Institute. He was already a substantial mathematician having, among other things, turned the sphere S2 inside out.

As Lima had predicted Steve showed immediately great interest in my work on structural stability on the 2-disk. I was delighted to see this in­terest; at that time, hardly anybody besides Lefschetz cared about structural stability.

In those days, one burning question that 1 had much discussed with Lefschetz was: How to express in n-dimensions the "no saddle connection" condition of Andronov-Pontryagin? Before the end of 1958, Steve had the right answer to this question: The stable and unstable manifolds of singu­larities and closed orbits should meet transversally.

Simple-minded and rather trivial as this may look to the specialist today, the transversality condition applied to stable and unstable manifolds was a very crucial step in Steve's work in both topology and dynamical systems.

In fact, in 1949, Rene Thorn in a seminal short paper considered a Rieman-nian manifold M together with a Morse function on it / : M -> R and to

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74 M.M. Peixoto

these objects he associated the decomposition A3 [A"] of M into cells given by the stable [unstable] manifolds of grad / . Thorn used this decomposition As to derive, among other things, the Morse inequalities.

The important remark that the transversality condition applied to this case implies that the boundary of one of the cells of A5 is the union of lower dimentional cells led Steve to his celebrated work on the high dimensional Poincare conjecture. Whereas Thorn considered only A5, Steve considered both A5 and A" and added the transversality condition.

The transversality condition in a more general setting than the one of grad / also played an important role in Steve's subsequent—and no less important—work on dynamical systems.

So these two major developments in topology and dynamical systems can be traced back to the way Steve handled the "no saddle connection" condi­tion of Andronov-Pontryagin. Very humble beginnings indeed!

Steve, his wife Clara and their kids, Nathan aged 2 and Laura aged 1, arrived in Rio in early January 1960. He was then working hard on the Poincare conjecture but kept an eye on dynamical systems. We used to talk almost every day, in the afternoons, about my two-dimensional structural stability problem.

During his 6-month stay in Rio, Steve made another important contribu­tion to structural stability, the horseshoe diffeomorphism on S2. This was a landmark in dynamical systems theory, showing that structural stability is consistent with the existence of infinitely many periodic orbits.

In mid-June 1960 at the end of his stay, he made a quick trip to Bonn and Zurich to speak about his work on the high dimensional Poincare con­jecture that he was then finishing.

During Steve's absence I discovered a major flaw in my own work, due to the fact that I had bumped into what nowadays is called the closing lemma.

So I became very anxious to discuss this with him. On his arrival back in Rio, I invited Clara and Nat to join me and go

to the airport to meet Steve at dawn. Even at a distance it was clear that something was wrong with Steve.

Haggard, tense and tired he had on his face what to me looked like a black eye.

In contrast with his appearance, his speech soon became smooth and even. Yes, he was in trouble, there were objections to his proof, some were serious. He hoped to be able to fix everything. He mentioned names but indicated no hardfeelings toward anyone.

Later, he wrote about this trip and called it "rather traumatic" and "dra­matic and traumatic."

These words of Steve fit very well the overall impression that he left on me in that singular meeting at Rio's airport late June 1960 as the day was just breaking in: cold, measured ferocity of purpose.

When I was told about the "Beaches of Rio" episode, with the implication

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7. Some Recollections of the Early Work of Steve Smale 75

associated to it of a frolicsome trip South of the Rio Grande, I found it a funny counterpoint to what I had actually witnessed.

So I was there and saw from a privileged position the birth of two of Stephen Smale's major contributions to mathematics.

As it is the case with many great mathematical achievments, one detects there the presence of that magic triad—simplicity, beauty, depth—that we mathematicians strive for. But simplicity is perhaps the more easily detect­able of these qualities and simplicity is also a striking feature of Steve's work mentioned above.

In this connection, it is perhaps appropriate to mention here some poe­tic reflexions of A. Grothendieck about invention and discovery. Translated from the French this is part of Grothendieck's poetical outburst about his craft:

A truly new idea does not burst out all made up of diamonds, like a flash of sparkling light, also it does not come out from any machine tool no matter how sophisticated and powerful it might be. It does not announce itself with great noise proclaiming its pedigree: I am this and I am that... It is something humble and fragile, something delicate and alive

I will stop here, well aware of the saying according to which "dinner speak­ers are like the wheel of a vehicle: The longer the spoke, the greater the tire."

Thank you for your attention.

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8 Luncheon Talk R. THOM

Dear Steve, Probably more than 35 years have elapsed since the first time I heard

about you. It was in Princeton in 1955 when Raoul Bott spoke to me about a very bright student, who had recently found a wonderful theorem about immersions. I was startled by the result, and by the straightforward geomet­ric aspect of the proof (of which I gave an account at the Bourbaki Seminar a little later). Moe Hirsch gave us, in his opening lecture, a beautiful review of all the consequences of your theorem. If you will excuse me for some extra technicalities, I would like to add to these a further one, in another—not mentioned —direction; namely, the connection with the theory of singu­larities of smooth maps, which started to attract my interest after the found­ing theorems of H. Whitney on the cusp and the cuspidal point. The problem of classifying immersions can be generalized as follows: Given two smooth manifolds X" and Y', and some "singularity type" (s) of local smooth maps: g; R" -> R", find conditions for a homotopy class h: X -> Y such that no smooth map / of the class h exhibits the singularity (s). Following Ehresmann's theory of jets, one may associate to any singularity of type (s) a set of orbits for the algebraic action of the group Lk(n) x Lkp) of fc-jets of local isomor­phisms of source and target spaces in the space Jk(n,p) of local jets from R" to R'. (See, for instance, H. Levine's Notes on my Lectures as Gastprofessor in Bonn.) If it happens that this set of orbits is an algebraic cycle in homology theory, then it was proved by A. Haefliger that the dual cohomology class to this singular cycle sf) for a given map / is some specific characteristic class (mod Z or Z2) of the quotient bundle (TX/f*(T(Y)) [this class is a polyno­mial characterizing the singularity (s)]. This may be seen as a faraway conse­quence of your immersion theorem.

Then came, a few years later, your astonishing proof of the Poincare con­jecture, which, with the preceding results, led to your winning the Fields Medal at the Moscow Congress of 1966. This was also the time of your engagement in the Berkeley Movement, and against the Vietnam War. Around that time also, I had more opportunities to get to know you. I remember happily the two trips we made in your VW Camping bus from

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8. Luncheon Talk 77

Paris to Eastern France, first to some Mineral Fair in the Vosges mountains, later on to Geneva. I was deeply impressed by the straightforward generosity of your political action. We, as old Europeans, are far more cautious about general political conditions; we know that things are not going to change so easily. May I seize this opportunity to express a personal comment on the present political situation? As a citizen of France, a nation which, in little more than 25 years, was twice rescued from disaster and annihilation thanks to the American Army, I look forward with apprehension to the day when the last G.I. will move out from European Soil...

In those years, this youthful enthusiasm of yours made a strong impression on me, and—excuse again this flowery expression—I had the feeling that you wanted to make of your life a "hymn to freedom."

Scientifically also, those were the years when, with your "horseshoe" exam­ple, you started a wonderful revival of Qualitative Dynamics, a revival which, for its global impact on general science, could be compared to the glorious period of the years 1935-45 for Algebraic Topology. Not quite, nevertheless, for the revival of Dynamics you started affected only the Western part of the World's Mathematics, whereas in the Soviet Union the Dynamical School with names such as Kolmogorov, Sinai, Arnol'd was still in full bloom.

In the years 1975-80, I cannot hide the fact that our relations went through a delicate passage: After the mediatic explosion which popularized Catastrophe Theory in 1975 came a backlash reaction in which you took part. I must say that of all the criticism that Christopher Zeeman and myself had to hear around that time, that which came from you was the most difficult to bear; for being criticized by people you admire is one of the most painful experiences there is. Now, after 15 years have passed, we may look at the matter more serenely. Had we chosen for "catastrophe theory" a less fragrant (or should I say flagrant?) name, had we called it "the use of Qualita­tive Dynamics methods in the interpretation of natural phenomena," proba­bly nothing of the sort would have happened. But even now, I do not feel too guilty about the mediatic aspect of the catastrophe terminology; I chose it because of its deep philosophical bearing: namely, any phenomenon entails an element of discontinuity—something must be generated by the local phe­nomenon, and fall upon your eye (or your measuring apparatus). In this light, the validity of Catastrophe Theory as an interpretative methodology of phe­nomena can hardly be denied.

Let me end with a general comment on your mathematical style. It could be said that there are basically two styles of mathematical writing: the consti­pated style, and the easy-going—or free air—style. Needless to say, the con­stipated style originated with our old master Bourbaki (and it found very early, as I discovered in Princeton in 1951, very strong supporters on this side of the Atlantic Ocean). Formalists of the Hilbertian School believed that mathematics can be done—or has to be done—without meaning. But when we have to choose between rigor and meaning, the Free Air people will decidedly choose meaning. In this we concur, and we shall leave the Old

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Guard to their recriminations about laxist writing. Rigor comes always soon­er or later; as General de Gaulle said: Vintendance, fa suit toujours, meaning that the ammunition always follows the fighters.

You always considered Mathematics as a game (this may justify some laxity in style), and wc enjoyed your playing. It is now your 60th birthday. For an Academic, this is just an end of youth. We sincerely wish you to continue in the same way, to gratify us with the wonderful Findings of your ever creative mind, and this for many years still to come.

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9 Banquet Address at the Smalefest E.C. ZEEMAN

It is a pleasure to wish Steve a happy birthday and to acknowledge the profound influence that he has had upon British mathematics. Prior to that influence, I remember the first lecture that I ever heard Henry Whitehead give way back in 1950: Henry was saying rather pessimistically that topolo-gists had to give up the homeomorphism problem because it was too difficult and should content themselves with algebraic topology. Marshall Stone said much the same thing in a popular article called "The Revolution in Mathe­matics," that topology had been nearly completely swallowed up by algebra.

Then, during the next decade, two spectacular results came whizzing across the Atlantic completely dispelling that pessimism. The first was Barry Mazur's proof of the Schonflies Conjecture (later beautifully refined by Mort Brown), and the second was Steve's proof of the Poincare Conjecture. The secret in each case was the audacity to bypass the main obstruction that had blocked research for half a century. In Barry's case, the blockage was the Alexander horned sphere which he bypassed by the hypothesis of local flat­ness, and in Steve's case it was dimension 3 which he bypassed by going up to dimensions 5 to infinity. Steve's result, in particular, gave a tremendous boost to the resurgent British interest in geometric topology.

But even more profound has been his influence in dynamical systems. British mathematics had suffered since the war from an artificial apartheid between pure and applied, causing research in differential equations in the UK to fall between two stools and almost disappear. But thanks to the influence of Steve and Rene Thorn, there is now a flourishing school of dy­namical systems in the UK, which is having the beneficial side effect of bring­ing pure and applied together again. In particular, Steve and many of his students came to the year-long Warwick symposia in 1968-69 and 1973-74 which had the effect of drawing widespread attention to the field, and in 1974 we were very proud to be able to give him an honorary degree at Warwick.

I would like to focus on three characteristics of Steve's work. Firstly, his perception is very geometric, and this has always enabled him to see through to the heart of the matter. It has given a unifying thread to all his work. His lectures are beautifully simple, and yet profound, because he always chooses

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exactly the right little sketch that will enable his audience to visualise the essence and remember it.

The second characteristic is his excellent mathematical taste. Or to put it another way, I happen to like the same kind of mathematics as he does. We both started in topology, and then both moved into dynamical systems. I remember at one point we both independently became interested in game theory, and when we turned up at the next conference—I think it was at Northwestern—we discovered we had both inadvertently advertised talks with the same title, dynamics in game theory, but luckily the talks were quite different.

Of course, the real evidence for the excellence of Steve's mathematical judgement is the number of mathematicians worldwide who now follow his taste. In fact, I have only known him to make one serious error of judgement, and that was his opinion of catastrophe theory.

The third characteristic of Steve's work to which I would like to draw attention is his audacity and courage. These two qualities are complemen­tary. His audacity is his desire to make a splash, to shock people, to get under their skins, and to make them confront themselves. But one can forgive his audacity because of his courage, his courage to stand by his beliefs even when swimming against the tide. These two personality traits pervade all his activi­ties, his mathematics, his sport and his politics.

In mathematics, he has the audacity to let his intuition leap ahead of proof, and the courage to publish that intuition as bold conjectures. In sport, he has the audacity to tackle mountains and set sail across the oceans, and the courage to carry through with these achievements. In politics, he had the audacity to rebuke the Soviet Union on the steps of Moscow University for invading Hungary, and the courage to face the KGB afterwards if necessary; and in the same breath the audacity to rebuke his own country for invading Vietnam, and the courage to face the consequences afterwards of having to defend his funding against attacks by politicians. This week he had the au­dacity to refuse to have his broken ankle set in plaster, and the courage to endure the resulting pain so that he would not disappoint us at the Smalefest.

Of course, much of Steve's courage springs from Clara, who has always stood by him through thick and thin, giving him a secure harbour within which to anchor, and from which he could then sail out to conquer the world. It was she who introduced him to the collecting of minerals, to which he has devoted so much of his enthusiasm over the years. I would like to say spe­cially to her tonight that we all include her in the celebrations. So I call upon you to drink a toast to Clara and Steve.

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Some Retrospective Remarks

Steve Smale

DEPARTMENT OF MATHEMATICS

CITY UNIVERSITY OF HONG KONG

KOWLOON, HONG KONG

The purpose of this section is to add a few odd thoughts and other remarks relative to the papers collected below. This addenda will be brief in view of the fine reviews prefacing each part by Moe Hirsch, Gerard Debreu, Jacob Palis, Mike Shub, Tony Tromba, Jerry Marsden, and Lenore Blum/Felipe Cucker. There are also the informal talks by Raoul Bott, Mauricio Peixoto, Ren6 Thorn, and Christopher Zeeman above, and discussions reprinted below which pre-empt things I could say. A biography of me by Steve Batterson is about to appear, published by the American Mathematical Society. I have not read it at the time these words are being written, but I have every reason to expect that this carefully researched work will portray accurately the environment in which my articles were created. I can recommend the proceedings of the Smalefest of Hirsch, Marsden, and Shub, for related further historical accounts. I take this opportunity to acknowledge, with great appreciation, the expositions of these many mathematicians towards the understanding of my work.

Remarks on Part 1. Topology

A particular example of the results of references 3, 6, and 7 has been subsequently named "Turning a sphere inside out" or "everting the 2-sphere". There were three movies made of this homotopy. The first was made by Nelson Max, the second was done by the Minnesota Geometry Center, and the third was finished in time for the Berlin ICM, in 1998. A photograph inspired by the second movie adorns the cover of the recent biography by Steve Batterson. There were two covers of the Scientific American, May 1966 and October 1993, picturing the evcrsion. (The last attributed the result to Bill Thurston and he made a correction in the following issue.)

As to Vietoris Mapping Theorem for Homotopy, reference 2, I recall Eldon Dyer telling me the following story: Chern as editor had given him the manuscript to referee with the suggestion he reject it. (At that time I was an unknown student at the University of Michigan, and as an editor I might well have taken the same action as Chern). However upon looking at it, KIdon liked it and recommended acceptance. It was published.

Both papers, reference 5, on self-intersections, and 9, on involutions of the 3-sphere, have fundamental mistakes in them.

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There is the story in reference 17, which chronicles my work on Poincart's conjecture in higher dimension, and the h-cobordism theorem, of references 10, 11, 12, 13, 14, 15. For a fine detailed mathematical account close to my original spirit, I like the book Differential Manifolds by Antoni Kosinski, 1993, Academic Press.

I have said on a number of occasions that I "left topology" in the summer of 19(51 upon com­pletion of the h-cobordism article (paper 15). I believe that at that time I felt that tbn outstanding problems in dimensions three and four represented exceptional cases in topology. The challenges of dynamics at that time seemed more exciting, and I said so publicly. Maybe some of the topologists never forgave me! In the meantime I have gained more respect for the problems in dimension 3 and 4.

The paper on gradient dynamical systems (number 11) played an important role in the devel­opment of my work in topology. Hut at the same time that paper (as well as paper 46) helped me see some ideas for dynamics, for example giving a special case of the Kupka-Smale Theorem and beginning the clarification of the global stable manifolds.

Remarks on Part 2. Economics In spite of a great support for me by the mathematical economics community, there was one

disappointment. I suggested that the proof of the existence of equilibrium, should be integrated into a framework where supply and demand are seen as maps from the space of commodities to the space of prices. Here 1 emphasize that these spaces are distinct, and also the natural source and target for these fundamental quantities of economic theory. Following Walras of the 19th century, one can show that this system (structured) of equations, supply equals demand, has a solution, to obtain the basic existence theorem of economic equilibrium theory.

The fundamental advance of equilibria theory occurred in the 1950's with the proof of existence of equilibria in great generality by Ken Arrow and Gerard Debreu. Their work transformed this problem of existence into a situation which utilized fixed point theory (of Kakutani) to solve. But this transformation by using an auxilliary map, destroys the natural structure of the supply/demand maps from commodities to prices.

In much of my work in economics I tried to reorient the Arrow-Debreu framework toward the original Walrasian. For example in my handbook paper, reference 31, I developed the existence theory via the supply-demand maps with the same generality of Arrow-Debreu, using degree theory and approximations. However I think that the economics community never accepted my point of view. Even though Gerard in his review discusses my work on existence with great generosity, I feel that this point must be made.

My approach to the "Prisoner's Dilemma" (reference 30) centered around a resolution using repeated games. This was followed by some popular lectures. Although I believe that I was not

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the first to see the connection between this game and the opera Toaca, I did give the UC Berkeley Faculty Research Lecture with the title, The Tragedy of Tosca and the Logic of the Arms Race.

Moreover I spoke on the same subject at a German meeting Natur- Witsentchafter Gegen Atom-rusting. My talk, reference 34, was received by the very large audience well enough, but Linus Pauling who spoke after me, received a long lasting standing ovation. I recall that during this applause, he graciously came to me and had me stand up as if I were a supporting musician!

Remarks on Part 3. Miscellaneous

An Extended version of my paper on mathematical problems for the next century, reference 37, is about to appear in a book edited by Arnold et al, published by the American Math Society.

Remarks on Part 5. Dynamical Systems

Here is some chronology on the horseshoe: It was 5 years between the discovery and the pub­lishing of the paper. See reference 69 for the story in Rio, spring 1960, of finding the horseshoe on the beaches there. This account gives some of my more recent perspectives on the meaning of the results. I did speak of my work on this subject in Berkeley in 1960, but until the next summer I was spending most of my time finishing the papers in topology. It was in late summer 1961 in Kiev at an international conference on non-linear oscillations when I began speaking systematically on the horseshoe and homoclinic points, and right after that were my lectures in Moscow on the subject. It was eventually at the Morse Symposium in Princeton, that I took the opportunity to publish the full paper with details. I did announce the results in the proceedings of the Kiev meeting and also mentioned them in my ICM Stockholm 1962 paper.

I think that an overlooked part of the history of dynamics has to do with foundational aspects related to the concept of global stable manifold. Before the early 60's, the definition of this object had not been satisfactorily given and for that reason progress in dynamics was slowed. In retrospect I see now my attention to the concept of the global stable manifolds, especially as in my Pisa lectures, reference 49, played this foundational role. The clarification of just how these objects were defined helped pave the way to make my further work in dynamics possible.

Mathematicians as Birkhoff could certainly recognize one-dimensional stable manifolds for sur­face diffeomorphisms, and basic text of Coddington-Levinson gave the difficult existence proof for the local objects, very generally. But this last book did not proceed to a definition of the global object. It stopped abruptly with the local result. I am reminded a bit by the situation in numerical analysis, where an "algorithm" is readily recognized, but a mathematician working in that subject can't define the object.

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Rene1 Thorn and Christopher Zeeman have been important and supportive friends for several decades. But I am afraid that my critique of Catastrophe Theory in the review of reference 64, made them quite angry with me. I can understand that, as I was pretty sharp in what could well be seen as an attack on their work. One can see some of their more recent perspectives expressed in their informal talks above. I recall that my feeling at that time was, I said what had to be said about an issue that was swirling about my head. I spent a long time trying to assess Catastrophe Theory and putting my thoughts into the words of that controversial paper.

A number of my papers in general were written in the midst of political turmoil, but those that appeared in the 1968 Berkeley global analysis conference organized by Chern and myself, papers 55, 56, 57, 58, were special even at that. David and Sue Elworthy arrived in Berkeley for the conference, in the middle of a street battle between students and police. They took the side of the students, and with their suitcases at their side, fought behind the barricades. Helicopters dropped tear gas on campus and the opening cocktail party had to be ended early because of curfew.

The conference in Salvador, Brasil, of reference 61, was held during a very repressive time under the military dictatorship. I was under attack by Grotendieck for even attending the meeting. And Paul Koosis wrote a critique of me for going to Brasil in the journal Mother Functor. In Brasil the situation became tougher as my Brasilian friends Mauricio Peixoto and Jacob Palis became upset with me and other mathematicians who supported a math student at IMPA, one who had been imprisoned and tortured by the military. They were concerned that our actions could have as a consequence the destruction of Brasilian mathematics. Mike Shub has a written letter on this, which appeared in the New York Review of Books. Mike and I also reported to Mother Functor an account of the trip. Serge Lang discussed the event at the Berkeley 1990 Smalefest. One notable aspect of that summer in Brasil was an automobile trip with Jacob, from Rio to Babia, where the conference was held. The great tensions between us were mediated by our very warm friendship. Incidently on that trip, we stopped in Teofilo Otoni and at the home of the famed mineral dealer "Jacinto", acquired our first important Brasilian mineral specimen.

One can see Batterson's biography for some context in which the article Differentiable Dynamical Systems, reference 54, was written.

Remarks on Part 6. Mechanics Jerry Marsden in his article below gives as reference for my lectures on elementary particles,

notes by Ralph Abraham. Actually the Abraham notes relate to my course on the calculus of variations. The notes of the elementary particle lectures were written up by Victor Guillemin (Columbia University, cerca 1962).

The topology of celestial mechanics has come a long way since my 1970 papers, ref 71, 72. The recent AMS Memoir of McCord, Meyer and Wang gives some picture of the developments.

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Remarks on Part 7. Biology, Electric Circuits, Math. Programming

Reference 79 is on my work with Bob Williams on the period doubling map, which I learned from Bob May and was made famous by Mitch Feigenbaum. In fact Mitch was in the audience at Aspen when 1 spoke on our result, and he has written how this influenced his own discoveries. Subsequently when he showed me what he had found, I put him in touch with Oscar Lanford who gave him useful support.

Another paper, reference 80, dealt with population biology. Later Moe Hirsch developed this whole area with a series of beautiful papers.

Remarks on Part 8. Theory of Computation

My efforts in the theory of computation are still going on after quite a number of years. I was supportive of the development of the new organization called Foundations of Computational Mathematics. There is the need for such a group because there are strong pressures against theory and foundational work in the area of scientific computation. I received demands from funding agencies and mathematicians for explicit applications and numerical evidence that what I was doing was useful. This contrasts very much with the situation in theoretical computer science where theoretical work as on the problem "does P=XP?1' is encouraged.

I end these words by expressing my great appreciation to Colette Lam for the onormous and effective job that she has done in assembling these papers. Especially I give my warm thanks to Felipe Cucker and Roderick Wong for making these volumes possible.

10 The Work of Stephen Smale in Differential Topology MORRIS W. HIRSCH

Background The theme of this conference is "Unity and Diversity in Mathematics." The diversity is evident in the many topics covered. Reviewing Smale's work in differential topology will reveal important themes that pervade much of his work in other topics, and thus exhibit an unexpected unity in seemingly diverse subjects.

Before discussing his work, it is interesting to review the status of differen­tial topology in the middle 1950s, when Smale began his graduate study.

The full history of topology has yet to be written (see, however, Pont [52], Dieudonne [8]). Whereas differentiable manifolds can be traced back to the smooth curves and surfaces studied in ancient Greece, the modern theory of both manifolds and algebraic topology begins with Betti's 1871 paper [4]. Betti defines "spaces" as subsets of Euclidean spaces define by equalities and inequalities on smooth functions.1 Important improvements in Betti's treatment were made by Poincare in 1895. His definition of manifold de­scribes what we call a real analytic submanifold of Euclidean space; but it is clear from his examples, such as manifolds obtained by identifying faces of polyhedra, that he had in mind abstract manifolds.2 Curiously, Poincarc's "homeomorphisme" means a C1 diffeomorphism. Abstract smooth manifolds in the modern sense— described in terms of coordinate systems—were de­fined (for the two-dimensional case) by Weyl [73] in his 1913 book on Rie-mann surfaces.

Despite these well-known works, at mid-century there were few studies of the global geometrical structure of smooth manifolds. The subject had

1 This is the paper defining, rather imprecisely, what are now called Betti numbers. Pont [52] points out thai the same definition is given in unpublished notes of Rie-mann, who had visited Betti. 2 It is not obvious that such manifolds imbed in Euclidean space!

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84 M.W. Hirsch

not yet been named.3 Most topologists were not at all interested in smooth maps. The "topology of manifolds" was a central topic, and the name of an important book by Ray Wilder, but it dealt only with algebraic and point-set topology. Steenrod's important book The Topology of Fibre Bundles was published in 1956. The de Rham theorems were of more interest to differen­tial geometers than to topologists, and Morse theory was considered part of analysis.

A great deal was known about algebraic topology. Many useful tools had been invented for studying homotopy invariants of CW complexes and their mappings (Eilenberg MacLane spaces, Serre's spectral sequences, Postnikov invariants, Steenrod's algebras of cohomology operations, etc.) Moreover, there was considerable knowledge of nonsmooth manifolds—or more accu­rately, manifolds that were not assumed to be smooth, such as combina­torial manifolds, homology manifolds, and so forth. Important results in­clude Moise's theory of triangulations of 3-manifolds, Reidemeister's torsion classification of lens spaces, Bing's work on wild and tame embeddings and decompositions, and "pathology" such as the Alexander horned sphere and Antoine's Necklace (a Cantor set in R3 whose complement is not simply connected). The deeper significance of many of these theories emerged later, in the light of the /i-cobordism theorem and its implications.

A great deal was known about 3-dimensional manifolds, beginning with Poincare's examples and Heegard's decomposition theory of 1898. The latter is especially important for understanding Smale's work, because it is the origin of the theory of handlebody decompositions.

The deepest results known about manifolds were the duality theorems of Poincare, Alexander, and Lefschetz; H. Hopf's theorem that the indices of singularities of a vector field on a manifold add up to the Euler characteristic; de Rham's isomorphism between singular real cohomology and the coho­mology of exterior differential forms; Chern's generalized Gauss-Bonnet for­mula; the foliation theories of Reeb and Haefliger; theories of Tiber bundles and characteristic classes due to Pontryagin, Stiefel, Whitney, and Chern, with further developments by Steenrod, Weil, Spanier, Hirzebruch, Wu, Thorn, and others; Rohlin's index theorem for 4-dimensional manifolds; Henry Whitehead's little-known theory of simple homotopy types; Wilder's work on generalized manifolds; P.A. Smith's theory of fixed points of cyclic group actions. Most relevant to Smale's work was M. Morse's calculus of variations in the large, Thorn's theory of cobordism and transversality, and Whitney's studies of immersions, embeddings, and other kinds of smooth maps.

3 The term "differential topology" seems to have been coined by John Milnor in the late 1950s, but did not become current for some years. The word "diffeomorphism" did not yet exist—it may be due to W. Ambrose. While Smale was at the Institute for Advanced Study, he showed me a letter from an editor objecting to "diffeomorphism," claiming that "differomorphism" was etymologically better!

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10. The Work of Stephen Smale in Differential Topology 85

No one yet knew of any examples of homeomorphic manifolds that were not diffeomorphic, or of topological manifolds not admitting a differentiable structure—Milnor's invention of an exotic 7-sphere was published in 1956. Work on the classification of manifolds, and many other problems, was stuck in dimension 2 by Poincare's conjecture in dimension 3 (still unsolved).

The transversality methods developed by Pontryagin and Thorn were not widely known. The use of manifolds and dynamical systems in mechanics, electrical circuit theory, economics, biology, and other applications is now common4; but in the fifties it was quite rare.

Conversely, few topologists had any interest in applications. The spirit of Bourbaki dominated pure mathematics. Applications were rarely taught or even mentioned; computation was despised; classification of structure was the be-all and end-all. Hardly anyone, pure or applied, used computers (of which there were very few). The term "fractals" had not yet been coined by Mandelbrot; "chaos" was a biblical rather than a mathematical term.

In this milieu, Smale began his graduate studies at Michigan in 1952.5

The great man in topology at Michigan being Ray Wilder, most topology students chose to work with him. Smale, however, for some reason became the first doctoral student of a young topologist named Raoul Bott. In view of Smale's later work in applications, it is interesting that Bott had a degree in electrical engineering; and the "Bott-Duffin Theorem" in circuit theory is still important.

Immersions

Smale's work in differential topology was preceded by two short papers on the topology of maps [56, 57]. His theorems are still interesting, but not closely related to his later work. Nevertheless, the theme of much subsequent work by Smale, in many fields, is found in these papers: fibrations, and more generally, the topology of spaces of paths.

Given a (continuous) map p: E-*B, a lift of a map / : X -* B is a map g:X-*E such that the following diagram commutes:

E

X " yB

That is, p o g = / We say (p, E, B) is a fibration if every path f:-* B can be

4 Thanks largely to Smale's pioneering efforts in these fields. 5 Smale's autobiographical memoir in this volume recounts some of his experiences in Michigan.

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86 M.W. Hirsch

lifted, the lift depending continuously on specified initial values in £.6 Fibra-tions, the subject of intense research in the fifties, are the maps for which the tools of algebraic topology are best suited.

In his doctoral thesis [65], Smale introduced the use of flbrations of spaces of differentiable maps as a tool for classifying immersions. This novel tech­nique proved to be of great importance in many fields of geometric topology, as will be discussed below.

Smale's first work in differential topology was about immersions. An immersion f:M-*Nis& smooth map between manifolds M, N such that at every x e M, the tangent map TJ~: TXM -» Tflx)N is injective. Here TM denotes the tangent vector bundle of M, with fiber Tfx over x e Af. A re­gular homotopy is a homotopy /„ 0 ^ t ^ 1 of immersions such that Tf, is a homotopy of bundle maps.7 An immersion is an embedding if it is a homeo-morphism onto its image, which is necessarily a locally closed smooth sub-manifold. A regular homotopy of embeddings is an isotopy.

Here is virtually everything known about immersions in the early fifties: In a tour de force of differential and algebraic topology and geometric intu­ition in 1944, Hassler Whitney [77, 78] had proved that every (smooth) n-dimensional manifold could be embedded in R2" for n ^ 1, and immersed in R2""1 for n > 2. On the other hand, it was known that the projective plane and other nonorientable surfaces could not be embedded in R \ and Whitney had proved other impossibility results using characteristic classes. Steenrod had a typewritten proof that the Klein bottle does not embed in real projec­tive 3-space. The Whitney-Graustein theorem [76] showed that immersions of the circle in the plane are classified by their winding numbers. As a student working with Ed Spanier I proved the complex projective plane, which could be embedded in R7, could not be immersed in R6.8

Immersions of Circles The problem Smale solved in his thesis is that of classifying regular homo­topy classes of immersions of the circle into an arbitrary manifold N. More generally, he classified immersions / : / -»N of the closed unit interval / =

6 Precisely: Given a compact polyhedron P and maps F: P x / - B, g: P x 0 - » £ such that pog = F\P x 0 -»£ , there is an extension of g to a map C : P x / ♦ £ such that p o G = F. 7 What is important and subtle here is joint continuity in (r, x) of df,x)/dx. Without it, "regular homotopy" would be the same as "homotopy of immersions." In the plane, for example, the identity immersion of the unit circle is not regularly homotopic to its reflection in a line, but these two immersions are homotopic through immersions, as can be seen by deforming a figure-eight immersion into each of them. 8 The proof consisted of computing the secondary obstruction to a normal vector field on an embedding in R7, using a fonnula of S.D. Liao [38], another student of Spanier. This calculation was immediately made trivial by a general result of W.S. Massey [40].

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10. The Work of Stephen Smale in Differential Topology 87

[0,1] having fixed boundary data, i.e., fixed initial and terminal tangent vec­tors/'(0) and/'l).

Smale's approach was to study the map p:E-*B, where

• £ is the space of immersions' f:I-*N having fixed initial value /(0) and fixed initial tangent /'(0);

• B is the space of nonzero tangent vectors to N; • p assigns to / the terminal tangent vector f'(\).

The classification problem is equivalent to enumerating the path components of the fibers because it can be seen that such a path component is a regular homotopy class for fixed boundary data.

Bott asked Smale an extraordinarily fruitful question: Is (p,E,B) afibra-tionl This amounts to asking for a Regular Homotopy Extension Theorem. In his thesis [65], Smale proved the following:

Theorem. Let u„0 < t < 1 be a deformation of f'\) in B, i.e., a path of nonzero tangent vectors beginning with f'( 1). Then there is a regular homotopy F'.S1 x / -»N, Fx, t) = f,(x) such that f0 = / , all f, have the same initial tan­gent, and the terminal tangent of f, is u,. Moreover, F can be chosen to depend continuously on the data f and the deformation u,.

This result is nontrivial, as can be seen by observing that it is false if N is 1-dimensional (exercise!).

It is not hard to see that the total space £ is contractible. Therefore the homotopy theory of fibrations implies that the klh homotopy group of any fiber F is naturally isomorphic to the (k + l)st homotopy group of the base space B. Now B has a deformation retraction onto the space T, N of unit tangent vectors. By unwinding the homotopies involved, Smale proved the following result theorem for Riemannian manifolds N of dimension B > 2:

Theorem. Assume N is a manifold of dimension n ;> 2. Fix a base point x0 in the circle, and a nonzero "base vector" v0 of length 1 tangent to N. Let F denote the space of immersions f:Sl-*N having tangent v0 at x0. To f assign the loop / # : S1 -* TjiV, where f0 sends xe Sl to the normalized tangent vector to f at x, namely / '(x)/||/ '(x)||. Then f, induces a bijection between the set of path components of F and the fundamental group 71,(71 N, v0).

For the special case where N is the plane, this result specializes to the Whitney-Graustein theorem [76] stated above.

9 A space of immersions is given the C1 topology. This means that two immersions are close if at each point their values are close and their tangents are close. It turns out that the homotopy type of a space of immersions is the same for all C topologies, 1 S r < oo.

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88 M.W. Hirsch

Immersions of Spheres in Euclidean Spaces Smale soon generalized the classification to immersions of the fc-sphcre S* in Euclidean n-space R". Again the key was a fibration theorem. Let E now denote the space of immersions of the closed unit fc-disk Dk into R", and B the space of immersions of S*"1 into R". The map p: E-> B assigns to an immer­sion / : Z)* - R" its restriction to the boundary. Smale proved that (p, E, B) is a fibration provided k <n. Geometrically, this says regular homotopies of f\Sk~l can be extended over Dk to get a regular homotopy off, and similarly for Ic-parameter families of immersions.

Using this and similar fibration theorems, Smale obtained the following result [58, 59]:

Theorem. The set of regular homotopy classes of immersions S* -»R" corre­sponds bijectively to nk(Vmk), the kth homotopy group of the Stiefel manifold of k-frames in R", provided k < n.

To an immersion / : S* -> R* Smale assigned the homotopy class of a map d: Sk -* VHk as follows. By a regular homotopy, we can assume / coincides with the standard inclusion Sk -+ R" on a small open Ac-disk in S\ whose complement is a closed fc-disk B. Let e(x) denote a field of Jc-frames tangent to B. Form a k-sphere £ by gluing two copies B0 and B, of B along the boundary. Define a map af): I -» VHk by mapping x to f+e(x) if x e B0, and to e(x) if x e B,. Here f+ denotes the map of frames induced by Tf. The homotopy class of af) is called the Smale invariant of the immersion / .

The calculation of homotopy groups is a standard task for algebraic to­pology. While it is by no means trivial, in any particular case a lot can usual­ly be calculated. The Stiefel manifold Vnk has the homotopy type of the homogeneous space 0(n)/0n — k), where 0(m) denotes the Lie group of real orthogonal m x m matrices. Therefore explicit classifications of immersions were possible for particular values of k and n, thanks to Smale's theorem.10

A surprising application of Smale's classification is his theorem that all immersions of the 2-sphere in 3-sphere are regularly homotopic, the reason being that n203)) = 0.u In particular the identity map of S1, considered as an immersion into R3, is regularly homotopic to the antipodal map. The analo­gous statement is false for immersions of the circle in the plane.

When Smale submitted his paper on immersions of spheres for publica­tion, one reviewer claimed it could not be correct, since the identity and

10 Even where the homotopy group nk(VnJl) has been calculated, there still remains the largely unsolved geometric problem of finding an explicit immersion / : Sk -»R" representing a given homotopy class. Some results for k = 3, n = 4 were obtained by J. Hass and J.Hughes [18]. 1 ' Always remember: rc2 of anv Lie group is 0.

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10. The Work of Stephen Smale in Differential Topology 89

antipodal maps of S2 have Gauss maps of different degrees!12 Exercise: Find the reviewer's mistake!

It is not easy to visualize such a regular homotopy, now called an eversion of the 2-sphere. After Smale announced his result, verbal descriptions of the eversion were made by Arnold Shapiro (whom I could not understand), and later by Bernard Morin (whom I could).13

One way to construct an eversion is to first regularly homotop the identity map of the sphere into the composition of the double covering of the projec-tive plane followed by Boy's surface, an immersion of the projective plane into 3-space pictured in Geometry and the Imagination [24]. Since this iden­tifies antipodal points, the antipodal can also be regularly homotoped to this same composition.

Tony Phillips' Scientific American article [49] presents pictures of an ever­sion. Charles Pugh made prizewinning wire models of the eversion through Boy's surface, unfortunately stolen from Evans Hall on the Berkeley campus. There is also an interesting film by Nelson Max giving many visualizations of eversions. Even with such visual aids, it is a challenging task to understand the deformation of the identity map of S2 to the antipodal map through immersions.

Smale's proof of the Regular Homotopy Extension Theorem (for spheres and disks of all dimensions) is based on integration of certain vector fields, foreshadowing his later work in dynamics.

There is no problem in extending a regular homotopy of the boundary restriction of / to a smooth homotopy of / ; the difficulty is to make the extension a regular homotopy. Smale proceeded as follows.

Since D* is contractible, the normal bundle to an immersion / : D" -»R" is trivial. Therefore, to each x e Dk, we can continuously assign a nonzero vector w(x) normal to the tangent plane to /(D*) at f(x).1* (Note the use of the hypothesis k < n.) Now f(Dk) is not an embedded submanifold, and w is not a well-defined vector field on /(D*), but / is locally an embedding, and w extends locally to a vector field in R". This is good enough to use integral curves of w to push most of/(£>*) along these integral curves, out of the way of the given deformation of / along S*"1. Because of the extra dimension, Smale was able to use this device to achieve regularity of the extension. Of course, the details, containing the heart of the proof, are formidable. But the concept is basically simple.

In his theory of immersions of spheres in Euclidean spaces, Smale introduced two powerful new methods for attacking geometrical problems:

12 The Gauss map, of an embedding f of a closed surface S into 3-space, maps the surface to the unit 2-sphcre by sending each point xeSlo the unit vector outwardly normal to f(S) at f(x). 13 Morin is blind. 14 More precisely, w(x) is normal to the image of Df(x), the derivative of/at x.

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90 M.W. Hirsch

Dynamical systems theory (i.e., integration of vector fields) was used to con­struct deformations in order to prove that certain restriction maps on func­tion spaces are fibrations; and then algebraic topology was used to obtain isomorphisms between homotopy groups. These techniques were soon used in successful attacks on a variety of problems.

Further Development of Immersion Theory I first learned of Smale's thesis at the 1956 Symposium on Algebraic Topol­ogy in Mexico City. I was a rather ignorant graduate student at the Univer­sity of Chicago; Smale was a new Ph.D. from Michigan.13 While I under­stood very little of the talks on Pontryagin classes, Postnikov invariants and other arcane subjects, I thought I could understand the deceptively simple geometric problem Smale addressed: Classify immersed curves in a Rieman-nian manifold.

In the fall of 1956, Smale was appointed Instructor at the Univcrstiy of Chicago. Having learned of Smale's work in Mexico City, I began talking with him about it, and reading his immersion papers. I soon found much simpler proofs of his results. Every day I would present them to Smale, who would patiently explain to me why my proofs were so simple as to be wrong. By this process I gradually learned the real difficulties, and eventually I un­derstood Smale's proofs.

In my own thesis [25] directed by Ed Spanier, I extended Smale's theory to the classification of immersions f:M-*N between arbitrary manifolds, provided dim N > dim M. In this I received a great deal of help from both Smale and Spanier. The main tool was again a fibration theorem: the restric­tion map, going from immersions of M to germs of immersions of neighbor­hoods of a subcomplex of a smooth triangulation of M, is a fibration.

The proof of this fibration theorem used Smale's fibration theorem for disks as a local result; the globalization was accomplished by means of a smooth triangulation of M, the simplices of which are approximately disks.

The classification took the following form. Consider the assignment to / of its tangent map TF: TM -* TN between tangent vector bundles. This defines a map 0 going from the space of immersions of M in N to the space of (linear) bundle maps from TM to TN that are injective on each fiber. The homotopy class of this map (among such bundle maps) generalizes the Smale invariant. Using the fibration theorem and Smale's theorems, I showed <1> induces isomorphisms on homotopy groups. By results of Milnor and J.H.C. Whitehead, this implies <1> is a homotopy equivalence. The proof of the classi­fication is a bootstrapping induction on the dimension of M; the inductive step uses the fibration theorem.

1 * For an account of the atmosphere in Chicago in the fifties, see my memoir [28].

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10. The Work of Stephen Smale in Differential Topology 91

Thus immersions are classified by certain kinds of bundle maps, whose classification is a standard task for algebraic topology. A striking corollary of the classification is that every parallelizable manifold is immersible in Euclidean space of one dimension higher.16

Several topologists'7 reformulated the classification of immersions of an m-dimensional manifold M into a Euclidean space R"+* as follows. Let "PM: M -»BO be the classifying map (unique up to homotopy) for the stable nor­mal bundle of M. An immersion / : M -* R"+* determines a lift of 4*M over the natural map BO(k) -» BO. Using homotopy theory, it can be deduced from the classification theorem that regular homotopy classes of immersions cor­respond bijectively in this way to homotopy classes of lifts of *PM.

Subsequently many other classification problems were solved by showing them to be equivalent to the homotopy classification of certain lifts, or what is the same thing, crosssections of a certain flbration. The starting point for this approach to geometric topology was the extraordinarily illuminating talk of R. Thorn at the International Congress of 1958 [70], in which he stated that smoothings of a piecewise linear manifold correspond to sections of a certain flbration.18

Other proofs of the general immersion classification theorem were ob­tained by R. Thorn [69], A. Phillips [48], V. Poenaru [50], and M. Gromov and Ja. Eliasberg [14. 15] (see also A. Haefliger [16]). Each of these dif­ferent approaches gave new insights into the geometry of immersions.

Many geometrically minded topologists were struck by the power of the flbration theorem and attacked a variety of mapping and structure problems with fibration methods.

The method of fibrations of function spaces was applied to submersions (smooth maps f:M-*Noi rank equal to dim N) by A. Phillips [48]. Again the key was a fibration theorem, and the classification was by induced maps between tangent bundles. This was generalized to k-mersions (maps of rank k > dim N) by S. Feit [10]. General immersion theory was made applicable to immersions between manifolds of the same dimension, provided the do­main manifold has no closed component, by V. Poenaru [50] and myself [26].

Fibration methods were used to classify piecewise linear immersions by Haefliger and Poenaru [17]. Topological immersions were classified by J.A. Lees [37] and R. Lashof [36]. *

M. Gromov [11-13] made a profound study of mapping problems amena­ble to fibration methods, and successfully attacked many geometric problems of the most diverse types. The article by D. Spring in this volume discusses

16 Exercise (unsolved): Describe an explicit immersion of real projective 7-space in R8! 17 The first may have been M. Atiyah [2]. 18 This may have been only a conjecture—Thorn was not guilty of excessive clarity.

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92 M.W. Hirsch

Gromov's far-reaching extensions of immersion theory to other mapping problems such as immersions which are symplectic, holomorphic, or isomet­ric; see Chapters 2 and 3 of Gromov's book [13].

R. Thom [69] gave a new, more conceptual proof of Smale's theorem. R.S. Palais [47] proved an isotopy extension theorem, showing that the restric­tion map for embeddings is not merely a fibration, it is a locally trivial fiber bundle (see also E. Lima [39]). R. Edwards and R. Kirby [9] proved an isotopy extension theorem for topological manifolds.

The 1977 book [35] by R. Kirby and L. Siebenmann contains a unified treatment of many classification theories for structures on topological, piece-wise linear, and smooth manifolds. Besides many new ideas, it presents devel­opments and analogues of Smale's fibration theories, Gromov's ideas, and Smale's later theory of handlebodies. See in particular, Siebenmann's articles [54] and [55], a reprinting of [53].

Diffeomorphisms of Spheres In 1956 Milnor astounded topologists with his construction of an exotic differentiable structure on the 7-sphere, that is, a smooth manifold homeomor-phic but not diffeomorphic to S7. This wholly unexpected phenomenon trig­gered intense research into the classification of differentiable structures, and the relation between smooth, piecewise linear, and topological manifolds.

Milnor's construction was based on a diffeomorphism of the 6-sphere which, he proved, could not be extended to a diffeomorphism of the 7-ball; it was, therefore, not isotopic to any element of the orthogonal group 0(7) considered as acting on the 6-sphere. His exotic 7-sphere was constructed by gluing together two 7-balls by this diffeomorphism of their boundaries. These ideas stimulated investigation into diffeomorphism groups.

Two-spheres In 1958 Smale [59] published the following result:

Theorem. The space Diff(S2) of diffeomorphism of the 2-sphere admits the orthogonal group 0(3) as a deformation retract.19

Again a key role in the proof was played by dynamical systems. I recall Smale discussing his proof of this at Chicago. At one stage he did not see

19 Around this time an outline of a proof attributed to Kneser was circulating by word of mouth; it was based on an alleged version of the Riemann mapping theorem which gives smoothness at the boundary of smooth Jordan domains, and smooth dependence on parameters. I do not know if such a proof was ever published.

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10. The Work of Stephen Smale in Differential Topology 93

why the proof did not go through for spheres of all dimensions, except that he knew that the analogous result for the 6-sphere would contradict Milnor's constructions of an exotic differentiable structure on the 7-sphere! It turned out that the Poincare- Bendixson Theorem, which is valid only in dimension 2, played a key role in his proof.

In 1958 Smale went to the Institute for Advanced Study. This was a very fertile period for topology, and a remarkable group of geometers and topolo-gists were assembled in Princeton. These included Shiing-Shen Chern, Ed Spanier, Armand Borel, Ed Floyd, Dean Montgomery, Lester Dubins, Andy Gleason, John Moore, Ralph Fox, Glenn Bredon, John Milnor, Richard Palais, Jim Munkres, Andre Weil, Henry Whitehead, Norman Steenrod, Bob Williams, Frank Raymond, S. Kinoshita, Lee Neuwirth, Stewart Cairns, John Stallings, Barry Mazur, Papakyriakopoulos, and many others.

I shared an office, with Smale and benefited by discussing many of his ideas at an early stage in their development. Among the many questions that interested him was a famous problem of P.A. Smith: Can an involution (a map of period 2) of S3 have a knotted circle of fixed points? He did not solve it, but we published a joint paper [29] on smooth involutions having only two fixed points. Unfortunately it contains an elementary blunder, and is totally wrong.20

Three-spheres Smale worked on showing that the space Diff(S3) of diffeomorphism of the 3-sphere admits the orthogonal group 0(4) as a deformation retract. Using several fibrations, such as the restriction map going from diffeomorphisms of S3 to embeddigs of D3 in S3, and from the latter to embeddings of S2 in S3, and so forth, he reduced this to the same problem for the space of em-beddings of S2 in R3. Although it failed, his approach was important, and stimulated much further research. Hatcher [20] proved Smale's conjecture in 1975.

In a manuscript for this work Smale analyzed an embedding in Euclidean space by considering a height function, i.e., the composition of the embedding with a nonzero linear function.21

Smale tried to find a height function which, for a given compact set of embeddings of S2 in R \ would look like a Morse function for each embedding, exhibiting it as obtained from the unit sphere by extruding pseudopods in a manageable way. These could then all be pushed back, following the height function, until they all became diffeomorphisms of S2.

20 I am glad to report that other people have also made mistakes in this problem. 21 This idea goes back to Mobius [45], who used it in an attempt to classify surfaces; see Hirsch [27] for a discussion. J. Alexander [1] had used a similar method to study piecewise linear embeddings of surfaces.

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94 M.W. Hirsch

At that point, appeal to his theorem on diffeomorphisms of S2 would finish the proof.

He had a complicated inductive proof; but Robert Williams, Henry White-head and I (all at the Institute then) found that the induction failed at the first step!

Nevertheless the idea was fruitful. J. Cerf [6] succeeded in proving that Diff(S3) has just two path components. Cerf used a more subtle development of Smale's height function: He showed that for a one-parameter family of embeddings, there is a function having at worst cubic singularities, but be­having topologically like a Morse function for each embedding in the family. Cerf's ideas were to prove useful in other deformation problems in topology and dynamics, and surprisingly, in algebraic K-theory. See Hatcher [19], Hatcher and Wagoner [21], and Cerf [5, 7].

Smale would return to the use of height functions as tools for dissecting manifolds in his spectacular attack on the generalized Poincare conjecture.

In using height functions to analyze embedded 2-spheres, Smale was grap­pling with a basic problem peculiar to the topology of manifolds: There is no easy way to decompose a manifold. Unlike a simplicial complex, which come equipped with a decomposition into the simplest spaces, a smooth manifold —without any additional structure such as a Riemannian metric—is a ho­mogeneous global object. If it is "closed"—compact, connected and without boundary—it contains no proper closed submanifold of the same dimen­sion, is not a union of a countable family of closed submanifolds of lower dimension. This is a serious problem if we need to analyze a closed manifold because it means we cannot decompose it into simpler objects of the same kind.

Before 1960 the traditional tool for studying the geometric topology of manifolds was a smooth triangulation. Cairns and Whitehead had shown such triangulations exist and are unique up to isomorphic subdivisions. Thus to every smooth manifold there is associated a combinatorial manifold. In this way simplicial complexes, for which combinatorial techniques and in­duction on dimension are convenient tools, are introduced into differential topology. But useful as they are for algebraic purposes, they are not well-suited for studying differentiable maps.22

Smale would shortly return to the use of Morse functions to analyze mani­folds. His theory of theory of handlebodies was soon to supply topologists with a highly succussful technique for decomposing smooth manifolds.

22 Simplicial complexes were introduced, as were so many other topological ideas, by Poincare. Using them he gave a new and much more satisfactory definition of Betti numbers, which had originally been defined in terms of boundaries of smooth sub­manifolds. It is interesting that while the old definition was obviously invariant under Poincare's equivalence relation of "homeomorphisme," which meant what we call "C1

difieomorphism," invariance of simplicially defined Betti numbers is not at all obvi­ous. (It was later proved by J. Alexander.) Thus the gap between simplicial and difTerentiable techniques has plagued topology from its beginnings.

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10. The Work of Stephen Smale in Differential Topology 95

The Generalized Poincare Conjecture and the /j-Cobordism Theorem

In January of 1960 Smale arrived in Rio de Janeiro to spend six months at the Instituto de Matematica Pura e Aplicada (IMPA). Early in 1960, he submitted a research announcement: The generalized Poincare conjecture in higher dimensions [60], along with a handwritten manuscript outlining the proof. The editors of the Bulletin of the American Mathematical Society asked topologists in Princeton to look over the manuscript. I remember Henry Whitehead, who had once published his own (incorrect) proof, struggling with Smale's new techniques.23

The theorem Smale announced in his 1960 Bulletin paper is, verbatim:

Theorem (Theorem A). / / M" is a closed differentiable (C00) manifold which is a homotopy sphere, and ifn±\ 4, then Af" is homeomorphic to S".

The notation implies M" has dimension n. "Closed" means compact with­out boundary. Such a manifold is a homotopy sphere if it is simply connected and has the same homology groups as the n-sphere (which implies it has the same homotopy type as the n-sphere).

Poincare [51] had raised the question of whether a simply connected 3-manifold having the homology of the 3-sphere is homeomorphic to the 3-sphere S3.24 Some form of the generalized conjecture (i.e., the result proved by Smale without any dimension restriction) had been known for many years; it may be have been due originally to Henry Whitehead.

Very little progress had been made since Poincare on his conjecture.25

Because natural approaches to the generalized conjecture seemed to require knowledge of manifolds of lower dimension, Smale's announcement was

23 Whitehead was very good about what he called "doing his homework," that is, reading other people's papers. "I would no more use someone's theorem without reading the proof," he once remarked, "than I would use his wallet without permis­sion." He once published a paper relying on an announcent by Pontryagin, without proof, of the formula n4(S2) = 0, which was later shown (also by Pontryagin) to have order 2. Whitehead was quite proud of his footnote stating that he had not seen the proof. Smale, on the other hand, told me that if he respected the author, he would take a theorem on trust. 24 As Smale points out in his Mathematical Intelligencer article [67], Poincare docs not hazard a guess as the the answer. He had earlier mistakenly announced that a 3-manifold is a 3-sphere provided it has the same homology. In correcting his mis­take, by constructing the dodecahedral counterexample, he invented the fundamental group. Thus we should really call it Poincare's question, not conjecture. 29 There is still no good reason to believe in it, except a lack of counterexamples; and some topologists think the opposite conjecture is more likely. Maybe it is undecidable!

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96 M.W. Hirsch

astonishing. Up to then, no one had dreamed of proving things only for manifolds of higher dimension, three dimensions already being too many to handle.

Nice Functions, Handles, and Cell Structures Smale's approach is intimately tied to his work, both later and earlier, on dynamical systems. At the beginning of his stay in Princeton, he had been introduced to Mauricio Peixoto, who got Smale interested in dynamical systems.26

Smale's proof of Theorem A begins by decomposing the manifold Af (dropping the superscript) by a special kind of Morse function / : Af -»R, which he called by the rather dull name of "nice function." He wrote:

The first step in the proof is the construction of a nice cellular type structure on any closed C manifold M. More precisely, define a real-valued / on W to be a nice function if it possesses only nondegenerate critical points and for each critical point P.JXP) = A(0), the index of p.

It had long been known (due to M. Morse) that any Morse function gives a homotopical reconstruction of Af as a union of cells, with one 5-cell for each critical point of index $.

Smale observed that the s-cell can be "thickened" in Af to a set which is diffeomorphic to D1 x D*~*. Such a set he calls a handle of type s; the type of a handle is the dimension of its core V x 0. Thus from a Morse function he derived a description of Af as a union of handles with disjoint interiors.

But Smale wanted the handles to be successively adjoined in the order of their types: First 0-handles (n-disks), then 1-handles, and so on. For this, he needed a "nice" Morse function: The value of the function at a critical point equals the index of the critical point. A little experimentation shows that most Morse functions are not nice. Smale stated:

Theorem (Theorem B). On every closed C°° manifold there exist nicefunctions.

To get a nice function, Smale had to rearrange the fc-cell handle cores, and to do this he first needed to make the stable and unstable manifolds of all the critical points to meet each other transversely.27

26 See Peixoto's article on Smale's early work, in this volume; and also Smale's auto­biographical article [66]. 27 If p is a singular point of a vector field, its stable manifold is the set of points whose trajectories approach p as t -» oo. The unstable manifold is the stable manifold for —/, comprising trajectories going to p in negative time.

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10. The Work of Stephen Smale in Differential Topology 97

Smale referred to his article "Morse Inequalities for a Dynamical System" [61] for the proof that a gradient vector field on a Riemannian manifold can be C approximated by a gradient vector Field for which the stable and unstable manifolds of singular points meet each other transversely. From this he was able to construct a nice function. The usefulness of this will be seen shortly.

In his Bulletin announcement [60] Smale then made a prescient observation:

The stable manifolds of the critical points of a nice function can be thought of as the cells of a complex while the unstable manifolds are the dual cells. This structure has the advantage over previous structures that both the cells and the duals are differ-entiably imbedded in M. We believe that nice functions will replace much of the use of C triangulations and combinatorial methods in differential topology.

With nice functions at his disposal, Smale could decompose any closed manifold into a union of handles, successively adjoined in the same order as their type. This is a far-reaching generalization of the work of Mobius [45], who used what we call Morse functions in a similar way to decomposed surfaces.

Eliminating Superfluous Handles The results about nice functions stated so far apply to all manifolds. To prove Theorem A required use of the hypothesis that M is a homotopy sphere of dimension at least five. What Smale proved was that in this case there is a Morse function with exactly two critical points—necessarily a maximum and a minimum. It then follows easily, using the grid of level surfaces and gradient lines, that M is the union of two smooth n-dimensional submani-folds with boundary, meeting along their common boundary, such that each is diffeomorphic to D".

From this it is simple to show that M is homeomorphic to S".28 Actually more is true. In the first place, it is not hard to show from the decomposition of M into two n-balls that the complement of point in M is diffeomorphic to R". Second, it follows from the theory of smooth triangulations that the piecewise linear (PL) manifold19 which smoothly triangulates M is PL isomor-phic to the standard PL n-sphere.

How did Smale get a Morse function with only two critical points? He used the homotopical hypothesis to eliminate all other critical points. To

28 In fact, it takes some thought to see why one cannot immediately deduce that M is diffeomorphic to S"; but recall Milnor's celebrated 7-dimensional counterexample [42]. 29 A piecewise linear manifold has a triangulation in which the closed star of every vertex is isomorphic to a rectilinear subdivision of a simplex.

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98 M.W. Hirsch

visualize the idea behind his proof, imagine a sphere embedded in 3-space in the form of a U-shaped surface. Letting the height be the nice Morse function, we see that there are two maxima, one minimum, and one saddle. The stable manifold of the saddle is a curve, the two ends of which limit at the two maxima. Now change the embedding by pushing down on one of the maxima until the part of the U capped by that maximum has been mashed down to just below the level of the saddle. This can be done so that on the final surface the saddle point has become noncritical, and no new saddle has been introduced. Thus on the new surface, which is diffeomorphic to the original, there is a Morse function with only two critical points. If we had not known that the original surface is diffeomorphic to the 2-sphere, we would realize it now.

The point to observe in this process is that we canceled the extra maximum against the saddle point; both disappeared at the same time.

Smale's task was to do this in a general way. Because M is connected, there is no topological reason for the existence of more than one maximum and one minimum. If there are two maxima, the Morse inequalities, plus some topology, require the existence of a saddle whose stable manifold is one-dimensional and limits at two maxima, just as in the U-shaped example earlier. Smale redefined the Morse function on the level surfaces above this saddle, and just below it, to obtain a new nice function having one fewer saddle and one fewer maximum. In this way, he proved [62] there exists, on any connected manifold, a nice function with only one maximum and one minimum (and possibly other critical points).

The foregoing had already been proved by M. Morse [46]. Smale went further. Assuming that M is simply connected and of dimension at least five, he used a similar cancellation of critical points to eliminate all critical points of index 1 and n — 1.

Each handle corresponds both to a critical point and to a generator in a certain relative singular chain group. Under the assumptions that homology groups vanish, it follows that these generators must cancel algebraically in a certain sense. The essence of Smale's proof of Poincare's conjecture was to show how to imitate this algebraic calculation vith a geometric one: By isotopically rearranging the handles, he showed that a pair of handles of successive dimensions fit together to form an n-disk. By absorbing this disk into previously added handles, he produced a new handle decompostion with two fewer handles, together with a new Morse function having two fewer critical points. To make the algebra work and to perform the isotopies, Smale had to assume the manifold is simply connected and of dimension at least five.

In this way he proved the following important result. Recall that M is r-connected if every map of an i-sphere into M is contractible to a point for 0 ^ 11 ^ r. The ith type number u, of a Morse function is the number of critical points of index i.

44

10. The Work of Stephen Smale in Differential Topology 99

Theorem (Theorem D). Let M" be a closed (m — i)-connected Cw manifold, with n ^ 2m and (n, m) # 4,2). Then there is a nice function on M whose type numbers satisfy u0 = u„= 1 and a, = 0 for 0<i<m,n — m<i<n.

This is a kind of converse to the theorem on Morse inequalities. Smale applied Theorem D to obtain structure theorems for certain simply

connected manifolds manifold having no homology except in the bottom, top, and middle dimensions. To state them* we need Smale's definition of a handlebody of type (n, k, s): This is an n-dimensional manifold H obtained "by attaching s-disks, k in number, to the n-disk and 'thickening' them."30

The class of such handlebodies Smale denoted by Jf(n, k, s). Notice that a manifold in Jtf (n,0,s) is a homotopy n-sphere that is the union of two n-disks glued along their boundaries.

For odd-dimensional manifolds, Smale generalized the classical Heegard decomposition of a closed 3-manifold [22, 23]:

Theorem (Theorem E). Let M be a closed (m — l)-connected C° manifold of dimension 2m 4- 1, m ^ 2. Then M is the union of two handlebodies H, H' e Jr*(2m + \,k,m).

For highly connected even-dimensional manifolds, Smale proved the fol­lowing result which generalizes the classification of closed surfaces:

Theorem (Theorem F). Let M be a closed (m — I reconnected C00 2m-manifold, m ^ 2. Then there is a nice function on M whose type numbers equal the Betti numbers of M.

For a surface m = 1, and one should additionally assume M is orientable (otherwise the projective plane is a counterexample). Suppose M is a con­nected compact orientable surface of genus g. If g = 0 then the first Betti number is zero, and Theorem F says there is a Morse function with only two critical points, which implies M is a sphere. For higher genus, one can derive from Theorem F the usual picture of sphere with g hollow handles.31

30 "Handlebody" is from the German "henkelkorper," a term common in the fifties (but J. Eells always said "Besselhagen"). Although it sounds innocuous today, at the time "handlebody" struck many people as a clumsy neologism—which only made Smale use it more. 31 Both Jordan [31] and Mobius [45] published proofs of the classification of com­pact surfaces in the 1860s. From a modern standpoint these are failures. The authors lacked even the language to define what we mean by homeomorphic spaces. It is striking that the following "definition" was used by both of them: Two surfaces are equivalent if each can be decomposed into infinitely small pieces so that contiguous pieces of one correspond to contiguous pieces of the other. While we find it hard to make sense of this, apparently none of their readers was disturbed by it!

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100 M.W. Hirsch

Theorem (Theorem H). There exists a triangulated manifold with no differ-entiable structure.

In fact he proved a significantly stronger result: There is a closed PL mani­fold which does not have the homotopy type of any smooth manifold.

Smale started with a certain 12-dimensional handlebody H e Jf(12,8,6) previously constructed by Milnor in 1959 [43]. Milnor had shown that the boundary is a homotopy sphere which could not be diffeomorphic to a stan­dard sphere because H has the wrong index. Smale's results showed that the boundary homeomorphic to S11 and a smooth triangulation makes the boundary PL homeomorphic to S u . By gluing a 12-disk to H along the boundary, Smale constructed a closed PL 12-manifold Af. Milnor's index argument implied that M did not have the homotopy type of any smooth closed manifold.

An entirely different example of this kind was independently constructed by M. Kervaire [33].32

The h-Cobordism Theorem In his address to the Mexico City symposium in 1956 [71], Thorn introduced a new equivalence relation between manifolds, which he called "./-equiva­lence." This was renamed "/i-cobordism" by Kervaire and Milnor [34]. Two closed smooth n-manifolds M0, M, are h-cobordant if there is a smooth com­pact manifold W of dimension n + 1 whose boundary is diffeomorphic to the disjoint union of submanifolds Vh i = 0,1, such that M, and N, are diffeomor­phic, and each N, is a deformation retract of W. Such a W is an h-cobordism between Af 0 and M — 1.

This is a very convenient relation, linking differential and algebraic topol­ogy. It defines an equivalence relation between manifolds in terms of another manifold, just as a homotopy between maps is defined as another map, thus allowing knowledge about manifolds to be used in studying the equivalence relation. Whereas the geometric implications of two manifolds being /j-cobordant is not clear, nevertheless it is often an easy task to verify that a given manifold W is a cobordism: It suffices to prove that all the relative homotopy groups of (W, N,) vanish, and for this the machinery of algebraic topology is available. In contrast, there are very few methods available for proving that two manifolds are diffeomorphic; and a diffeomorphism is a very different object from a manifold.

32 Kervaire's example is constructed by a similar strategy from a 10-dimensional handlebody in Jf(l 0,2,5). It has an elegant description: Take two copies of the unit disk bundle of S3 and "plumb" them together, interchanging fiber disks and base disks in a product representation over the upper hemisphere. In place of the index, Kervaire used an Arf invariant.

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10. The Work of Stephen Smale in Differential Topology 101

For these reasons there was great excitement when, shortly after the an­nouncement of the generalized Poincare conjecture, Smale proved the follow­ing result [64]:

Theorem (The h-Cobordism Theorem). Let W be an h-cobordism between M0 and Mx. If W is simply connected and has dimension at least 6, then W is diffeomorphic to M0 x /. Therefore, M, and M0 are diffeomorphic.

So important is the /t-cobordism theorem that it deserves to be called The Fundamental Theorem of Differential Topology.

Kervaire and Milnor studied oriented homotopy n-spheres under the rela­tion of /t-cobordism. Using the operation of connected sum, they made the set of h-cobordism classes of homotopy n-spheres into an Abelian group 0„ [34]. They proved these groups to be finite for all n # 3 (the case n = 3 is still open), and computed their orders for 1 ^ n ^ 17, n / 3. For example, the order is 1 for n = 1,2,4, 5,612; it is 2 for n = 8,14, 16; and it is 992 for n = 11. In this work, they did not use the h-cobordism theorem. Use of that theorem, however, sharpens their results, as they remark: "For n # 3, 4, 0„ can be described as the set of all diffeomorphism classes of differentiable structures on the topological n-sphere," where it should be understood that the diffeo-morphisms preserve orientation.

From Milnor and Kervaire's work Smale proved, as a corollary to the /t-cobordism theorem, that every smooth homotopy 6-sphere is diffeomorphic toS6.

The Structure of Manifolds In this paper "On the Structure of 5-Manifolds" [63], Smale puts handle theory to work in classifying certain manifolds more complicated than ho­motopy spheres, namely, boundaries of handlebodies of type (2m, k, m).

Using Milnor's surgery methods he is able to show, for example, that a smooth, closed 2-connected 5-manifold, whose second Stiefel-Whitney class vanishes, is the boundary of a handlebody of type (6, k, 3). He then shows that such a 5-manifold is completely determined up to diffeomorphism by its second homology group, and he constructs examples in every diffeomor­phism class.

Another result Smale states in this paper is that every smooth, closed 2-connected 6-manifold is homeomorphic either to S6 or to a connected sum of S3 x S3 with copies of itself

In the same issue of the Annals, C.T.C. Wall has a paper [72] called "Classification of (n — l)-connected 2n-manifolds" containing a detailed study of the smooth, combinatorial and homotopical structure of such manifolds. The h-cobordism theorem is the main tool (in addition to results of Milnor and Kervaire, plus a lot of algebra). Wall proves:

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102 M.W. Hirsch

Theorem. Let n ^ 3 be congruent modulo 8 to 3, 5, 6 or 7.33 Let M, N be smooth, closed (n — l)-connected 2n-manifolds of the same homotopy type. Then M is diffeomorphic to the connected sum of N with a homotopy 2n-sphere. If n = 3 or 6 then they are diffeomorphic.

Milnor had shown in 1956 that there are smooth manifolds homeomor­phic but not diffeomorphic to S7. Kervaire and Milnor's work [34], plus the )i-cobordism theorem, showed that up to orientation-preserving difleomor-phism there are exactly 28 such manifolds. Wall [72] proved a surprising result about the product of such manifolds:

Theorem. The product of two smooth manifolds, each homeomorphic to S1, is diffeomorphic to S7 x S7.

The s-Cobordism Theorem There is no room to chronicle the all consequences, generalizations, and applications of the /i-cobordism theorem and its underlying idea of handle cancellation. But one—the s-cobordism theorem—is worth citing here for its remarkable blend of homotopy theory, algebra and differential topology.

As with much of topology, this story starts with J.H.C. Whitehead. In 1939, he published a paper with the mysterious title "Simplicial Spaces, Nu­clei and m-Groups" [74], followed a year later by "Simple Homotopy Types" [75]. In these works, he introduced the notion of a simple homotopy equiva­lence between simplicial (or CW) complexes. Very roughly, this means a homotopy equivalence which does not overly distort the natural bases for the cellular chain groups. He answered the question of when a given homotopy equivalence is homotopic to a simple one, by inventing an obstruction, lying in what is now called the Whitehead group of the fundamental group, whose vanishing is necessary and sufficient for the existence of such a homotopy. Whitehead proved that his invariant vanishes—because the Whitehead group is trivial—whenever the fundamental group is cyclic of order 1, 2, 3,4, 5 or oo. See Milnor's excellent exposition [44].

Several people independently realized that Whitehead's invariant was the key to extending Smale's /i-cobordism theorem to manifolds whose funda­mental groups are nontrivial: D. Barden [3], B. Mazur [41], and J. Stallings [68]. The result is this:

Theorem (The s-Cobordism Theorem). Let W be an h-cobordism between M0 and Mx.lfWhas dimension at least 6, and the inclusion of M0 (or equivalently, of A/,) into W is a simple homotopy equivalence, then W is diffeomorphic to M0 x I. Therefore, Mj and M0 are diffeomorphic.

These are the dimensions for which 7i„_, (SO) = 0.

48

10. The Work of Stephen Smale in Differential Topology 103

Using Whitehead's calculation we immediately obtain:

Corollary. The conclusion of the h-cobordism theorem is true even if W is not simply connected, provided its fundamental group is infinite cyclic or cyclic of order ^ 5 .

The s-cobordism theorem has been expounded by J. Hudson [30] and M. Kervaire [32].

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York, 1956. 25. M.W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242-

276. 26. , On imbedding differentiable manifolds in euclidean space, Ann. Math. 73

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105-109. 32. M. Kervaire, A manifold which does not admit any differentiable structure, Com­

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38. S.D. Liao, On the theory of obstructions of fiber bundles, Ann. Math. 60 (1954), 146-191.

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41. B. Mazur, Relative neighborhoods and the theorems of Smale, Ann. Math. 77 (1936), 232-249.

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52

A NOTE ON OPEN MAPS

STEPHEN SMALE

In [ l ] E. E. Floyd proved that if X and Fa re Peano continua and f: X—>7 is light, open, and onto, then the covering homotopy prop­erty for a point holds. Here we will prove (Theorem 1) that under different (roughly speaking, more general) conditions the covering homotopy property for a point holds up to a homotopy. This has cer­tain implications on the induced homomorphism of the fundamental groups (Theorem 2).

1. Definitions. We consider a triple (X, p, Y) to consist of two topological spaces X and Y and a map p from X into Y. We denote by / the closed unit interval.

A triple (X, p, Y) has the covering homotopy property for a point if given a m a p / : I—*Y and a point <?£/>_1/(0), there exists a map / : I-+X with^O) ==q and pj=f. The covering homotopy property for a point holds up to homotopy if given (X, p, Y),f and q as above, there exists a map / : I—*X with /(0) = q, pj(l) = / ( l ) and pf is homo-topic t o / with the homotopy fixed on the end points of / .

A map is called proper if the inverse images of compact sets are compact. If A is a subset of a space X, then TI(A/X) will denote the image of iri(A) in -w\X) under the homomorphism induced by inclu­sion. A space S will be called semilocally 1-connected if for each point x £ S , there exists a neighborhood U of x such that m( U/S) = 1.

A triple (X, p, Y) will be said to have Property A if the following is true:

Property A. The space X is locally arcwise connected and Haus-dorff; Y is semilocally 1-connected and metric. The map p is open, proper, and onto.

The main theorem of this note is the following.

THEOREM 1. A triple (X, p, Y) having Property A has the covering homotopy property for a point up to homotopy.

2. Proof of Theorem 1.

LEMMA 1. Let Y be a metric space andf: X—*Y be proper and onto. Suppose y(EY and U is an open set of X containing f~*y). Then there exists a neighborhood V of y such that f~l(V)CU.

Presented to the Society, April 14, 1956 under the title The effect of an open map on the fundamental group; received by the editors May 1, 1956.

391

53

392 STEPHEN SMALE [April

PROOF. Suppose the lemma is not true. For each integer »>0 , let Vi be a neighborhood of y of diameter 1 /»'. Then for each i there exists a pointyiGVtwithf~1(yt)r\(iX-U)?£0. Choosex<Gf- l(y i)nX- U) for each *'. The set K — \Jyi\Jy is compact hence f~l(K) is. Therefore the set \xi] has a limit point, say x. By the continuity of/, x£J~l(y). Since X — U is closed, x^X—U, contradicting the previous state­ment. q.e.d.

REMARK. R. L. Wilder has pointed out to me that if instead of Y being metric, Y is Hausdorff and has a countable basis of neighbor­hoods at each point then the proof for this lemma is still valid. In all the theorems in this paper, metric may be replaced by possession of this property.

The proof of Theorem 1 proceeds as follows. For each y £ Fchoose by the semilocally 1-connectedness property

of Y a neighborhood Uy of y such that ri(Uy/X) = 1. Then for each x(zp~l(y) let Py(x) be an arcwise connected neighborhood of x such that p \Pvx) ] C Uy. By the compactness of p~l(y) choose a finite number of these neighborhoods, say P\v, • • • , P*v, covering p~*(y). Let Py = \JPiy and Vy = fi/>(P,y). Since p is open Vy is an open set of Y containing y. Choose by Lemma 1 a neighborhood Vy of y con­tained in 7 , such that p~l\vy)CPy.

Let g: I—*Y be given with g(0) =y<> and SoG^'Ovo). We will con­struct a map g: /—>X such that £(0) =Xo, £1(1) =g( l ) and pg is homo-topic to g with fixed end points.

Let 8 be the Lebesgue number of the covering of / , g - ,( Vy)/y€ Y. If / o= [tGI/0<t<S, g(Io) is contained in one of the Vy's say V with VCVCtf , F -=n^(P , ) . P = U/ \ , and p - 1 (V)C/ > . The open sets here are to correspond in the obvious fashion to those of the previous paragraphs. Then xa will lie in some Pi say Pi. Let xi be an arbitrary point in the nonempty intersection P i fV - l [g (8 ) ] . For 0<t<5 define !(/) to be an arc in Pi with f (0) = x0 and |(5) =xi. Then by the choice of U, for Q<t<8, g(t) is homotopic to Pgit) with the homotopy fixed on t = 0 and t = &. Iteration will yield a satisfactory definition of g(t) for all < £ / . This proves Theorem 1.

REMARK. Examples show that Theorem 1 is false if either of the local connectedness conditions is dropped. Also the qualification "up to homotopy" is necessary.

3. Applications. COROLLARY 1. / / (X, p, Y) is a triple possessing Property A, y G K,

and x£.p~l(y), then the induced transformation prir^X, />_,Cy), x) —nri( Y, y) is onto.

This is an immediate consequence of Theorem 1.

54

'9571 A NOTE ON OPEN MAPS 393

COROLLARY 2. Suppose under the conditions of Corollary 1 the single inverse image p~*y) is arcwise connected. Then the induced homo-morphism pn wiX, x)—MTI( Y, y) is onto.

Corollary 2 follows from Corollary 1 and the exact homotopy se­quence of the pair X, £ - I(y))-

THEOREM 2. Let (AT, p, Y) be a triple having Property A. Then the quotient Ti(Y, y)/pt[iri(X, x)] is finite.

M. L. Curtis [2, p. 239] proved a somewhat weaker theorem using the previously mentioned result of E. E. Floyd. The proof in his paper applys directly here to yield Theorem 2 from Theorem 1. It will not be repeated.

Theorem 2 is roughly a generalization of a result obtained by T. Ganea [3, p. 195]. Ganea proved that under local connectedness con­ditions, given an open map of a compact Hausdorff space X onto a Hausdorff space Y, if vi(X) is finite then iri(Y) is. Ganea uses Cheval-ley's definition of fundamental group.

In the following, the homology is the singular theory, Q denotes the rational group and Z the group of integers.

THEOREM 3. If X, p, Y) is a triple possessing Property A and X is arcwise connected then the induced homomorphism p*:H\(X; Q) -*Hi(Y; (?) is onto.

PROOF. The Hurcwicz homomorphism h: vi(Y)—*H\Y) induces a homomorphism ht: Tci(Y)/p*[ir1(X)]-+H1(Y; Z)/p*[Hi(X; Z)]. Since h is onto, hf is also onto. Then from Theorem 2 it follows that Hi(Y; Z)/p*[Hi(X; Z)\ is finite. Theorem 3 follows from the univer­sal coefficient theorem for homology.

It might be mentioned that Whyburn [4] proved a result similar to Theorem 3 using rational Vietoris homology. The only differences are that his spaces were compact metric without local connectedness conditions and he assumed that the Vietoris group Hi(X; Q) was finitely generated.

BIBLIOGRAPHY

1. E. E. Floyd, Some characterizations of interior maps, Ann. of Math. vol. 51 (1950) pp. 571-575.

2. M. L. Curtis, Deformation free continua, Ann. of Math. vol. 57 (1953) pp. 231-247.

3. Tudor Ganea, Simply-connected spaces, Fund. Math. vol. 38 (1951) pp. 179-203. 4. G. T. Whyburn, The mapping of Betti groups under interior transformations,

Duke Math. J. vol. 4 (1938) pp. 1-8. UmvBRjrnr or MICHIGAN

55

A VIETORIS MAPPING THEOREM FOR HOMOTOPY STEPHEN SMALE

Let X and Y be compact metric spaces and let a map / : X-* Y be onto. The Vietoris Mapping Theorem as proved by Victoria [8] states that if for all 0£r£n-i. and all y£Y, Hr(f-l(y))-0(aug­mented Vietoris homology mod two) then the induced homomor-phism ft:Hr(X)—*Bf(Y) is an isomorphism onto for rj»n —1 and onto for r—». Begle [l; 2] has generalized this theorem to nonmetric spaces and more general coefficient groups. Simple examples show that an analogous theorem does not hold directly for homotopy. However by imposing strong local connectedness conditions, results can be obtained. That is the idea of this paper. We prove:

MAIN THEOREM. Letf: X-* Y be proper and onto where X and Y are ^-connected, locally compact, separable metric spaces, X is LO, and for each yGY,f~l(y) is LO~l and n-\)-connected. Then

(A) YisLCand (B) the induced homomorphism ft: rr(X)-*rT(Y) is an isomorphism

onto for allO £r£n — l and onto for r—«. We recall that a map is called proper if the inverse image of a com­

pact set is compact. Clearly any map between compact Hausdorff spaces is proper. A space X is said to be n-connected if T,(X) —0 for O^r i n . As above we often suppress the base point of a homotopy group.

Part (A) of the Main Theorem is a homotopy analogue of a theorem of Wilder [9, p. 31 ]• The proof of the Main Theorem can be pieced together from Theorems 8 and 9 of 2 and Theorem 11 of §3. These theorems taken together in fact say a little more than the Main Theorem. It should be mentioned that the Vietoris Mapping Theorem has been generalized using proper maps of noncompact spaces; for example see [10].

1. It will be assumed that all spaces are locally compact, separa­ble, and metric. A proof of the following may be found in [7].

LEMMA l.Letf: X—*Y be proper and onto. Suppose y&Y and U is an open set of X containing /"'(y»). Then there exists a neighborhood Vof y,such thattKV)QU.

Presented to the Society in part, February 25,1956; received by the editors Au-gtutS.1956.

604

56

A VIET0RIS MAPPING THEOREM FOR HOMOTOPT 60S

The following theorem may be found in [S, p. 82] where the terms are defined.

THEOREM 1. If X is compact and LC*, then given any c>0, there exists ij**t*(X, <) >0 such that every dense partial realisation of mesh <tl of a finite complex of dimension i « + l can be extended to a full realisation of mesh <t.

If / and g are two maps of a compact space X into a space Y, by <*(/. g) we will mean max <*(/(*), g(x))\ x£X. The next theorem is a special case of one which may be found in [6, p. 48].

THEOREM 2. Given a compact set F in an LC* space X and <>0, there exists an i?-i?"(«, F) with the following property: If K is a poly­hedron of dimension £n, and if ft, /i map K into F satisfying d(/o,A) <ij, there exists a homotopy ft: K-*X between ft and f\ such that for each x£K the curve ft(x) has diameter <t.

We will use the same symbol to denote a polyhedron and one of its underlying complexes. If K is a complex, K' will, as usual, mean the rth skeleton of K. If XQ Y, the symbol Tr(X/Y) denotes the image of Tr(X) in T,(Y) under the homomorphism induced by inclusion. We say that X iB semi-r=LC if for every * £X there exists a neighbor­hood V of x such that rr(V/X)=0. Obviously if X is r-LC it is semi-r—LC

THEOREM 3. Given a compact set Fin a semi-n=LC, LC**1 space X, there exists an v=ri*(F) with the following property: If K is a poly­hedron of dimension ^n, and if ft, f\ map K into F satisfying d(f«, f\) <n, there exists a homotopy fc K-+X between ft andfi.

PROOF. By the local compactness of X, choose a > 0 so that Cl (U(F, a))*=F is compact. Since X is semi-«—LC we can find « with 0 < e < a so that if V is a neighborhood in F of diameter <t then Tn(V/X) - 0 . It will be shown that no-if—'('A F) of Theorem 2 may be taken as the n"(F) demanded by our theorem.

Let/o, h and K be as given with i(/a, /i) <??«. Take a subdivision Sd of K so fine that if o£Sd then max diameter (/.(a), fio)) <e/3. The choice of ijo yields a homotopy ht: Sd*~1—*X between /» and f\ restricted to Sd*~l with ht(x) having diameter <«/3 for each xEStf"-1. Since t«x, A«(Stf"-1) C ** for each tEI.

If <r*£.Si, the maps/o,/i, and ht define in an obvious fashion a map J7of A -<r»X0VJ<r»XlUcr"X/ into F. From the choice of v, and Sd it follows that the diameter of H(A)<t. Then by the choice of e,

57

606 STEPHEN SMALE Quaa

H may be extended to <r*X7. Thus it obtained our desired homotopy U q.e.d.

By S* is meant the n-sphere. THEOREM 4. Let X be LC* and contain a compact LC*~X subset M

and an open set PDM. Then there exists an open set Q—C?(P, M) such that PDQDM, with the following property: If g: SM-*Q is given, then there is a homotopy *»: S*-*P of g with gi(S*)QM.

PROOF. Choose a > 0 so that Cl (U(M, a)) — F is compact and con­tained in P. Let i;o— *7"(a, F) be given by Theorem 2. Let 171 ri'r~l(,M, ijo/3) be given by Theorem 1. It will be shown that ()— U(if, tii/3) may be taken as the Q of our theorem.

Let g: S*—*Q be given. Take a subdivision Sd of S* so fine that for <r£Sd, the diameter of g(c) is less than iji/3. A map f: Sd—*M is con­structed as follows. If v is a vertex of Sd let |(v) be a point of M at a distance less than iji/3 from f («). This defines a dense partial realiza­tion of 5" in M which is easily shown to have mesh less than ift. The choice of Hi yields a full realization I of S* into M with mesh less than 170/3. It is easily checked that for every xG~S*, dg(x), J(x)) <ijo-Then by the choice of IJ0 we obtain our desired homotopy between g and £. q.e.d.

THEOREM 5. Let X be LC*-1 and semi-n-LC and let M be a compact LC*-1 subset of X. Then there exists a Q-'Q'iM) containing M with this property: For every map g: S*-*Q there is a homotopy g,: S*—*X of gwithgi(S*)QM.

Theorem 5 is proved in the same way as the proceeding one only this time using Theorem 3 instead of Theorem 2.

2. THEOREM 6. Let f: X~* Y be proper and onto. Suppose X is LC-1, and for each yE Y.f-1^) is LC— and (n-l)-connected. Let be given

(1) v>0, (2) a mbcomplex L of an n-dimensional or less) complex K, (3) amopg:K-+Y, (4) a map J: L-+X such Ptatfl-g\z..

Then there exists an extension G of \toK such that d(JG, g) <rj.

PROOF. We use induction on n. The theorem is trivially true for n—0 (we interpret LC~l and ( —l)-connected to mean no condition is implied).

Now suppose the theorem is true for » — * — !. We will show that

58

19571 A VIETORIS MAPPING THEOREM FOR HOMOTOPT 607

then it is true for n — k. Let ij, L, K, g, and f be given as in (1), (2), (3), and (4) (with n = k). Choose 0, O<0<JJ, so that Cl (U(g(K), /3)) —5 is compact.

For each y£B, a system (y, U„ P„ F„ Q„ Vt) is defined as fol­lows: XJ% is a neighborhood of y of diameter less than 0, P, —/"'(t/»), and F,=f-ly). As defined in Theorem 4, Q, is 0*_1(-P„ ^»). Finally 7, is a neighborhood of y with/ - , (^)C0» as given by Lemma 1.

Let i be the Lebesgue number of the covering V,\yeB of B. Take a subdivision Sd of X so fine that for every simplex a of K, g(<r) has diameter less than 5/3.

The induction hypothesis can be applied to yield an extension (still denoted by | ) of | to LUStf*-1 such that d(g',fi) <«/3 where g' de­notes g restricted to L^JSd*"1.

If a4 is a ^-simplex of Sd which is not in L, then from the last two sentences it follows that the diameter of fl(e*) is less than 5. Then some Vo of Vt\y£B contains/|(ff*). Let the corresponding system as defined above be denoted by (y«, Utt P«, Ftf Q«, V9).

By the choice of V«, l(ok)CQ»- Then by the choice of Q9, there is a homotopy | ,: <rk—*Pt of £» (g» denotes | restricted to oh) such that gi(cr*)C^o. Since T»_>(F») -=0, | i can be extended to v*. Then £* can be extended to a map £»': <r*—»P«. From the choice of Pt it follows easily that dJli, gk) <n (f* denotes £ restricted to «r*). The desired extension is obtained by repeating the above process on each k-simplex of Sd.

THEOREM 7. Letf: X-+Y be proper and onto where X is LO~% and semi-(n-l)-LC. Suppose for each yEY, f~*(y) is LC~-*, (n-2) -connected and **-i(f~l(y)/X) - 0 . Let be given

(1) a subcomplex L of an n-dimensional complex K, (2) a map g: K-* Y, and (3) a map g: L-*X such thatfl-g\L.

Then there exists an extension of I to all of K. The proof of Theorem 7 is very similar to that of the preceding

theorem except that Theorem 5 is used in place of Theorem 4. It will not be given.

THEOREM 8. Let f: (X, *)—»( Y, y0) be proper and onto where X is LC-* and semi-n-\)=LC. Suppose for each yG Y,f~l(y) " LC—*, (n-2)-connected, and T_i(f~l(y)/X)-0. Then the induced homo-morphism ft: r*-i(X, x.)-+x»-i( Y, y8) ** one-to-one.

PROOF. Let g: (/", P)-*(X, *.) represent an element of T _ I ( X , X,) such that/f: (/", p)~*(Y, ye) can be extended to J". It is sufficient

59

608 STEPHEN SMALE (JOM

to show that g can be extended to 7s. Theorem 7 says that this in­deed can be done, q.e.d.

THEOREM 9. Let/: (X, xt)—*(Y, yt) be proper and onto where X and Y are LC^1 and Y is also semi-n-LC. Suppose for all y£ Y, /^(y) is iC" - 1 and (» — l)-connected. Then the induced homomorphism ft: vn(X, Xt)-**m(Y, y,) is onto.

PROOF. Let g: (S», p)-*(Y, y») represent an element of wn(Y, y9). Choose <x>0 so that Cl (UhiS*), a))-P is compact. Choose by Theorem 3 ijo~ij*(^) with ijo<a. Theorem 6 yields a map f: (5", p) -*(X, xt) with d(t,fl) <ijo. By the choice of ij#, g and fl are homotopic in Y. This proves Theorem 9.

3. THEOREM 10. Let f: X—*Y be proper and onto and suppose for each yEY,f~l(y) is Q-connected and 0-LC. Let X be LC1. Then Y is LO.

PROOF. That Kis 0-I.Cia well-known. Let ^G^and W, a neigh­borhood of p be given: Let P-f-^W) and F - / - 1 ^ ) . Choose by Theorem 4, Q"Ql(P, F). From the construction of Q and the fact that P is O-connected it follows that we may assume Q is 0-connected. By Lemma 1 let 7 be a neighborhood of p withf~l(V)QQ. To prove the theorem it is sufficient to show Ti(V/W)-l. Let g: S»-*V be given.

For each tES1 and <>0 we define a system (W(t, «), Pt, e), F(*i <)> Q(U <)> Vfo ')) similar to the one used in the proof of Theorem 6 and in the preceding paragraph. Let W(t, «) - U(g(t), t/2), P(t, t) -tKW(t, «)), and F(t, t) - /^ ( iW) . Then Q(t, e) is chosen by Theo­rem 4 equal to Ql(P(t, «), F(t, «)). As in the previous paragraph we will assume Q(t, t) to be 0-connected. Choose V(t, <), a neighborhood of g(t), by Lemma 1 so t h a t / ^ W . «))C<?C «)•

Take «i>0 so that UtgiS1), ei)CT. Define Vi to be the collection V(t, tJltES1. Take a subdivision Sdi of S1 so fine that for each trESdi, g(v) is contained in an element of Vi say V,. Denote the cor­responding system as defined above by (W„ Pr, F„ Q„ VJ). Note that by the choice of <i, W. and V. are contained in V and Q,CQ for each <rE-Srfi-

We now define a map fi: Sdi-*Q with the property gi(v)CQ* ^m

each cESdi. If v is a vertex of Sa\ let |I(P) be an arbitrary point of /"*(£(»))• Then if a is a 1-simplex of .%, by the choice of V„ fi(r) CQ-Extend & to all of a by the O-connectedness of Q,. This defines fi. Let gi—fli. It is seen readily that d(j, fi) <ci.

60

i9J7) A VIETORIS MAPPING THEOREM FOR HOMOTOPT 6 0 9

By the choice of Q, fc is homotopic in P to a map of Sl into F. This implies that £1 is homotopic in W to p.

Choose <i such that 0<e»<min \dCV„ f(<r))|<r65tfi where CV, is the complement of V, in K. Let Sdt be a subdivision of 5tfi so fine that for every vGESdt, g(a) is contained in some element, say V„ of V , - V(t, tJltES1. Denote the corresponding system by (W„ P., F„ Q„ Vr) as before.

A map ft: Sa\-*Q is defined as follows. If v is a vertex of Sdi, let £«(*) K|i(v). The rest of the definition of | i is analogous to that of & using 5it and elements of V» instead of Sdi and Vy. Then for each vESdt, |i(<r)C0». Let ft-/ft- Then dfa, g)<t,.

We will now construct a homotopy *i: 5 1 X/-*^ between # and f t. Let ciGSii, «iG-Stfi and tfiOi- From the choice of <t it follows that TTMC V.V Then (?„C(>*i "ince QMCTK W.t) C/-i( Kfl)C0«r This im-plies it(<ri)CQu- Let A -»iXOU<riXlUaxXJCyiX/ and define fc: A-»Q„ by *i(/, 0) =1,(0, *"i(<, 1) - f t (0 and for <€«-i, *i(<.*) =*i(0 -«ft(*). By the choice of Qtl, h\ is homotopic in Ptl to a map of A into F,,. This implies that hi—fki can be extended to OiXl in W,,. This yields the homotopy Ai between gi and ft with the property that for each (t, t»)G5»X/, d(hi(t, v), g(t)) <«i.

Continuing as above one obtains for each natural number t a map f<: S1-^V and a homotopy A<: S1XI-*V, with A<<<, 0) =*<('), *<('. 1) -«<+i(0. and for all t, v)£SlXl, <*(*<(/, *), *(/)) <e< where we may assume that the u converge to zero.

A homotopy H: S1XI—*V between gi(t) and g(t) is defined as fol­lows:

H(t,v)-kl(t,2v) 0 ^ » ^ l / 2 , 2»-i _ i 2* - 1

tf(*,p) - **(*,2»» - 2 » - 2 ) ^ — £ » S - ^ - , * = 2,3, • - • ,

F ( M ) - i < 0 . From the facts in the previous paragraph it is easily checked that

H is well-defined and continuous. As we have already shown that f i is homotopic to p in W, this proves Theorem 10.

For homology in the rest of the paper we will use augmented Cech theory with compact carriers over the integers. The following theorem is the goal of thiB section. It generalizes Theorem 10.

THEOREM 11. Let f: X-+Y be proper and onto. Suppose for each yGY, /-!(>) is (» -1)-connected and LC—1. Let X be LO. Then Y is LO.

61

610 STEPHEN SMALE

PROOF. First, an argument that Y is Ic* will be roughly sketched. For the case of field coefficients this would follow from a theorem of Wilder [9].

By a local theorem of Hurewicz [4], since X is LC* it is Un. For each yG-Y,f~l(y) is (» —l)-connected. Then by the Hurewicz Theorem, the augmented singular homology groups oif~ly) vanish up through dimension » —1. By a theorem in [5], since/-1(y) is LC*-1 this im­plies that the Cech homology groups H,(/~l(y)) vanish for 0 £ r £ n — 1. Thus/is (» —l)-monotone over the integers in the sense of Wilder [$>].

Let pG Y and U, a neighborhood of p, be given. Let F—f~l(p) and P-/ - 1(17). By an easily proved homology analogue of Theorem 4 one chooses QDF so that an r-cycle (r fixed less than n+1) on Q is homologous in P to one in F. Choose a neighborhood V of p so that f~KV)QQ by Lemma 1.

Let s, be an r-cycle of V. By the Begle-Vietoris theory [l; 2; 3] using the fact that X is k*, one can find an r-cycle w, of Q so that f(wt) is homologous to %,. By the choice of Q this implies that *v is homologous to zero in U. Thus Y is k*.

By Theorem 10 Y is LC1. Then by the previously mentioned theorem of Hurewicz in [4] it follows that Y is LC*. q.e.d.

BIBLIOGRAPHY 1. E. G. Begle, The Vutori* napping fJuoremfor bicompaet spout, Ann. of Math.

vol. 51 (1950) pp. 534-543. 2. , The Vietoris mapping theorem for bicompaet spaces, II, Michigan Mathe­

matical Journal vol. 3 (1956). 3. , A fixed Point theorem, Ann. of Math. vol. 51 (1950) pp. 544-550. 4. W. Hurewicz, Homotopie, Homologi*, und lohaler Zusammenhang, Fund. Math.

vol. 25 (1935) pp. 4«7-t85. 5. S. Lefachetz, Topics in topology, AnnaU of Mathematics Studies, Princeton,

1942. 6. M. H. A. Newman, Local connection in locally compact spaces, Proc. Amer.

Math. Soc. vol. 1 (1950) pp. 44-53. 7. S. Sraale, A note on open maps, Proc. Amer. Math. Soc. vol. 8 (1957) pp. 391-

393. 9, L. Vietoris, Cher den htheren Zusammenhang kompahter Rlume und tine Klaste

von fusammenhangstreuen Abbildungen, Math. Ann. vol. 97 (1927) pp. 454-472. 9. R. L. Wilder, Mappings of manifolds, Summer Institute on Set Theoretic

Topology, Madison, 1955. 10. , Some mapping theorems with applications to nonAocaUy connected

spaces, Algebraic Geometry and Topology, Princeton, 1956.

UmvBJtsrrr or CHICAGO

62

REGULAR CURVES ON RIEMANNIAN MANIFOLDS^) • Y

STEPHEN SMALE

Introduction. A regular curve on a Riemannian manifold is a curve with a continuously turning nontrivial tangent vector. (*) A regular homotopy is a homotopy which at every stage is a regular curve, keeps end points and directions fixed and such that the tangent vector moves continuously with the homotopy. A regular curve is closed if its initial point and tangent coin­cides with its end point and tangent. In 1937 Hassler Whitney [17] classified the closed regular curves in the plane according to equivalence under regular homotopy. The main goal of this work is to extend this result to regular curves on Riemannian manifolds.

THEOREM A. Let x« be a point of the unit tangent bundle T of a Riemannian manifold M. Then there is a 1-1 correspondence between the set x« of classes (under regular homotopy) of regular curves on M which start and end at the point and direction determined by x9 and xi(7\ xt).

This correspondence may be described as follows. If/€«"« let/be a repre­sentative of 7 and let $f at t be the vector of T whose base point is f(t) and whose direction is defined by f(t), the derivative of /at t. Then £/is a curve on T which represents an element of xt(7\ XC). The correspondence of Theo­rem A is that induced by $.

If / is a closed regular curve in the plane then its rotation number y(f) is the total angle which f(l) turns as t traverses I. The Whitney-Graustein Theorem says that two closed regular curves on the plane are regularly homotopic if and only if they have the same rotation number. Using the fact that the unit tangent bundle of the plane is E*XSl, this theorem follows from Theorem A.

Let xe be a point of the unit tangent bundle T of a Riemannian manifold M. The space of all regular curves on M starting at the point and direction determined by x» is denoted by E. A map x from E onto T is defined by send­ing a curve into the tangent of its endpoint at its endpoint. The following can be considered as the fundamental theorem of this work.

THEOREM B. The triple (E, x, T) has the covering homotopy property for polyhedra.

Presented to the Society, August 24, 1956; received by the editors September 29, 1956. 0) The material in this paper U essentially a dissertation submitted in partial fulfillment

of the requirements for the degree of Doctor of Philosophy in the University of Michigan, 1956. (*) These definitions will be made precise in the body of the work. Also, theorems stated

will be proved later.

492

63

REGULAR CURVES ON RIEMANNIAN MANIFOLDS 493

Let T be the fiber over xt of (£, T, T) and let fl be the ordinary loop apace of T at xp.

THEOREM C. The map $ is a weak homotopy equivalence between T and Q. Theorem B is used to obtain Theorem C and, in turn, Theorem A follows

from Theorem C. If / is a regular curve on M, $f is a curve, as we shall say, a lifted curve,

on T—T(M). Clearly, not all curves on T are lifted curves. In particular, every lifted curve must be an integral curve of a certain 1-form «o on T. If the integral curves of at were exactly the lifted curves, Theorems A, B, and C could be proved by proving theorems on integral curves. Unfortunately, however, u« admits as integral curves some non-lifted curves. Nevertheless, these considerations raise questions concerning the loop space of integral curves of u» on T.

THEOREM D. Let a be a l-form of Class A on a three dimensional manifold M such that wAdwstQ on M and let x«GAf. Denote by fl« the loop space at xt of piecewise regular curves on M which are integral curves ofw and by 0 the ordinary loop space of M at x„. Then the inclusion i: fl„—»0 is a weak homotopy equiva­lence.

I would like to express my appreciation to my adviser, Professor Raoul Bott, for the encouragement and advice which he gave throughout the preparation of this thesis. I would also like to thank Professor Hans Samelson for reading the manuscript and for suggesting several corrections.

1. Fiber spaces. A triple (22, p, B) will consist of two arcwise connected spaces E, B and a map p from E into B. A triple will be said to have the CHP if it has the covering homotopy property for polyhedrons [ l l j .

If g is a map from a space X into a space Y, then the restriction of g to a subset A of X will be denoted by *u or sometimes just g. P will always denote a polyhedron. The set \t real|a £t £b] is denoted by [a, b]. A cube P is the Cartesian product of k copies of / , the closed unit interval.

The following proposition is well-known. It is a special case of a theorem proved in [6, p. 136].

PROPOSITION 1.1. Let (£, p, B) be a triple which has the covering homotopy property for cubes. Then it also has the CHP.

LEMMA 1.2. Let a triple (E, p, B) have the CHP. Let a be a simplex of some dimension n and let g: aXl—*B be given. Suppose o is the pointset) boundary of 9, A »ffXPJoXQQoXl andf: A-+E covers gu- Then there exists an exten­sion F off to all of9XI covering g.

The proof is immediate. There is a homeomorphism from 9X1 onto PXl which sends A homeomorphically onto I"X0. Then the application of the CHP yields the desired map F.

64

494 STEPHEN SMALE [Much

The statement of the following theorem is quite similar to what Hurewicz calls the Uniformitation Theorem in [5]. It will be found useful in proving the CHP for a triple whose base space is a manifold.

PROPOSITION 1.3. Suppose a triple (£, p, B) has the CHP locally; that is, for each point x£B, there exists a neighborhood V of x such that (p~* V), p, V) has the CHP. Then E, p, B) has the CHP.

Proof. Let H: PXI-+B be a given homotopy and h: PX0—*E a covering of H\rxt. We will define a covering homotopy H:PXl-*E. For each yGH(PXl) let 7, be a neighborhood of y so that (p-l(Vt), p, Vw) has the CHP. Assume PXI has been given some definite metric. Denote by S the Lebesgue number of the covering U,-H-i(V,)\yEH(Pxr) of PXI. Put Jo — [0, 3/3], It is sufficient to define 27 on PXlt for then iteration will yield a full covering homotopy.

Take a simplicial complex K such that | K\ - P . Let Sd(K) be a subdivi­sion of K such that the diameter of any simplex of Sd(iT) is less than 3/3, and let Sd(K)r be the r-skeleton of Sd(K). If v is a vertex of Sd(K) the choice of Jo yields that HvXh) is contained in some neighborhood V where P~l(V), P, V) ha» the CHP. This fact immediately gives a definition of 27 on Sd(iT),| Xl». Proceeding by induction suppose 27 has been defined on SdC-K")*"-1! Xl», and a* is an r-simplex of Sd(.K). From the choices of Sd(£)

and U it follows that <r*Xi» has diameter less than i. Then Hio+Xh) is con­tained in some V such that (pr^V), p, V) has the CHP. Already 27 has been defined on o^Xl^Jo+XO. Then application of Lemma 1.2 yields 27 on o'Xl*. In this manner 27 is defined on all of | Sd(2Qr| Xh and then by induction on all of |Sd(X)| XU. This proves 1.3.

2. Regular corves. By a manifold we shall mean a connected Riemannian manifold of class 3 and dimension greater than 1. There are no assumptions such as completeness or compactness. If i f is a manifold, T,(M) or often just r , will be the space (or bundle) of all tangent vectors of M. By T(M) or T is meant the sub-bundle of Tt which consists of the unit tangent vectors of AT.

As usual, a curve on a space is a map of / into the space. Let i f be a mani­fold and let/be a curve on M whose derivative [l, p. 46] exists at the value t»GI- This derivative is an element v(tt) of if/(i«>, the tangent vector space of M at/(<o). By convention in this work the derivative f (t,) off at <o will be the pair (/(<,), v(t,)). Thus,/*(/») will be an element of T,(M). By the magni­tude oif(t») or \ft»)\ we mean the magnitude of »(*»).

A parametrised regular curve on a manifold i f is a curve/ on M such that f(t) exists, is continuous and has positive magnitude for each <£/ . Two parametrized regular curves / and g will be called equivalent if there exists a homeomorphism h of I onto itself such that for all <£f, h'(t) exists, is con­tinuous and positive, and/(() =f (&(/))• It is an easy matter to check that this

65

1958] REGULAR CURVES ON RIEMANNIAN MANIFOLDS 495

E(M) <£ is a regular curve on M . . — *»>

is a true equivalence. A regular curve is an equivalence class of parametrized regular curves.

The arc-length of a regular curve g, defined in the usual way, exists, and is independent of its representative. It will be denoted by L(g). Implicit use of the following proposition will be made throughout this work.

PROPOSITION 2.1. If g is a regular curve on a manifold then there exists a unique representative of g still denoted by g such that |g'(t)1 =L(g) for all * £ / . L(g) is the only possible constant value here.

The proof for the plane is in [17]. The same proof holds for the general case of a manifold. The representative of a regular curve given by 2.1 will be called distinguished. A distinguished representative is just a parametrization proportional to arc-length. Unless we note otherwise, a regular curve will be identified with its distinguished representative.

Let i f be a manifold and xt a fixed point of T(M). We denote by E(M) or sometimes simply E, the space of regular curves on M whose normalized initial tangents are x9; in other words,

^(0)1 Let i: T9XT9—*R+ be any metric on Tt (R+ is the space of non-negative

real numbers). Then for/ and g&E, let d(J>g) = mAxi\f'(t),g'(t)]\tGl-

From the fact that J is a metric, it follows easily that d is a metric on E. We will suppose E to have the topology induced by d.

Let /< be a sequence of points of E converging to a point / of E. Then for each <£/,/(/) converges to f(t). If a sequence xn of points of Tt converges to *o then from the topology of Tt it follows that the base points of *» in M converge to the base point of x». Thus/<(f) converges to/(») for each <£/ .

The map x: E—*T of Theorem B may be defined by x(fl - / ( 1 ) / | / ( 1 ) |. To speak of (£, x, F) as a triple, E must be arcwise connected. This is proved later (Lemma 6.2).

3. The reduction of the proof of Theorem B to 3.1 and 3.2. The proof of Theorem B depends essentially on Propositions 3.1 and 3.2.

PROPOSITION 3.1. Let M be a manifold and f: E(M)-*M be the map f(g) =*(1). Then (E, f, M) has the CHP.

Suppose If is a manifold. Let xi: Tt(M)-*M be the map which sends a tangent vector into its base point. A homotopy / , : P—+Tt(M) will be called vertical if for all p^P and v£I, x^.(£)-Xi/,(/>). A homotopy / , : P-*X (X any space) is said to be stationary on a subpolyhedron A of P if f.(J>) =/o(/>) for all PGA and v£I.

66

496 STEPHEN SMALE [M»rch

PROPOSITION 3.2. Let M be a manifold andf,: P—*TM) be a given vertical homotopy with J: P—*E(M) covering /0- Then there exists a covering homotopy J,: P—*E. Furthermore iff, is stationary on a subpolyhedron A of P then , will also be.

Propositions 3.1 and 3.2 will be proved in the following sections. Now we will show how Theorem B follows from 3.1 and 3.2.

LEMMA 3.3. Let P be a polyhedron and A be a subpolyhedron which is a strong deformation retract [2] of P. Let g and h be maps of P into a space X which agree on A. Then there exists a homotopy H-.PX I—*X between g and h which is stationary on A.

Proof. Since A is a strong deformation retract of P there is a homotopy K: PXI->P such that K(p, 0) -/>, K(p, l)EA, and if p£A, K(p, t) =p. The desired homotopy H: PXl—*X may be defined as follows:

B(P, t) = gK(p, 2/) 0 £ / £ 1/2, B(p, t) - hK(p, 2 - 20 1/2 g * £ 1.

The fc-sphere is denoted by Sk. LEMMA 3.4. Let P and A be as above with P contractible, and M be an n

dimensional manifold. Let F: P-*M be given and g: P->T- T(M), hi P^T be two covering maps of F which agree on A. Then there exists a homotopy h,: P—*T between g and h such that h, is stationary on A and for each » £ / , h, covers F.

Proof. Let E' be the induced bundle F-^T) [13, p. 47].

E'-^T

I'1 I"' * F * P >M

By definition:

£' = IfoOePxrl *(»-*,<*). qi(f, 0 =- t, qi(p, t) - p.

Since P is contractible, E' is a product PXS" - 1 [13, p. 53] with qx being the projection of E' onto P. Let x': E'—*SH~1 be the other projection.

Define l:P-+E'CPXT by i(p)-(p, g(p)) and let f*:P->S>-' be the composition r'g. Similarly, define k and h~* from h.

Apply the previous lemma to obtain a homotopy J£:P-*S~~l between i* and k* which is stationary on A. Define h\:P-*E'-PXS*-1 by h\(p) "(J>, #(£)) and h.:P->T by h.-qik\. It can be quickly checked that A,

67

1958] REGULAR CURVES ON RIEMANNIAN MANIFOLDS 497

satisfies the lemma, q.e.d. To prove Theorem B it is sufficient by Proposition 1.1 to show that

(£, T, T) has the covering homotopy property for cubes. Suppose, then, we are given a homotopy/,: P—*T and a map/: P-*E covering/o where P is a cube. We will construct a covering homotopy/.: P—*E.

Application of Proposition 3.1 yields a covering map h: PXI-+E of xi/» such that k(p, 0) =/(£). Lemma 3.4 then yields a homotopy Hu: PXI—*T such that (1) H,(p, v)=Tk(p, v), (2) Hxip, »)- / . (£) , (3) Hu(p, v) covers *if,(P) for each u£I, and (4) Hm is stationary on PXO. By (3) H„ is a vertical homotopy so Proposition 3.2 applies to yield a homotopy 27.: PY.I-+E of h~ which covers HM.

We assert that 27i(p, c) can be taken as the desired covering homotopy /,(/>). From Hi(p, v) =/.(/>) it follows that 27i(», p) covers /,(£). Since Hu is stationary on PXO, 2T. is also. Then Z7i(£, 0) = 27,(/>, 0)=JJ(p, 0) =-/(/>) or 27i(p, r) is a homotopy of J(p). This shows that Theorem B follows from Propositions 3.1 and 3.2.

4. Proof of Proposition 3.1. We will need two lemmas. By v ±w it is meant that the vectors v and w are perpendicular.

LEMMA 4.1. Let n> 1 and S*~l be the unit vectors of Euclidean n-space £" considered as a vector space. Suppose P is a cube and a map w: P—tS*-1 is given. Then there exists a map u: P-+S—1 such that for all P&P, up)±w(p).

This lemma is not true for a general polyhedron. In particular, if P =» S"_1

and u> is the identity, the existence of such a map u implies the existence of a unit vector field on 5"~l. It is well known that this is impossible for odd n.

Proof of 4.1. Let V„,t be the Stiefel manifold [13, p. 33] of ordered orthogonal unit 2-frames in £". With a projection Pi sending a 2-frame into its first vector, Vn,* becomes an (n—2)-sphere bundle over S"-1. Let E' be the induced bundle ur'( V*,t).

Since P is contractible E' is a product. Let s: P—£' be any cross-section, and let Pt: V*,t—*Sn~l send a 2-frame into its second vector. Then the composition u—pifs has the desired property, q.e.d.

LEMMA 4.2. Given yB, 0<y0£l/2, there exists a real continuous differenti-able function &(y) defined on I such that (1) 0(0) -/3(1) - 0 , (2) F(0) =0, and (3) for yZyfl, ?(y)Z4.

Proof. Consider the function:

68

498 STEPHEN SMALE [March

Ky) = 0 0 £ y £ y«/2,

r(y) (1 - y.)y + r(l - y.) y/2 £ y * y., y»

r(y) = 4y - 4 yo £ y S 1.

Note that r(y) could be taken as /9(y) except for the fact that it has corners at y™yo/2 and y »yo. By 'rounding off the corners" of r(y) the desired func­tion can be obtained.

In order to prove 3.1 it is sufficient by Proposition 1.3 to show that (j-1(Z7o), p, U%) has the CHP where U» is a coordinate neighborhood of M. Since U» is homeomorphic to £" we can identify the two spaces under this homeomorphism. Thus Ut*=E*QM. Ti(U«) is a product space £"X£" where the first factor comes from the base point and the second from the direction and magnitude of a vector. We identify each of the two factors of T%( Ut) with a single £" whose elements we consider as vectors. If 5*-1 is the space of unit vectors of £", T(U9) =£"X5" -1. The magnitude of a vector v of £* is writ­ten \v\.

For convenience the following new convention is used in this section and the next. The derivative of a regular curve at a point of Uo will not carry the base point. That is, it is now the projection of the old derivative onto the second factor of T»(Uo). This is possible since most of the analysis in these sections is concerned with Tt(Ut) and U0.

By Proposition 1.1 it is enough to show that (f~l(U»),f, Ut) has the cover­ing homotopy property for cubes. Let A,: P-*Ut be a given homotopy with R: P—*f~\Ut) covering *• where P is a cube. We will construct a covering homotopy k.: P->f-l(U,).

Choose /with 0 £ / < l such that for all p£P and <G [/, 1], Z(p)(t)eU,. Then choose / , with JS J» < 1 so that for all />GP and *G [J,, 1J,

„<„<„_,<,)(1)I < i » . The following choices are motivated by the need to insure the regularity

of the covering homotopy curves we are constructing. Let K - max | k.(p) - h,(p) | | « G / , p G P).

If K-0, yo-1/2. Otherwise let

. n I^KDld-/ .) A__\ »-m\l «E *eP)-

The compactness of P yields that y»>0. Taking yo as above, let /3(y) be the function given by Lemma 4.2. By taking v>p) -*'(/>)(!)/1 i'(P)(l)\, Lemma4.1 yields a map u: P -»S- 1

69

1958] REGULAR CURVES ON RIEMANNIAN MANIFOLDS 499

such that u(p)±Ji'(p)(l). We define the desired covering homotopy A"»: P—*f~iU9) as follows. For

OS<S/o »et *".(#(<)-*(*)(0. For 7 ,£<£1 let s = st)-t-J,)/(l-Jt); then set

*.(*)(') = *W(0 + ' ^O) - *•(*)] + /*(*) I UP) - W I u(P). Here J(p) is to be taken distinguished (see (2), but k\(p), in general

will not be. Note that for / Js/0 all the terms used to define h\p) lie in U* and hence the additions make sense.

The following properties of k\ can be readily checked: (1) k\(P)(t) is continuous in v, p, and t. (2) *.(/>)(l)-*.(£)• (3) *.(/>)-%). The derivative of h\(P)(t) for t£J» can be computed to be:

U (p)(t) - *'0)«) + SbfciP) - W)] + J V « I *.(>) - *•(#) I «(Ph Then it can be seen:

(4) *.(£) » differentiable. (5) hi p)(0)/\ ki (p)(0) | «*„. The derivative meant here is in the sense of

§2. The following requires proof: (6) h(P) i» a regular curve. For this it is sufficient to show that k,'(p)(t)r*Q for t^J9. For such t we

can write ki (/»)(/) — Ai+At where Ai - h'(p)(t) + Wls) | k.(p) - ko(p) | u(p)

and A, - 2J'*[A.(» - k,(p)].

We will divide the proof into two parts. CASE I. s*y,: We claim | At\ £ (9/10) | k'(p)(l) |. For a certain number A,

Ax - k'(p)(l) + Au(p) - (J'(tf(l) - k'W)) and then by the triangle inequality

Mil fc I ^wa) + A««I - l ma) - n'ipmi. By the choice of /» we obtain

Mil * l *W) + «Wl -^\*<P)M\-Finally, since u(p)±k'(p)(1) we have |Ai| fc(9/10)| k'(p)(1)| as claimed.

On the other hand, by the choice of y« (since siyt)

70

500 STEPHEN SMALE (March

M«I s *'(i/3)! *'(»(D! (i - Ja) - d/3)! A"'(»(I) i. From the triangle inequality it follows that h\'p)(t)^0.

CASE II. s&y,: We use a lemma.

LEMMA 4.3. Let a, b, and c be vectors in £" such that \b\ < ( l / 1 0 ) | o | and cLa. Let v be a scalar, p>4. Then the inequality \a+b+vc\ > | 3c| holds.

Proof. Since

| a + b + vc\ £ | o + vc\ - \bl

it is sufficient to show

\a + vc\'Z ( | 3 c | +\b\)*

or using the fact that cLa

| . | « + | « | ' 2 9 | c | ' + 6|ft | \c\+ |6 |«.

Since v 2j4 it is sufficient to show

|o|»fc - 7 | c j « + 6 | 6 | C | + |*|». This is easily checked considering separately the two cases

\c\ 5 \b\ and \ c\ > \b\ . It follows from 4.3 that

M i l £ 3*'| k,(p) - ht(p)\ taking Ji'(p)(l) = a, *"'(*>) (1)-*"'(£) (*)=*. s'\K(p)-ht(p)\u = c, and P'(s)=v. By the choice of /o, |&| < l / 1 0 | a | and since s^y0, f ^ 4 .

On the other hand | 4 , | S25 ' |A . (£ ) -A , (» | . Then by the triangle in­equality A7 (p)(t) ?*0. This finishes the proof of (6).

Properties (1), (4), (5) and (6) imply that K,(p) is really an element of E, (2) says that A", covers A, and (3) that A", is a homotopy of ft. Therefore we have proved 3.1.

5. Proof of Proposition 3.2. Let C/ebea coordinate neighborhood on M, and let V= T(Uo). By the argument used to prove Proposition 1.3, it is suffi­cient to prove 3.2 for the case where /,(/>)|ti£J, PE.P) C.V. The notation and conventions of the last section will be continued.

Let n: T^Ut) = £"X £"-»£" be the projection onto the second factor. Then T»(r(t/0)) C-S"-1. The angle between two vectors of 5*-' is a continuous function of the vectors. This fact, together with the compactness of P, justifies the following choice. Pick e>0 such that for all PE.P and \v—v*] <«, the angle (measured in radians) between Ttf,(P) and x*/,'(i>) is less than 1/10.

For v <e let a,(p) be the oriented angle from icift(P) to T»/.(J>). By our choice of t, a.(/>) < 1/10.

For v£t, and < £ / , we will define Q,(P, t) to be the following rotation of

71

1958] REGULAR CURVES ON RIEMANNIAN MANIFOLDS 501

E%; that is, we are defining a map Q,: P XI~*R» where 2?» is the rotation group of £". If Ttftp)~rtfrp) let Q,(p, t) = e, the identity rotation. Otherwise let Q*i.P, 0 rotate V, the unique plane determined by *tf»(p) and ftfwip), through the angle ta,p) and leave Vx, the orthogonal complement of V, fixed.

That (?.(/>, 0 is continuous in r, p, and * and has a continuous first deriva­tive in t, Qt'(p, t), is easily seen. Later, in fact, we will have occasion to com­pute this derivative.

Choose J,0£J<1, such that

/(»(<) I P e P, t e [J, i] c ut. LEMMA 5.1. There exists a Jt with J£J«<1 such that for all PEP and

/ € [ /o , 1],

7(#)<0 -KPKD Jt- 1

Proof. It follows from the definition that

7(0(0

SylTWDl

/ ( 0 ( 1 ) = lim- / ( ♦ ) ( ! )

/ - 1 so by the compactness of P there exists a J» with / 5J/o<l such that for /G[/o, 1],

7(0(0 -7(0(D * - 1

Then by the triangle inequality KPW) -7(0(i)

np)w sj\rwM\.

t-1

Also clearly for Jo it ^ 1,

7(0(0 -7(0(D

* yl7'(0(Dl

TW« -;wd) 7 f l - 1

These last two inequalities yield the lemma. Choose Jt by 5.1 and such that also

|7'(0(D - T W O I < (V10) min | / ( 0 ( 1 ) | | p e P holds for all p£P and <G [J», 1 ]• Then for v£t,J, is defined as follows.

For OS** / , set7,(0(0 -7 (0 (0-For J9£t£l let 5 - J ( 0 - ( < - / . ) / ( 1 - / O ) ; then set

7.(0(0 - [7(0(0 - 7(0(i)]0.(>, s)[e + (5« - *)g.'(>, o ) ] + 7 ( 0 ( D . Here Q,(p, s), Q,' (p, 0) and e are to be considered as transformations acting

72_

502 STEPHEN SMALE (Much

on the right. The curve J(p) is taken distinguished, but in general, 7»(£) will not be.

The following properties of the covering homotopy can be quickly checked.

U) 7»(£)(0 » continuous in 0, p, and t. (2)/ . (*)»/(*). (3) JwiP) is stationary on a subpolyhedron A of P if/,(/>) is. The first derivative of/,(£) can be computed as follows for <£./»:

?.'O)0) - T(P)(t)Q*(P, »)[« + (*' - *)Qi (P, 0)] + -WHO -7W(i)JG/(#, *)[« + (* - *)#(>, 0)1

Using this it can be further checked that: (4) 7. (/»)(<) »» differentiable. (5)7.'(P)(0)/7.'(*)(0) (6)7/(P)(i)/7^)(D The following requires proof: (7) The curve*p)(t) is regular. For the proof of (7) we will use:

x« in the sense of §2. f*(P) again in the sense of §2.

LEMMA 5.2. Let 0£<£1. Then (a) *A« transformation Qi(f, t) reduces the magnitude of a vector to less than 1/10 of its original magnitude and (b) the transformation e+t*—t)Qi (P, 0) does not change the magnitude of a vector by a factor of more than 1/10.

Proof. For given p and v let coordinates x\, • - • , x" of En be chosen so that V (from the definition Q,(p, t)) is the Xi —x% plane and the direction of r%f»(p) coincides with the x% axis. Then with this system suitably oriented Q,(J>, t) can be represented in the matrix form,

cos [ta,(p)] sin [l*.(p)]

0

-sin[ta,(#)]0-cos [ta,(#)J 0 •

0 1

• 0 •0

0

0 0 0 •• • 1

Then Qi p, t) will be of the form, sin [Uu(p)] cos [ta.(p)] 0 • • • 0

-co»[ta,(p)] an[ta,p)]0 0 -cn(p) 0 0 0 0

• • • • • • • • 6 6 6 • • • 6.

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1958] REGULAR CURVES ON RIEMANNIAN MANIFOLDS 503

If ftf Pt, • ' • , P% are the components of a vector p* in the above system then

I PQiiP, 01 = I °.(P) I [0»i)f + W] 1 ' 1 £ I « .«I I P\-This yields (a) since |a.(/>)| <1/10. (b) follows from (a) and 5.2 is proved.

To prove (7) it is clearly sufficient to show that /,'(£)(/) 3*0 for t^J». Let ?.' (/>)(<) « J 4 I + 4 , where

^i -'(P)(t)Q-(P, *)[• + (*" ~ ' ) # ( > , #)1 and

+C.f>,*)(2*-l)0/O,0). From Lemma 5.2(b) and the fact that Q,(p, s) does not change the mag­

nitude of a vector, it follows that | J!I| 2 (9/10) \J'(p) (*) | . Then by the choice of/„|A»Ifc(8/lO)l7'<0(l)I.

On the other hand, by Lemma 5.2 one easily obtains

\A,\s3no\m"J-_Jfm Then from the choice of / . (see 5.1) it follows that \A,\ ^(4/10)|7'O)(l)|. By the triangle inequality the inequalities on |i4i| and \Ai\ yield / / ( £ ) ( 0 ^ 0 and hence (7).

Properties (1), (4), (5) and (7) imply that /,(/>) really belongs to E, (2) says that/• is a homotopy of /and (6) that/, covers/,. Lastly, (3) is the sta­tionary property demanded by 3.2. Thus / , is a satisfactory covering homo­topy fort>3«.

The above construction may be repeated if < < 1 using /,(/>) instead of /(/>) and using a new value for /«, if necessary. This yields a covering homo­topy for r^2«. Iteration yields a full/, and the proposition is proved.

6. On the topology of the liber I\ We recall some definitions and theorems of [11 ]. Let A* be a space and x*£X. The path space of X written E^X) or sometimes Em, is the space of all curves (or paths) on X which start at x», with the compact open topology. Define p: EH—*X by sending a path onto its endpoint, i.e., let pf) - / ( l ) . The loop space of X at x0, P~l(xt) is denoted by Q(X) or 0. It is shown in [11, pp. 479-481] that (£„, p, X) has the CHP and that E„t is contractible.

Using the notation of the previous sections let M be a manifold, T= T(M), E=E(M) and r-r(JI)-«r- |(»»). Define a map 4>:E-*E.tT) by *(f)(<) a e f / (0 / | f ' (0 | for i£E. It can be seen that <t> is continuous as follows. Let E' be the set E,,(T) endowed with the metric topology

<*•(/, t) - max d\f®, t(t)] | / € / •

74

504 STEPHEN SMALE [Match

Then £ can be factored through £ ' by maps &: E—*E' and <h' £'—♦£«„ where <£i is the identity. It is well known that the metric topology and the compact open topology are equivalent on £,,. See for example [4, p. 55]. Hence fa is continuous. From the topology of E it follows that 4n is con­tinuous. Thus ^ is continuous. Let £ be restricted to T. Then $(T) CO. This map is the same as $ of the Introduction.

A map between two spaces is a weak homotopy equivalence [after 10, p. 299] if it induces an isomorphism of the homotopy groups of the spaces. The following theorem is well-known. A proof can be found in [14, p. 113].

THEOREM 6.1. A weak homotopy equivalence induces isomorphisms of the singular homology groups of the spaces involved.

Theorem C and 6.1 yield that $ induces isomorphisms of the singular homology groups of Y and Q. This is of interest because a certain amount of attention has been given to the problem of determining the singular homology of loop spaces. For example see [ l l ] and [15].

The proof of Theorem C requires the following lemma. LEMMA 6.2. If M is a manifold the space E(M) is homotopically trivial. Proof. Consider first the case where M is Euclidean n-space E". Assume

xi*o to be the origin of a coordinate system of En and let Tixe=*0 where T§ is the projection of T=E»XS*-1 onto S"~'.

For some k^0 let/ : 5*—£ be given. To prove the lemma for E" it is suffi­cient to show that / is homotopic to a constant. Since S* is compact we can choose 7 > 0 close enough to 0 so that for all £ES* and <£ [0, J],

\rm)-f(p)(o)\ < l/'(»(o)|. Then for t 6 [ 0 , 1/2] let/,(£)(I) -f(p)(t-2(1 -J)vt). The curve/(/>) is to

be distinguished, but/,(/>) will not be, in general. This homotopy merely con­tracts f(p) into a curve whose tangent is fairly close to a constant.

Define e(f) as the fixed path of E given by *tt. Then for *G [1/2, 1 ] define

/ . (»«) - (2 - 2t)fllt(p)(t) + (2» ~ !)«(')• where fyip)(f) is the nondistinguished curve given by the previous homotopy.

It can be checked that/.(£) is really contained in E, that/0(f>) =/(£) and that fi(p) =e. It is the selection of / that yields the necessary regularity of MP) for v% 1/2.

We have proved the lemma for M =»£*. The proof for a general M goes as follows. As before, let/: &-+E. Now for v£ 1/2 let/,(£) be a "shortening" oif(p) so that for all p(!i&,fi/t(P) lies in a certain coordinate neighborhood about *#. For P E [ 1 / 2 , l ] the homotopy is the same as the total homotopy for £". q.e.d.

Theorem C is proved as follows.

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1058] REGULAR CURVES ON RIEMANNIAN MANIFOLDS 505

From the definition of # it follows easily that ^ commutes with the identity of 7*, i.e., ^ « i or

U' T >T

commutes. Then, ^ induces a homomorphism of the homotopy sequence of E into that of E,r We have the following commutative diagram with the horizontal sequences exact.

»rt(T) >rt(B) — *k(D -> • • •

1* i « 41* r 4 ( ( D ) - T i ( i E N ) - f » » ( 2 0 - • •

From T * ( £ ) T*(£,^ = 0 for all ft (using 6.2), it follows that $f is an iso­morphism for all k. This proves Theorem C.

7. Classes of regular curves on a manifold. Two regular curves on a manifold M are said to be regularly homotopic if they are homotopic and the homotopy f«: I—*M can be chosen such that for each « £ / , g, is a regular curve, f.'(0)«g§ (0), f / ( l ) - f o (1), and ,'(<) depends continuously on v. A regular curve f on i f will be called closed if f'(O) —^(1). It will be said to be at a point y% of T if ^,<0>/| JT'(O) | - > i . Two closed regular curves on M are /ree/y regularly homotopic if they are homotopic and the homotopy g,: I—*M can be chosen so that for each v£I, f, is a regular closed curve. Regular homotopy (free regular homotopy) is an equivalence relation and a class (free class) of regular curves on M will mean an equivalence class with respect to this relation.

M. Morse has investigated the behavior of locally simple sensed closed curves (or L-S-curves) under Z.-5-deformations. For definitions and discus­sion see [7; 8; 9] . In these articles he classified L-S-curves on closed 2-mani-folds and E* into equivalence classes under L-S deformations. He has noted the similarity between this study and the classification of closed regular curves with free regular homotopies playing the role of L-S-deformations. The results of this section are parallel to Morse's.

From the definition of regular homotopy it follows that if M is a manifold two curves of T(M) are regularly homotopic if and only if they lie in the same arcwise connected component of I \ i.e., in the same element of r,(T). Using this fact Theorem A is the case » « 0 of the following:

THEOREM 7.1. If M is a manifold, there exists an isomorphism q from x.(T(Af)) to Xn+iiTiM)).

76

506 STEPHEN SMALE Much

Proof. Let ft: T%(T(M))~*T%(QT)) be the isomorphism of Theorem C, and s:xnQ(T))—»wvn(r) be the Hurewicz isomorphism [16, p. 210]. The composition of these two maps q~s$t gives the isomorphism demanded by the theorem. The rest of this section will be devoted to special cases of Theo­rem A.

THEOREM 7.2. Let the dimension of a manifold M be greater than 2 and *•£ T(M). Then two regular closed curves on M atxt are regularly homotopic if and only if they are homotopic with fixed end points.

As we shall see, this theorem is far from true for 2-manifolds. Proof of 7.2. The "only i f part is immediate from the definition of regular

homotopy. Consider the exact homotopy sequence of T. Then

r,(5-1) - ri(r) -^» ndo - T.(S-»)

is exact. Since xi(S»-1) =r«(5""1) =0 for n>2, xl#: TI(T) »xi(Jlf). With this, Theorem A yields that Xi^:x»(r)«Ti(if). Moreover, from the definitions of xi and q we can consider r\fi to be just the map induced by sending a curve into itself. Then 7.2 follows immediately from the definitions of r»(T) and

Regular curve classes on some 2-manifolds will now be investigated. (a) The plane. The following definition is due to Whitney [17]: If / is a

regular closed curve in E*, its rotation number y(j) is the total angle which T«/'(0 turns as t traverses I. The function

/*(0 = »v"(0/IV(0l is a map of I into the unit circle. y(f) is 2x times the degree of this map.

THEOREM 7.3 (WHITNEY-GRAUSTEIN). TWO regular closed curves on the plane are freely regularly homotopic if and only if they have the same rotation number.

Proof. Because a translation of a regular closed curve in the plane is a free regular homotopy and preserves its rotation number it is sufficient to consider curves of r(E*).

Consider the isomorphisms

where rt is the projection of T-E*XSl onto 5». Let / b e the element of T represented by eMt in complex coordinates such that the base point x% of T is the vector 2x» whose base point is the complex number 1. If h^X, h will be the element of x0(T) containing h. Then rtfiQ) will be a generator of

77

1958] REGULAR CURVES ON RIEMANNIAN MANIFOLDS 507

Ti(Sl) say e. If g is any element of I \ T*fg(f) will be of the form me. From the definition of rotation number it follows that tn is the rotation number of g. Since rtfi is an isomorphism onto, this proves 7.3.

For the case of L-5-curves in the plane see [7]. (b) The 2-sphere S*. It is known that Ti(T(S*)) is cyclic of order 2; for

example see [13 ] . Hence, by Theorem A we can put regular closed curves of S* at a point xt into two classes under regular homotopy equivalence. For the case of -L-S-curves on 51 , see [8].

(c) The torus T*. From the exact homotopy sequence of the tangent bun­dle T(T>), it can be deduced that xi(r(r»)) is Z+Z+Z (Z is the infinite cyc­lic group). Then similar remarks to those of (b) apply.

(d) The protective plane P*. It can be proved that r^TiP1)) is cycle of order four. Hence, there are four classes of regular closed curves at a point *c on P'. For the case of L-S-curves see [9].

REMARK. By taking T as a fiber over a different point of T one can obtain results similar to those of Sections 6 and 7 for nonclosed curves.

8. Regular curves perpendicular to a submanifold. Let N be a regularly imbedded submanifold of a manifold M and let T be the unit tangent mani­fold of M. Let V be the normal bundle of N with respect to M; that is, V is the subspace of T which consists of all vectors which have their base points in N and are normal to N. Let Xo be a point of V and Q be the loop space of T at x%. Denote by Or the subspace of EHT) which consists of the paths ending in V.

Let IV be the subspace of E(M) of curves whose final tangent is in V; i.e., IV—T- l( V) where r is the map of Theorem B. Let T restricted to IV be still denoted by x. Then we have:

THEOREM 8.1. The triple (IV, T , V) has the CHP.

Proof. This theorem is an easy consequence of Theorem B. In fact, let the homotopy h,:P—*V be given with h~:P—*Tif covering h». Theorem B yields a covering homotopy A,: P—*E. But since h~, covers A, we have that J , ( £ ) € I V for all vEI and p£P and so h\: P-*TH. This proves 8.1.

Similarly, (£V, P, V) has the CHP. Let $ be the map 4 of $6 restricted to IV. Then

TH >Qr

I', I' V V

commutes so £ induces a homomorphism of the exact sequence of IV into that of Or. We have

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508 STEPHEN SMALE (Much

» Ti(r) - ♦ Tk(TK) -* T*(V) - » • ' •

Iff if* H/ > *»(Q) — T*(QT) - » r * ( 7 ) - • • •

By Theorem C, $t is an isomorphism onto and, of course, If is the identity isomorphism. From this and the "Five" Lemma [2, p. 16], it follows that ft is also an isomorphism onto. We have proved:

THEOREM 8.2. The map $: I\—Or defined above is a weak homotopy equiva­lence. Hence, by Theorem 6A it induces an isomorphism between the singular homology groups of IV and Sly.

9. Integral curves of * 1-form. Let T\M) be the unit tangent bundle of a 2-manifold M and T : T(M)—*M the projection. If dr is the differential of x and vE.Mm, the tangent space of M at m, then dr^ (v) spans a two dimen­sional subspace containing the vertical. Hence there exists a 1-form utr*0 on T(M) annihilating this distribution of planes. I f / i s a regular curve on M then $(f) (see the Introduction and Section 6) is an integral curve of w0. One might hope to get a characterization of regular curves this way. Unfortu­nately, however, w0 admits integral curves which are not the images under <t> of regular curves. A curve lying in a single fiber of T(M) is such an example. Thus it is not sufficient for the study of regular curves on a 2-manifold M to study integral curves of ut on T(M). However, there is still the question as to what can be said about integral curves of a». This section was written as an attempt to answer this question.

Throughout the rest of §9 we will assume that M is a given manifold of dimension three. It seems very likely that the theory here generalizes to manifolds of higher dimension. However, because the treatment of 3-mani-folds is so much simpler, we confine ourselves to this case.

A kind of curve essentially the same as the "stuckweise glatt" curves of [12] is considered here. A curve/on M is called a parametrized piecewise regu­lar curve if there exist real numbers *,- for *'«(), 1, • • • • , * with /o-0, 4 ~ 1 , and *<<*<+i such that for each i<k,f restricted to [tit <<«] is either constant or regular in the sense that \f(t)\ **0 for t€[tit tt+i]. We say that such a curve is distinguished if its parameter is proportional to arc-length [12]. By changing the parameter one can associate to each parametrized piecewise regular curve a unique distinguished parametrized piecewise regular curve [see 12]. Two parametrized piecewise regular curves will be called equivalent if their associated distinguished curves are the same. A piecewise regular curve is an equivalence class of parametrized piecewise regular curves. Each such curve will have a unique distinguished representative. Oftentimes we will identify a piecewise regular curve with its distinguished representative.

As will be shown by an example at the end of the section, the theorems here hold only if some restriction is placed on the 1-forms.

79

1958) REGULAR CURVES ON RIEMANNIAN MANIFOLDS 509

Let jo be a fixed point of M, and let« be a 1-form on M such that aAda^O on M. Denote by £» the space of all piecewise regular curves on M which start at go and are integral curves of w. Let Em be a metric space under the metric

dU, g) - max i(Jt), tt)) \tei) where i is any fixed metric on M. Define a map p: Em-*M by £(*) =£(1). Let £-1(j») be denoted by ft«.

THEOREM 9.1. T*« triple (£. , p, M) has the CHP. Roughly speaking, Theorem 9.1 is proved as follows. First, by a classical

theorem, there are local coordinates (x, y, s) about a point of M such that in them a assumes an especially simple form. Here the fact that uAi . i 'O is used. By Proposition 1.3 we reduce the proof, in a sense, to this local situa­tion. Then the local coordinates thus obtained are used to write down ex­plicitly the desired covering homotopy equations.

We break the definition of the covering homotopy curves into four parts according to values of the parameter t. In general, this curve will turn out to have a comer at f «= 1/4, J-1/2, and <-3/4. The first part of the constructed curve is merely a reparametrization of the given covering curve. Then the construction is such that at t —1/2, the s coordinate of the covering homotopy has moved to a position over the s-coordinate of the given homotopy. At f—3/4 the as-coordinate has undergone a similar motion, and finally, at t — 1, the ^-coordinate of the covering homotopy projects into the y-coordinate of the given homotopy in M.

Proof of 9.1. Let qGM. Take a coordinate neighborhood U of q with coordinates (*, y, s). In U we can write w-Pdx+Qdy+Rd* where P, Q, and R are differentiate functions of x, y and s. Then

u A <fc» - ( W + Off + RR^dx AdyAdn

in U where

P-**-Q*, V-P.-R., R'-Q.- Py. Hence PP'+QQ'+RR'i*0 in U since n»Adwf*Q. Then by a classical result of the theory of differential equations (see for example [3, p. 58]) there exist differentiate functions u, v and w of x, y, and % defined in a neighborhood of q such that w-du+vdv>. Furthermore Ow*oAdw—duAdvAdw so that w, v and w form a coordinate system in a neighborhood say V of q. By 1.3 it is sufficient to prove the CHP for the triple (£-I( JOi p, V). For convenience we will change the coordinates u, v, and w into x, y, and s respectively. So now x, y, and s are coordinates of V such that w = dx+ydz.

Let h,:P—*V be a given homotopy with k:P—*p~xV) covering ht. We will construct a covering homotopy it,: P—»£-,(V). To describe these maps

80

510 STEPHEN SMALE [Much

in the coordinate system (*, y, «) we use the following notation: UP) = MP), y*p), s.(p)),

Up)® - (*.(»(<), MP)(t), i.O)(0). Then: for 0£< £1/4 let

fhipW) = Kp)(4t)

for 1/4 £ / £1/2 let 5=5(0 = 4<-l and let

*.(*)(*) = *•(» - [*0) ~ *»(P)]y(P)s, J.WW = v(P),

for 1/2 £< £3/4 let 5 = 5(0 = 4 / - 2 . Then let

*.(>) - «•(#) ~ *(*)[«.<#) - *.(#)], *.(*)« = (365* - 455* - 205* + 30*«H*(f) - *,(?)]

+ 601 *.(» - t,(p) |1"y.(#)(5» - 5) + *.(/), * ( * )» - sg | *.(» - * ,w |«/«(,«- *) + * (* ) .

Sg = sign of x,(j>) - U(P), i,(p)(t) = - 601 x.(p) - *.(#) |"«(5» - 5) + «.(»

for 3 / 4 £ £ l let * = ,(/) =4<-3 and

*.(#)(<) - *.(#), MP)(t) - b*(P) ~ y(P)]s + *(P), I.WW = «.(#)•

In order to be sure that these equations define a satisfactory covering homotopy it must be checked that (1) k,(J>) is a curve in Em (for each p and v), (2) £,(/>) is a homotopy of k(p) or £,,(£) = *(£), and (3) kv(p) covers AK(/>) or*.(rt(l)-A.(p).

We will check (1) first. It is easy to note that *,(£)(0) -g 0 . Also, A.(£)(i) is clearly continuous and piecewise regular between the values f = 0, 1/4, 1/2, 3/4, and 1. It is necessary to check that £,(/>)(/) is well-defined at t = 1/4,1/2, and 3/4 since at each of these values Ju(P)(t) is defined in two different ways. By substituting these values of t into the appropriate equations it can be seen that where the definitions overlap they agree. To complete the proof of (1) it needs to be shown that £,(£)(*) satisfies « - 0 or, in other words,

it at

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1958] REGULAR CURVES ON RIEMANNIAN MANIFOLDS 511

identically for all v, p, and t. This is trivial for t£ 1/4. For 1 / 4 £ t£ 1 to prove that this differential equation is satisfied it is sufficient to make three com­putations, one for l / 4 ^ * £ l / 2 , one for l / 2 £ * £ 3 / 4 and one for 3 / 4 ^ / g l . These are not difficult and will be left for the reader.

To show (2) we set v = 0, getting

*o(#)(0 - x,(p), MP)(Q = y»(P) 2o(»(0 - «.(#)•

This is a parametrized piecewise regular curve whose associated distin­guished curve is exactly the given curve i(p)(t).

It is trivially checked that (3) holds, q.e.d. The space Em is contractible to a point. The deformation accomplishing

this is D: EUXI-*E* denned by D(f, v) -/(»<)• Define a map t: E*-+EU(M) (see §6) by letting *'(/) be the distinguished representative of / . Then t is continuous by the argument used to show that <b was continuous in §6. Let i be i restricted to fi» where Rm = p~l(qt)CEm. The argument used in §6 to show that $: T—*tt(T) was a weak homotopy equivalence may now be used to show that i: C„—»8(A/) is also a weak homotopy equivalence. This proves Theorem D.

The following example shows how Theorems 9.1 and D fail for a form wo which is completely integrable. Let M — E* and ut = xdx+ydy+zdz in a given Cartesian coordinate system (x, y, t) of E*. Take for qo any point at distance d>0 from the origin of E*. Then any integral curve of uo starting at q0 stays on the surface of the 2-sphere xi+y*+z***dt. Clearly, the conclu­sions of Theorems 9.1 and D fail in this case. Actually, Theorem 9.1 is false for any 1-form which is completely integrable at a certain point qt&E*. For then short curves at qo must lie on a surface of E* and the covering homotopy property cannot possibly hold.

BIBLIOGRAPHY

1. S. Chern, Differentiable manifolds. Notes at University of Chicago, 1955. 2. S. Eilenberg and N. Steenrod, Foundations of algebraic topology, Princeton, 1952. 3. A. R. Fonyth, A treatise on differential equations, London, 1951. 4. S. Hu, Homotopy theory, Note* at Tulane, 1950. 5. W. Hurewicz, On the concept of a fiber spate, Proc. Nat. Acad. Sci. U.S.A. vol. 55 (1955)

pp. 956-961. 6. I. M. James and J. H. C. Whitehead, Note on fiber spaces, Proc. London Math. Soc.

vol. 4 (1954) pp. 129-137. 7. M. Morse, Topolopcal methods in the theory of functions of complex variables, Princeton,

1947. 8. , L-S-homotopy dosses of locally simple curves, Annales de la Societe Polonaise de

Mathematique vol. 21 (1948) pp. 236-256.

0 | ( g 1/4,

l / 4 S « i 1,

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512 STEPHEN SMALE

9. , L'S-komotopy classes on the iopologic&l image of a protective plant, Bull. Amer. Math. Soc. vol. 55 (1949) pp. 981-1003.

10. H. Samelson, Groups and spaces of loops, Comment. Math. Helv. vol. 29 (1954) pp. 278-287.

11. J.-P. Serre, Homologie singuliere des espaces fibres, Ann. of Math. vol. 54 (1951) pp. 425-505.

12. Seifert and Threlfall, Variationsrechnung in Grossen, New York, 1948. 13. N. Steenrod, The topology of fiber bundles, Princeton, 1951. 14. G. W. Whitehead, Homotopy theory, Notes at Massachusetts Institute of Technology,

1954. 15. , On the homolcsy suspension, Ann. of Math. vol. 62 (1955) pp. 254-268. 16. , On the Freudenthal theorems, Ann. of Math. vol. 57 (1953) pp. 209-228. 17. H. Whitney, On regular closed curves in the plane, Compositio Math. vol. 4 (1937) pp.

276-284.

UNIVERSITY or CHICAGO, CHICAGO, I I I .

UNIVERSITY or MICHIGAN ANN ARBOR, MICH.

83

AlflfAUi OF MATHEMATICS Vol. 68. No. 3. November. 1958

Printtd in Japan

ON THE IMMERSION OF MANIFOLDS IN EUCLIDEAN SPACE BY R. LASHOF AND S. SHALE

(Received June 20,1957)

By an immersion f: Af * -* E**' of a k dimensional manifold in k + I dimensional euclidean space, we mean a differentiate map (for conveni­ence all manifolds and differentiate maps will be assumed C~), of M = M* into E*** which is regular; i.e., the induced map on the tangent space at each point of Af is one-one. We will assume Af is oriented and con­nected. We let BT be the tangent sphere bundle of the closed manifold Af, and B, be the normal sphere bundle of Af induced by the immersion / . We let Wt (resp. W,) be the integral Stiefel-Whitney characteristic classes of Br (resp. B-,) of dimension 4. / induces a map of B, into the unit sphere S**'~l in E**1 by translating the unit normal vectors to M in E"*1 to the origin. This map is called the normal map (Chern [3]), and since dim By = k + I — 1, we can define the normal degree of / as the degree of this map. Similarly, we can define a map by translating unit tangent vectors to the origin, and if / : Af* -* E1" and hence dim BT = 2k — 1, we can define a trangential degree.

In Section 1 we Btudy the relations between the Gysin homology se­quence of the Whitney sum of two sphere bundles and Gysin sequences of the components. In Section 2, we apply this result to the Whitney sum of BT and B., to show that:

(a) If/: Af* -*• E1* is an immersion of a closed manifold with orienta­tion Me Ht(Mk) then the trangential degree of / is Wt-M (i.e., the Kronecker index <Af, W^)).

(b) If/: Af* - E*¥l with I > 1 is an immersion of a closed manifold with orientation Af then the normal degree of / is —Wk'M.

Further, if / : Af -> Af' is an immersion of Af in any connected oriented manifold Af', not necessarily closed, of dimension k + I, I > 1, then the concept of normal degree may be generalized to be an integer mod W't+t • Af'. Here WUi is the Stiefel-Whitney clasB of the tangent bundle of Af', Af' represents the basic class if Af' is closed, and if Af' is not closed Wl+, • Af' is defined to be zero. Then we obtain

(b') The normal degree of an immersion / : Af *- Af'**', I > 1, is - Wt • Af mod (W'k+l • Af'). Since - Wk • Af = Euler characteristic of Af, (a) is the known result that the normal degree is the Euler characteristic (Chern [3]).

562

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ON THE IMMERSION OF MANIFOLDS 563

In Section 3 we show that for completely regular immersions (see Sec­tion 3 for definition), / : Mk -* f , k even, the tangential degree is twice the algebraic intersection number of Whitney [13]. Using results of Whitney, this enables us to prove :

Let Mk be a closed oriented manifold of dim k, k even. For any immer­sion f: Af * -* E**, Wt • M is even ; and for every even integer n there exists an immersion f: Mk -* Ek such that Wk • M = n.

As a corollary we obtain a theorem of Milnor [8] that there exists an immersion of real projective 3-space P' in El.

In Section 4 and 5 we study the tangential map t: M* -* G(k, I) where G(k, I) is the Grassmann manifold of oriented A-planes in Ek*i, associated to the immersion / : Af* -* E*>1 by assigning to each point of Mk the tangent plane at that point translated to the origin. If I > k, then it is well known that t* : H*(G(k, I)) -* H*(M) is determined by the charac­teristic classes of M. For the case Z ^ k, we get the following results :

1. L e t / a n d g be immersions of Mk in Ek*1 with I > 1 if k is odd, and with the same normal Stief el-Whitney class W, with integer coefficients. Then the induced tangential map «* : H*(G(k, I)) ->• H*(Af) of / and g are the same. Furthermore, if I is odd o r / i s an imbedding the condition that the classes Wt are the same is unnecessary. This theorem iB true if coefficients are the integers, Z, or the rationals.

2. If Af* may be imbedded in E"*1 or immersed in E***, where k is of the form 4(2r - 1), then in H*(Mk) we have Pt = Wk mod 2. (For k = 4, this is essentially a theorem of Pontrjagin.) The same result holds, say, if Mk may be immersed in £** with a (k — 3)-normal frame.

In general we give a complete review of the results on the character­istic classes of Af * obtainable from the cohomology of G(k, I) and the fact that Af * may be immersed or imbedded in ^*+l. We obtain a number of known results, for example, a result of Kervaire (Theorem 5.5).

Unless we say otherwise, the coefficient group for homology and coho­mology will be the integers. All the Stief el-Whitney classes in the first four sections will have integral coefficients. All manifolds will be con­nected and oriented.

1. Gysin sequence of a Whitney sum

Let (B„ St, M), pt: Bt ->• Af, i = 1, 2, be two sphere bundles, <S4 a sphere of dim dt — 1, with structure group Rt, the rotation group on the euclidean space Et of dim dt. We consider ft, x R^dR, where R is the

85

664 R. LASHOF AND S. SMALE

rotation group on the euclidean space E of dim dj + d,. The Whitney gum B, S, M), dim S = d, + d, — 1 of the two bundles iB the sphere bundle with group R defined by taking as coordinate functions the direct sum of the coordinate function of the two given bundles, considered as having values in R, Then we have natural fibre preserving inclusions/»: Bt-*B, i = 1, 2. ft may be defined by considering the associated vector bundles with fibres Et and E respectively, then / , is induced by the natural in­clusion Et-* Ei + Et = E of the fibres, restricted to the unit sphere. This gives a global map since by the definition of Whitney sum, the co­ordinate functions of the sum bundle act on each factor of the fibre separately in the fashion given by the component bundles.

THEOREM 1.1. / , induces a map of the Gysin cohomology sequence of (B, S,M) into that of (Blt St, M):

HT-*i-\M) HUM) > Hr(B) > H'^-'i-'tM) >

Hr-'t(M) H'(M) H r(Bt) H'+'-'i (M) where I is the identity, and letting Wt be the Stiefel-Whitney close of Bit St, M) of dim d„

Gf(z) = (-l)dt*««U TT„ G?(x) = xl)W1. REMARK. Wt, i = 1, 2, are integral Stiefel-Whitney classes, but we

may use any coefficient group for the terms in the Gysin sequences, then the cup product is under the natural pairing of the integers with the group. In particular, we may use real numbers mod 1 and topologize our coho­mology groups. Then under Pontrjagin duality we get a map of the homology sequence of Bt into that of B, both with integer coefficients ; and cup product goes over into cap product with the class Wt under du­ality.

In proving this theorem we use a number of results from Thorn's thesis [11]. Following Thorn we let A be the mapping cylinder of B -* M and let A' = A — B, then we have the maps

j : Hr(M)-*-Hr(A) fi : Hr(A') - Hr(A) <P* : Hr(M) -* Hr-'i-dH.A')

where j is an isomorphism induced by the projection A -* M, ft by the inclusion A -* (A, B), and <p* is the isomorphism obtained by Thorn by considering a carapace on A' as a carapace on M by redefining supports. We have corresponding maps of Bt-+ M. Then for example, / , : (Au 2?,) -

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ON THE IMMERSION OF MANIFOLDS 666

(A, B) induceB (where we have again used / , for all maps defined by /« : B<^B)

H'-'i-'iM) Hr(M) Hr*l-'*-**(M) L» j L»

Hr(A') Hr(A) » Hr(B) Hr*\A') |/r \f? [/? \/r

HT(A[) »H'(i4,) >Hr(B,) >Hr+\A[) !#>* pi j»,*

ff-'i(Af) HT(M) Rr*l-*iM) This gives a map of the Gysin sequence of (B, S, M) into that of (£„ St M); it is only necessary to identify the maps. Since

H'(A)-£>H'(AJ

HTM) Hr(M)

commutes, jilf*j — I: Hr(M) -* Hr(M). It remains to identify the map

Gr = <p?-lf?9* As shown by Thorn [11], if w is a fixed generator of H\M) and if we let

<P*(w) = UeH'i+SA'), ?,*(») = U(eH't(A't), we have:

f*(x)=j*(x)\jU, W = j~W, W the Stiefel-Whitney class of dim d^ + d, in (B, S, M) and similarly for (Bt, Sif M).

Now f?<p*(x) = f*j(x) Uf*U = Ux) U f?U. Hence if we can prove

(1.2) f?U = (-lY*jl(WJ\jUl, we have

/,V*(x) = ( - D V y ^ u i ^ ^ u ^ = ( -D'^^u TFJU tf. and

Gf(») = ( - l ) 4 ' ' « U l f J . REMARK 1. The proof for G,* is identical except for order of terms in

the cup product in (1.2) (and hence the difference in sign) and will not be repeated.

REMARK 2. If we use compact coefficient groups for the cohomology groups of the Gysin sequences, U and Ut are taken with integer coeffi-

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566 R. LASHOF AND S. SMALE

cients and the cup products j^x) Uf*U are under the natural pairing of the integers with the compact coefficient groups. In (1.2) on the other hand, everything is with integer coefficients.

It remains to prove (1.2): Let

li: H'*\Alt BO x H»'*(A„ Bt) - ff •♦•♦«.-<(A1> Bx x (A,, B,)) be the natural pairing. Note that H'((A„ BJ x (A„ B,)) = Hr(A, x A,, A, x Bt\jB, x A,) = Hr(A[ x A,) .

Let U be the class f*(w) in H'I^AI x A|) of the sphere bundle (At x5 ,u5 ,x AJ-+M x Afcorresponding to the generator w € H"(MxM) where w = ^(w, <g> to,), th.: #*'(Af) <g> H^M) -* H'I+IM x Af). Then Thom [11] shows that /i(I/, <g> Ut) = U. But the Whitney sum bundle A' -* M is induced by the diagonal map d: Af-* M x Af; i.e.,

A' -^-» A[ x A;

1 , 1 M M x M

commutes, where d' iB the induced map, and hence d'*U = U. Consider the sequence of maps

(A1( B,) - ^ (A„ B,) x A, -?U (A„ BJ x M - ^ (A1( £ t) x (A,, £,) where

», is induced by the inclusion of Af in A, (identity on first factor) p, is induced by the projection of A, on Af (identity on first factor) d, is induced by the diagonal map A1-*A1 x Aj

Then this sequence of maps is the same as the following :

(Au BJ -A» (A, B) -2L (A1( B>) x (A„ £,) since they are both fibre preserving and correspond to the diagonal map Af -+ Af x Af in the base, it is sufficient to check them on each fibre. Let Sx be a fibre in B2 over x € Af and Yj the mapping cylinder of S, -»• x, then both fibre mapB are induced by :

Y , — » Y, xx— Y, x y , . Hence

dfpntrtUi <g> C7.) = fTd'*U = ffU . To compute the left hand expression, we compute on each component of the above products separately, and we have by " abuse of notation ":

if(Ut ® Ut) = Ut ®j;%Ut = U1®Wt, since £'&: H(A,, Bt) -> .ff(Af) is the same as that given by the inclusions

_

ON THE IMMERSION OF MANIFOLDS 567

**

M-* A,-* (A,, Bt). Further P?(Ut®Wt) = Ul®jl(WJ)

d?(U1®j1Wt)) = U1Uj1W1) i.e., this last map actually is

m(A, Bt) ® H'tA) > H'ISU:, A) x A,) -2-» H'i^(Au Bt) Hence

fTU = Ux\jJx(WA = (-lrWWJvU* q.e.d.

2. Application* to immersed manifolds APPLICATION TO NORMAL DEGREE.1 Let Af be a compact oriented n-

dimensional manifold and/ : M-* M' be an immersion of M in an oriented manifold of dim n + N, N 2. Let Sv be the normal bundle of M in Af' and BT be the tangent bundle of M; then dim B-, = n + N — 1 and dim BT = 2n — I. The map/ induces a map/v : B^-*T, where T is the tangent bundle of AT. Then/,.: fl,**.,^,) -»• iJ„+*-I(r). Now consider the Gysin sequence of T. Note that the right square is commutative.

H.UM') ^ S HJLM') H^.AT) > H...-M)

#»+jr-i(-Bv) Hn+s-i(M) .

Since Hn+*-i(M) is zero, the image/v.(i?>) of the basic class B* of Hn+*-i(B») is contained in the kernel of flm*ir_,(r) -»• //,+,_1(Af') and hence in the image of #o(Af'). Let Wu+N - Af' be the value of W'„+ll on the basic class Af' of Hn+S(M').* The image of H^M') is isomorphic to the integers mod (W'%+g • AT); if Af' is compact W'H+N-M' =— O*,, wbere I2*» is the Euler characteristic of Af'. Hence the immersion defines a normal degree mod W'n+„ • Af'). We use the homology version of Theorem 1.1 (see re­mark following theorem) to compute this degree. Consider

0 0

1 1 #„♦,.,(£>) #„♦*-!(£, ® 5>) Hn.„.x(T)

H. (Af) 5 HIM) *HJ,M')

0 1 The definition of normal degree given below is due to S.S. Chern who suggested the pro­

blem solved here. » If Mis not closed let Wn*s • M' = 0.

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568 R. LASHOF AND S. SMALE

The vertical maps are part of the respective Gysin homology sequences. For the tangent bundle on M' we use singular homology theory and spectral sequences to define the Gysin homology sequence. (The Gysin sequences for non-triangulable space using compact supports are in gen­eral distinct from thoBe using singular theory ; e.g., if M' is a euclidean space. For a sphere bundle over the compact manifold M, the two Gysin sequences coincide.) The top line of horizontal maps are induced by in­clusions. The left hand square is commutative by Theorem 1.1. Since BT©.By -+• Mand T ->• M' are Bphere bundles of the same dimension, we have a map of their spectral sequences and hence of their Gysin sequenc­es using singular homology theory. Hence we see that if we choose the orientation of B properly, we have the degree of the normal map is — Wn-M = 0,M modulo W'n+N'M', where £lM is the Euler characteristic of M. Hence we have proved :

THEOREM 2,1. The degree of the normal map induced by the immersion of a compact orientable manifold M into an orientable manifold M', dim M = n, dim M' = n + N, N^2, is nM mod W^N • M'. If M' is compact, the degree of the normal map is ClM mod ClH,.

REMARK. If M' is euclidean space, the degree of the normal map is simply the integer il„ (since W'n+„ = 0), which agrees with previous known results.

APPLICATION TO TANGENTIAL DEGREE. Let / : M -*• E" be an immersion of a oriented manifold of dim k into 2A:-dim euclidean space. Then / in­duces a map <p: BT-+ S" _ 1 of the tangent bundle of M into the unit sphere of E**, by translating the unit tangent vectors of M in E* to the origin of E*. Since Br is of dim 2k — 1 we can talk about the degree of <p. This degree is called the tangential degree of the immersion/.

THEOREM 2.2. / / Wk e Hk(M) is the normal characteristic class of dim k of the immersion f, then the tangential degree offis Wk • M.

PROOF, Let B, be the normal bundle of M induced by / , and let Bj£)Bv be the Whitney sum. Then B, ©i?v is the bundle induced by / over M by the tangent bundle T of E*. The natural injection Br -* B, © Bv (see Section 1) and the induced map Br®B*-*T are bundle maps and induce maps of the corresponding Gysin sequences. We again use singular theory for the Gysin sequences of T. <p* is the composite map H^^Br) -* Hu-l(Br(£)By)^Htt-1(T)^Hik-1(S't-1). Hence consider the following diagram, the commutativity of which follows as in the proof of Theorem 2.1. The vertical sequences are the Gysin sequences for the correspond­ing bundles.

ON THE IMMERSION OF MANIFOLDS 569

0 = ff„_1(M) >Hit.,(M) * ff„-;(£») = 0

HtUBr) - + #«-,(.BT©Bv) > HtUT)

I-, 1 H,M) ^ HJLM) * Htf?*)

Hu-riS"-1)

0 = tfu(M) H»(M)- # « ( £ " ) = 0

It follows immediately that deg <p = Wt'M. REMARK 1. As in Theorem 2.1, we could use any other oriented mani­

fold M' of dim 2k in place of Eik and obtain results modulo the character­istic class of the tangent bundle of M'.

REMARK 2. In the general case of an immersion / : M* -* E"*e, e^2, one may obtain a generalization of Theorem 2.2. In fact, let <p : BT-* gk+t-i i^ the map obtained by translating unit tangent vectors to the origin. Then the following diagram this is the dual of the diagram used in the proof of Theorem 2.2, see also Theorem 1.1)

H'+*-\B,) « #•♦*-'(£,©£»)« He+*-\T) » He♦*-1(S<+*-,)

He(M) ,w, H°(M) H\E°*k) shows that <p*(S) = f ' f , , where S is the generator of H'+k-'(S'*k-1). Hence if we knew which homomorphisms <p* : H'*k-l(Se+k-l)->H,,+k-'(BT) were realizable from immersions we would know which normal classes are realizable. However, we are able to obtain information on this only in the case e = k (see Section 3).

3. The intersection number of an immersion

Unless otherwise stated, all manifolds in this section will be even di­mensional, closed and oriented. In the first part of this section we recall some of the theory of Whitney [13] related to the intersection number If.

An immersion f:Mk-* Eik of a A;-dim manifold M = M", has a regular self-intersection at /(p,) = /(p,) if the tangent plane of f(M) at /(p,) and /(p,) have only the point /(pO = /(p,) in common. If / has only regular self-intersections and no triple points then / is completely regular.

Consider M imbedded in Eik*x and let B, be the unit tangent bundle of the manifold M in En*\ Then a manifold with boundary, ^~, is defined as follows. S~ is the disjoint union of Br and all pairs (p, q) e Mx M with p =£ q. If qn -* p in M in the direction of a unit vector u at p then we let

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570 R. LASHOF AND S. SMALE

(p, Q*)-*(Pf «)• This defines the topology on ^~, and denning the differen­t iate structure in the obvious fashion makes S~ into a manifold with boundary Br.

Still considering Af in £ u + 1 let \q — p\ be the distance from p to q in E***1. Returning to the immersion / above and considering E* to be oriented with origin 0, we define a map F: ^~-* E" as follows :

F<r,q) = fM-f<*>. P*Q \Q-P\

F(p, u) = <p(p, u), (p, u)eBr,

where <p : BT -* Eu is the map induced by / taking the tangent vector to an arc through p into the tangent vector to the image of the arc in En

at/(p), and translating this last vector to the origin. It is easily shown that F is continuous [13]. Furthermore F maps no point of BT into O and maps a point (p, q) of ^~' — B, into 0 if and only if / (p) = f(q).

Suppose /(p) = f(q). Let uu • • •, ux be k independent unit tangent vectors of Af at p, v„ • • •, v* at q, each determining the positive orienta­tion of Af. Then the system of 2k vectors, A = <p(p, w,), • • •, f(P, «»)»• • , ?(Q> vt) will be independent at 0 ; and the orientation determined by A will not depend on whether we write the vectors at p or the vectors at q first, since k is even. The self-intersection f(p) =f(q) is positive or negative according to whether A determines the positive or negative orientation of En. The intersection number lf is the algebraic number of self-intersections.

THEOREM 3.1. Iff: Af * -»• Eu is a completely regular immersion of a closed oriented manifold, k even, then the tangential degree off is twice ls.

PROOF. Let S"~l be the sphere of unit vectors of E* at the origin, then S""1 is the boundary of the unit disc D of £™. Let e = max \F(x) \x € ^ ~ and e' = min \F(x)\xeBT. Since Br and ^"are compact, e and e' are well defined positive real numbers. Let h: E* -» E** be map which sends vectors v in the ring e' ^ |v| ^ e into S" - 1 radially by their direction, and stretches the rest of E"c in an obvious fashion such that h is a homeomorphism on the complement of this ring and is contin­uous on all of E». Thus hF: (^", BT)-+(D, S""1). It is clear that hF cut down to BT is just the tangential map defined in Section 2, and the degree of this map is the tangential degree of / .

Let &~ be the space obtained from &~ by identifying Br to a point 6 in J7~, and D by identifying S""1 to a point * in D. Then hF induces a map 0 : (^~, b) -* (D, 8). Consider the commutative diagram :

92

ON THE IMMERSION OP MANIFOLDS 571

\(hF), «,

#»(£>, S"" ) >Htk(D,x) The horizontal homomorphisms, being induced by relative homeomor-phisms, are isomorphisms onto. All the groups in the diagram are infinite cyclic. Further, using the exact sequences of the pairs we obtain the commutative diagram

# « ( j n >HlkJ',p) 9* i'l

Hik(D) >H3k(D,s) where again the horizontal maps are isomorphisms. Finally we have the following commutative diagram:

1UJT,BT) >H,UBr)

H«(D,S"->) fl^-1(S»-1) where again the horizontal maps are isomorphisms and the group are all infinite cyclic. From these diagrams it follows that the tangential degree is the same as the degree of 0* : Hik(^) -* Htk(D).

Since the map F is a homeomorphism on the components of F~\ V) for a sufficiently small neighborhood V of O, kF and hence 6 is a homeomor­phism on the components of 0-\V). According to the Hopf theory (e.g., Whitney [12]) the degree of 0 is the sum of the degrees of dj Vp, where the V„ p = 1, • • •, r, are the oriented components of d~\V). By the definition of F we get one component for each pair (p, q) such that F(P,Q)=Q i .e.,/(p) = /(«)• But this differs from the definition of the in­tersection number, I„ only in the fact that (p, q) and (q, p), / (p) - f(q), give two distinct components (both with the same orientation) and hence a given self-intersection is counted twice in the degree of 9; i.e., we have : tangential degree of / = degree of 0 = 21 f.

From Theorems 3.1 and 2.2 we get :4

COROLLARY 3.2. Jf f: Mk -* Eu is a completely regular immersion of a closed oriented manifold, k even, and Wk e H"(M) is the normal charac­teristic class of dim k of the immersion f, then

Wk-M = 2If.

Further, from the result of Whitney [13, Theorem 3] on the existence of 6 This result is essentially due to Whitney (see note at end of bibliography).

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572 R. LASHOF AND S. SMALE

completely regular immersions with given /,, and from the fact that Wk= 0 (mod 2) (Chern, [2, Theorem 2, p. 94]) we have :

THEOREM 3.3. Let M* be a closed oriented manifold of dim k, k even. For any immersion f: Mk -* Eu, W* • Af is even ; and for every even in­teger n there exists an immersion f: M* -* Eu such that Wt • M = n.

THEOREM 3.4. (Milnor [8]). There exists an immersion of real protective Z-space P' in E*.

PROOF. By Theorem 3.3, there is an immersion / : S'-+El with Wt - M = 2. By the bundle classification theory (Steenrod [9, Sections 26.2, 35.11]) there is only one bundle space over S* whose characteristic class is 2, and that is P1. Consider a small tubular neighborhood (a tubu­lar neighborhood may be defined in the case of an immersion as it usual­ly is for an imbedding, e.g., Thom [10]) about/(S1) in E*. The boundary (for an immersion the boundary of a tubular neighborhood will have self-intersections of course) of this tube is an immersion of f".

4. On the homology of Grawman manifolds Let G(k, I) be the Grassman manifold of oriented fe-planes in Ek*1. For

n > m, the inclusion Em*k -* E"*k induces a map i : G(k, m) -+• G(k, n). One may take the limit of these spaces in a certain sense to obtain the classifying space G(k, oo) for the rotation group RK for all manifolds. There are natural maps i : G(k, n)-+G(k, oo) for all n. As UBual Wt de­notes the Stiefel-Whitney class of Hl(G(k, oo)), l^k and I odd, or I = k. We shall use the same symbol to denote i*Wl in H'(G(k, n)) when there is no ambiguity.

Let V»+Ii, be the Stiefel manifold of Z-frames in E*+l and p: V*+u -> G1, k) send a l-trame into the Z-plane which is spanned by it. We will prove the following:

THEOREM 4.1. For k even and i^k

0 > H(( V,„.t) -^ Ht(G(l, k)) -X Ht(G (I, k + 1)) — - 0 is exact. For i = k and k even, the image of p+ is generated by the Schubert cycle (e.g., see [2]) * = (0 k)* - (0 k)~. The cycle <bisa genera­tor, or twice a generator of Hk(G(l, k)) mod torsion according to whether Wk*\ in Ht+1(G(k + 1,1)) is zero or not. Jfkis odd, I > 1 and i^k,im: H((G(l, k)) -* Ht(G(l, k + l))isan isomorphism onto.

For cohomology we have THEOREM 4.2. IfiiLk,k odd or even

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ON THE IMMERSION OF MANIFOLDS 573

H\Vkthl) JLH\G(l, k)) «-£- H\G(l, fc + D)« 0 is exact. If kis odd and Z>1 or i<k, Hi(Vt+ltl)=0. If i=k, k even, then p* is onto when Wk*i in H**1 (G(k+1,1)) is zero. If this class is not zero, the image of p* is generated by twice a generator of H*(F1+M). Since H\V„,J is free cyclic H\G(l, k)) = H\G(l, k + 1)) + Z when k is even. The class Wt together with the generators of Hk(G(l, k + 1)) generate Hk(G(l, k)) when Wk„ * 0 in Hk*l(G(k + 1,1)).

For the above theorem it is important to know, for k even, when Wk+1 is zero.

THEOREM 4.3. Let k be even. The Stiefel-Whitney class Wt*i in H"+\G(k + 1,1)) is not zero if k = 2 mod 4 and I > 2 or k = 0 mod 4 and l>4. If 1 = 1 or 1 = 2, Wk+l = 0. Ifl = 3orl = 4andk = 0 mod 4, FF»+i = 0 when k = 4(2r — 1), r any positive integer; otherwise Wttl =t 0.

To prove these theorems we introduce certain auxiliary spaces and maps as follows. The rotation group Rn of £* may be thought of as the space of n-frames of £*. Let V* = RnlRr, n > r ;> 2 where Rr is con­sidered as acting on the first r-vectors of an n-frame. Then V* is the Stiefel manifold of in - r) frames in E*, V»,,_r. Wherever mapB in the rest of this section are not mentioned explicitly, they refer to the maps defined in these paragraphs.

For 2 ^ 8 ^ n — r, let R, acting on the last 8 vectors of a frame define an action of R, on V*. We denote the quotient space RnIRr x R, = V'/R, by V",. If r + 8 = n, V",, may be considered as the GrasBman manifold of oriented 8-planes in b?, Gs, r), and F? - V?t, the map which Bends an 8-frame into the 8-plane spanned by it. In this way F? is a principal bundle over F " , with groups R,. We define inclusions V* -* V'*1 and F?,, -* V?*,1 by adding a fixed orthogonal vector to the (r + l) a t place. From the definitions one can check that the following diagram commutes.

V* F ? + 1

I I V " » VH+l ' r,» ' r.t

A map from F?,, to F?+1>, is defined by sending the (r + l) s t vector together with the first r-plane into the (r + l)-plane which they deter­mine. In this way F?,, becomes an r-sphere bundle over F?+J>,.

Similarly, a map from Fj? t to F?>t+1 is defined by sending the (» — * ) * vector together with the last 8-space into the (s + l)-plane which they span. Then F?,, is an r-sphere bundle over F?+lt, and (for r + s < n) the following diagram commutes

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574 R. LASHOF AND S. SMALE

' r.« * ' r , i+l

J 1 V» + l k V n + 1

r,» ' " r,»+l

LEMMA 4.4. Tfo map* V? -* V**1 and V?,, -*• V**1 induce isomomor-phisms in homology through dimension rfor any s. Thus by the naturality of the universal coefficient theorem the induced homomorphisms in cohomol-ogy are isomorphisms for the same dimension.

PROOF. First observe that Ht(Vt)-* Ht(V*t,) is an isomorphism for i <r because the groups vanish. For i = r the maps & = V;*1-* V*-* Vrl generate nr(V?) and nrV»r*x) [9, p. 132]. Thus nr(V*) -> «jyyl) is an isomorphism and by the naturality of the Hurewicz Theorem HTV*) -* Hr(Vrl) is also.

Now consider the sequence of Bphere bundles : V r * ' r,i * * v r.l ^T* v r.i*i ' ' " * v r,i II II 1

' T ' " r.J * * * y r,\ ^* y r.l+l ' ' ' ' * * r,t

Corresponding to the /-dimensional Bphere bundles in the above diagram, we have the Gysin sequences,

- HU VXli) - H,( V-V) - HAVtf.d - jyt_I_1(V-tVI) - HUV7?) By induction asBume (we have proved the case I = 0)

HAVldamy*?) » = 0 , - , r HA V?.,+0 « «J(Vtf.d j = 0, • •, i - 1 .

then by the 5-lemma, #<( V£,+J) ss fli(V"*ii) and the lemma follows. Theorem 4.1 for i < k follows immediately from the Gysin sequence of

VJ*,'*1 over VJt'.Y' and the preceding lemma. To prove the theorem for » = k, write down a portion of this sequence.

Z Z II _, II

I-H.«(VS:J.V) — HjLvm?) - u H,(

z The map r7 is induced by the projection of VI*1*1 into VJt!*1 which takes

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ON THE IMMERSION OF MANIFOLDS 575

the I + 1 frame into the I frame consisting of the last I- vectors of the I + 1 frame. Then the middle square of the previous diagram commutes.

For all k, q* iB onto. We consider now only k even. From exactness y' must be an isomorphism onto and certainly r is also. This implies that the image of y is the image of p*. The image of ^ is zero since p is de­fined by cap product with an order two class. Hence y and p* are 1-1 , We have proved that in the following diagram the bottom horizontal sequence is exact.

Hk(V^)-^-*Hk(V^)

1 , 1 . The vertical maps are isomorphisms onto by 4.4. This proves the first

sentence of Theorem 4.1 where we have let i^ = qj+. We will now compute the kernel of i+ : Hk(G(l, k)) -* H„(G1, k + 1)).

Let Ck(Gm, n)) be the group of Schubert fc-chains (e.g., see Chem [2]) of G(m, n), ZkGm, n)), the Schubert A;-cycles, etc.

It follows from the definition of the Schubert cells that CkGl, k)) = Ck(G(l, k + 1)) (under identification of map induced by inclusion), and that Ct*i(G(l, k + 1)) contains exactly the linear combinations of the cells (0 k + 1)+ and (0 k + 1)" in addition to the cells of C*+I(G(J, k)).

Now let zt belong to both Z(G(l, k)) and Bk(G(l, k + 1)). Then z, = 8e»„ where c l+I e Ck+1(G(l, k 1)). From the above observations we can write c*+i = <£« + m(0 * + 1)* + n(0 k + 1)" where c£+16 Ck+1(G1, k)). Hence

z* - &£♦! = m#(0 k + 1)* + n0(O k + 1)-By the boundary formulas for Schubert chains (e.g., Chern [2]) one ob­tains keven:

*» - 9c'*+i = # * . * = (0 k)+ - (0 k)-, N^m-n k odd:

zk - dcl.i = N'(0 k)+ + N'(0 *)", N' = -m-n. Thus for k even, 4> generates the kernel of i0.

Furthermore if k is odd, I > 1 and , , . ,„ . a(r) = + 1 if r = l o r 2 m o d 4

^r-1 ar) = - 1 if r = l o r 3 m o d 4

it may be checked that dc = (0 *)♦ + (0 k + 1)". Thus for k odd, I > 1, the kernel of «# is zero in Hk(G(l, k)).

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576 R. LASHOF S. SMALE

To finish the proof of 4.1 consider the Gyein sequences in cohomology, A; even,

0 H'iVl****) > mvttl*1) H'^Vill*1)

1 !P* 1 I o —> H\VH\?)-* Jf •( vjy«) — Hxrxi?) — HMvn\?) Here 12 s ift+I(FJt!*«') is the order two characteristic class of the Sk-

bundle VJ*,ut over Vlilf. If XI = 0 it follows that p* is onto (see the previous discussion for homology) or in homology if g is a generator of Hk(Vk

k*'+1) then p*(g) is a generator of Hk(Vty) mod torsion. If 12 0, V+id) is twice a generator of Hk(Vi\l) mod torsion.

To identify O, consider the following diagram where n is larger than max (2k + 3,1).

*.l * ' I,* ' ' ».*

1' . 1° r 1"' " k*l,t * r »,»+! ' ' » ,*+l

Here a is the homeomorphism which takes an Z-plane [»»♦„ • • •, e*+i+J into the orthogonal A: + 1 plane [e^ • • •, e»+J such that e„ • • •, ek+1, e*+n • • • i «*+i+i has the given orientation of E**1*1. Then /? can be defined so as to make the diagram commute. The maps y and rf are compositions of maps 7?,V+t -»• VJ.i'*' - V?+i.V which were defined previously. It can be checked that this diagram commutes and furthermore that VZ*t*i over VZ+*£ is the associated S* bundle of Vl+'+l over Vl\k£. This implies that if Wk+1 is the Stiefel-Whitney class of i?*+I(G(fc+l, n)), then f2; d*JWM. Since a* is an isomorphism we have 12 = 0 if and only if TP»+1 in H**l(Gk + 1, J)) is zero. This finishes the proof of Theorem 4.1.

The proof of 4.2 follows from arguments dual to those used in proving 4.1. We merely add that since Wk = (0 k)* - (0 k)~ has the value of 2 on <f>, it has the value 1 on a homology generator, hence the last statement of 4.2.

The proof of 4.3 proceeds as follows. Since A;+l is odd, W»+1 is defined with integer coefficients and is of order two. Generally if H *(X) has only order two torsion an element of H*(X) is zero if and only if its rational and Z, reduction are zero. Since H*(G(m, n)) has only order 2 torsion, alj m,n', TT*+, is zero if and only if its Z, reduction is zero. Thus for the proof of 4.3 we use coefficients Z, for all Stiefel-Whitney classes. A

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ON THE IMMERSION OF MANIFOLDS 677

formula of Chern [2] takes the following form,"

W, W, 0

Wt.> =

Wt W, WB 0 • • • Ws Wt W,W, 0

0 ; there are no relations be-

W„ W^ W0

Wk+1Wk Wl

In G(k + 1, /)), FT, = 0 for i > I and W1

tween the other Wt 'a. This determinant is symmetric with respect to the 45° axis. Hence all

non-symmetric terms appear twice and drop out. Now suppose k=4n+2 and Z>2. It is clearly sufficient to show Wktl^0

in G(k + 1, 3) for this case. But from the above determinant one notes that the symmetric term WJLWf^Q, hence Tr»*,*0, in H**\G(k+l, 3)). On the other hand, if A; = 4n and I > 4 it is sufficient to show that Wk+l + 0 inG(fc + 1, 5). There the symmetric term WAWtf*-* * 0 hence in this case Wk+1 =£ 0.

For I — 2, Wk+l is a polynomial in W„ but k is even so W»+, =- 0. The last sentence of the theorem also follows from the properties of the above determinant but it involves a long computational argument that does not seem worthwhile here.

5. The tangential map of an immersion

Let / be an immersion of an orientable manifold M* = M in Eh*\ T h e n / defines a tangential map t: M-* G(k, I) by translating a tangent plane at a point of/(Af) to the origin of E**'. The purpose of this section is to investigate the induced homomorphism in cohomology. If I > k, then it is well known that t* : H*(G(k, l))-+H*(M), the characteristic homomorphism, is determined by the characteristic classes of M. There­fore we confine ourselves to the case I k.

THEOREM 5.1. Let / : Mk-+E**1 be an immersion of an orientable manifold M — M* with k even and vrith I and k such that Wkti = 0 in H"*\Gk + 1,1)) (see 4.3).

Case I. If 1 = 1, one can choose a generator Sk of Hk(Gk, 1)) = Z so that Wt = 2 A*.

1 See also: S. S. Chern, On the multiplication in the characteristic ring of a sphere bundle, Ann. of Math., 49 (1948), 362-372.

99

578 R. LASHOF AND S. SMALE

Case II. If I = 2, one can choose a A* such that A, and (W,)*'1 generate H"(Gkt 2)) = Z + Z, Wk = 2 At + (fT,)*" and tf A; = 0 mod 4, P t = (Wt)kl\

Case III. If l = S,k = 4(2* - 1), Hk(G(k, 3)) mod torwon i* generated by a cocycle At and P», and Wk = 2\k + Pt mod torsion.

Case IV. ]fl = 4,k = 4(2" - 1), Hk(G(k, 4)) mod torsion w generated by a cocycle At, and all possible cup products of Wt and Pt with total degree k. Wk = WiGiWi, P,) + Pk + 2Ak mod torsion, where GiWt, Pt) is a polynomial.

Let a : Gk, I) -* G(l, k) be the homeomorphism defined in § 4 and a* : H*(G(l, k), G)^>H*(G(k, I), G) the induced isomorphism where G = Zt

or Z. Then if Wt and P, are Stiefel-Whitney or Pontrjagin classes of G1, k), Wt = a* Wi and P, = a*Pt are the dual classes of G(k, I).*

From 5,1., we obtain : Case I yields Hopf's theorem on the curvatura integra of an immersion

of an even dimensional manifold. Case II is essentially a generalization of the Chern-Spanier result [4]

on immersions of 2-manifolds in E*. Case IV yieldB: COROLLARY 5.2. Suppose a closed orientable manifold M" may be imbed­

ded in Ek** (or immersed in Ek*') where k is the form 4(2r — 1). Then in Hk(Mk), we have Pk = Wk mod 2.

For k = 4 this is essentially a theorem of Pontrjagin (see [2J). Corol­lary 5.2 follows from Case IV, since Hk(M") = Z has no torsion and Wt = 0 in Mk, and thus Wk = Pk + 2A» or Wk = Pk mod 2.

Theorem 5.1 is proved as follows. We first prove Case II. The coho-mology ring H*(G(2, <»)) is generated by Ws, so (W,)kl* is a generator of Hk(G(2, oo)). Applying 4.1 and 4.2, since the cycle * = (1 1)* -(1 1)- is a free generator of Ht(G(2, k)), (W,)k» and a cocycle Ai with value 1 on * generate Hk(G2, k)). Therefore the corresponding classes Ai' = a*A', and Wt)kl* = a*(Wf generate H*(G(k, 2)).

Let Wk = m\k + n(WJkl1. From 4.1 and 4.2 it follows that (Wt)kl%

must have the value 0 on a** = 4> = (1 1)+ — (1 1)". Then since Wk has the value 2 on 4> and Ai' has the value 1 on <J>, TO must be equal to 2. From the Whitney duality theorem one obtains Wk =

* Our a: Gk, t) -* G(l, k) corresponds to Wu's d". Wu shows that a*Wj = Wl mod 2 and a*(P«) = (-1)'P« with rational coefficients. In general for any class Z, (.a*?Z= ±Z with rational coefficients and (o*)*Z= Z rood 2. Hence (o*)»Z= ±Z with integer coef­ficients. In our work the sign does not matter.

100

ON THE IMMERSION OF MANIFOLDS 679

Wtfnzno&2, so n must be odd. Let At = A ' + (l/2)(w - 1)( W - Then Wt = 2A, + (W,)'11. Since (WJ*1* has the value 0 on * , A, will have the value 1.

We will now Bhow that (JP,)*" = Pk in Hk(G(k, 2)) if k = 0 mod 4. Since P, = 0 in Hk(G(k, 2)), i > 4 (e.g., [2]) by the duality Pontrjagin classes mod torsion (e.g., [6]) we can write P» = (P>". But P4 = W\ or P« = PPJ, hence P» = (J7,)*". This prove Case II of 5.2. Case I is proved the same way.

To prove Case IV, note by the previous arguments that Hk(G(k, 4)) mod torsion is generated by cup products of W, and Pt with total degree k and a cocyle Ai which has value 1 on <t> = (1 1)* — (1 1) ' . Then Wt = WtGiWt, P4) 4- (P»)"4 + 2At mod torsion exactly as in Case II, where GiWt, Pt) is a polynomial in Wt and P4. By the duality theorem for Pon­trjagin classes mod torsion, P» = (WtYF(W\, P4) -r »i(P4)*'*mod torsion (since (Wf = P„). By pulling Pt into H*(G(k, 2)) we see that Pt = ul(Pi)tli = ^t1(^r,)*', so «, = 1. Therefore we obtain Wt = 2A» + P* + WtG(Wt, Pt) mod torsion proving Case IV. Case III is immediate from the preceding. This proves 5.1.

In the following theorem integer coefficients are meant. THEOREM 5.3. Let f and g be immersions of M * in E**' with l> lifk

is odd, and with the same normal Stiefel-Whitney class W,. Then the in­duced homomorphisms of the tangential maps t* : H*(G(k, I)) -* H*M) of f and g are same.

The exceptional case referred to in Theorem 5.3, k odd and I = 1, has been studied by Milnor [8]. We will not consider it here.

COROLLARY 5.4. Let f and g be immersions of Mk in Ek+l with I > 1 if k is odd, and suppose that f and g are imbeddings or that I is odd. Then the conclusion of 5.3 holds (i.e., without any assumption on W,).

That the corollary follows from the theorem may be seen as follows. First, if I is odd then Wt = S*W,.l where 8* is the Bockstein operator and Wi-i does not depend on the immersion, hence Wt also does not depend on the immersion. On the other hand, if / is an imbedding W, — 0 (e.g., see Chern-Spanier [4]).

REMARK. All the results obtained in this section for immersions of M* in E"*1 can be generalized to the case where M* is immersed in E**'*' with a field of normal p-frames. Here the induced map is from M* to

101

C T f t

580 R. LASHOF AND 8. SMALE

VI*,'*' (see Section 4). By 4.4, H'iViX1*') is naturally isomorphic to H'(G(k, D) for» £ k.

We now prove 5.3. We need the following well known lemma. LEMMA 5.5. Let X be a space such that H*X; Z) has only order 2 tor­

sion. The torsion subgroup of H*(X; Z) is 8*(H*-\X; ZJ) where 8* is the Bockstein operator.

PROOF. Let u e C*'\X) be an integral cochain which is a cycle mod 2, i.e., 9u = 2c where c e C(X). Then since 88u = 0 = 2dc, 8c = 0, and c is a cycle representing an order 2 cohomology class.

Conversely, let c be any chain representing an order 2 cohomology class. Then 2c = 8u for some u e Ck~\X). But then u is a cycle mod 2.

We consider for the proof of 5.3 the following diagram :

H'(G(k, l);G)^-- Hr(G(k, ~ ) ; G) (°) L

mG(l, k) i G)S-H'(G1, «,) ; G) Here »* and »'* are induced by the inclusions » and i'.

LEMMA 5.6. JfG — Z%, then i* is onto for r < k. PROOF. For r < k, by 4.2, i'* is an isomorphism onto (this is also a

well known fact). Thus, cup products of W, mod 2, j = 1, • • •, J, generate Hr(G(l, k); ZO for r < k. Then cup products of a*W, = W, generate Hr(G(k, I): Z^. By the Chern cup product formula (see the end of Sec­tion 4) these W can be written as polynomials in the classes W, of G(k, I). Since H*(G(k, oo); Zt) is generated by such classes, this proves 5.6.

LEMMA 5.7. For r < k, Hr(G(k, I); Z) mod torsion is generated by the image ofi* and the class Wt = a*W, where W, is the Jth Stiefel-Whitney class of G(l, k).

The proof is Bimilar to 5.6 and uses the diagram (D). For r < k, Hr(G(l, k); Z) mod torsion is generated by JF, and Pontrjagin classes P. Then one uses the theorem that a*P = Ps mod torsion can be expressed as polynomials in the Pontrjagin classes P mod torsion of G(k, oo) (see [6, Bemerkung p. 68]). Then since H*(G(k, oo); Z)mod torsion is gen­erated by these Pontrjagin classes and Wt, this proves 5.7.

LEMMA 5.8. For r<k, Wt and the image ofi* generates HT(G(k, l); Z). PROOF. Let gu • • •, g», h, • • •, h* generate Hr(G(k, I); Z) where the g,

are free and the h are of order two. Then by 5.5 and 5.6 h, = 8h', =

102

ON THE IMMERSION OF MANIFOLDS 681

8i*dk„ where h' e HT~\Gk, I); Zt) and k, e Hr"l(G(&, «>); £,)• Hence h, is in the image of i* for each j . On the other hand, from 5.7 it follows that the g are generated by Wx and the image of i*, proving 5.8.

Until this point in the proof of 5.3 we have not significantly used our previous results.

For the proof of 5.3 we have only to consider the case r = k. If A; is odd and I > 1, by 4.2 and using the diagram (D), the same argu­

ments as above go through to yield 5.3 for this case. We now suppose k is even. Since M" is oriented we can assume

H"(M) = Z (otherwise H\M) = 0 and there is nothing to prove). From 5.1 one obtains immediately

LEMMA 5.9. If k is even and Wk+1 in H^Gik + 1,1)) is zero, then H*(G(k, I)) mod torsion is generated by the image of i* and W,. Then in this case we have 5.3.

Lastly we prove 5.3 when k is even and Wk+l ^ 0. Since H*M) = Z, we ignore toreion. By 4.2 we obtain that H"Gk, I), Z)moA torsion is generated by Wk = a* Wk, Wt = a*W, and P, = a*P,. Then by the rea­soning of Lemma 5.7 we obtain 5.3.

Lastly, we prove a theorem of Kervaire [7]. Let M* = M be a closed oriented manifold, k even, and let f: M-*- E**1

be an immersion with a cross-section in the bundle of normal l-frames. Then f induces a map <p:M-*- V,*,,, of M into the SHefel manifold [1], Let the induced homomorphism be denoted y% : Hk(M) -* Hk( V,+M).

THEOREM 5.10. There is a generator v of Hk(Vl+kl) = Z such that <Pi,(M) = l/20«v where £lt is the Euler characteristic and M the funda­mental cycle of M.

PROOF. Consider the commutative diagram

Hk(G(k,l))-^Hk(G(l,k)) where t and a are the tangential map and dual homeomorphism respec­tively, both defined earlier. Choose v by Theorem 4.1 so that P*(v) can be represented by (0 k)+ - (0 k)~. Let ?*(M) = nv and let W be the Stiefel-Whitney class in H*(G(k, I)). Then

W'lccMnv)] - IP W « P „ ( J f ) ] = W%(M) = t*W'(M) . On the other hand

103

582 R. LASHOF AND S. SMALE

W'[a+P+(nv)] = nWkamP+(v) = na*W*P*(v) = nW*[(0 A;)* - (0 k)~] = 2n .

Thus n = X2,/2 and the theorem is proved. REMARK 1. By similar techniques one may prove Theorem 5.3 for

coefficients modulo 2. We do not include a complete proof, but simply remark that it is well known that the cohomology ring of the Grassmann manifold Gk, I) of non-oriented A-planes in k + I space is generated by the classes Wu ••-, WK modulo 2 (see [2]). Since the manifold Gk, I) of oriented planes double covers Gk, I), one may use the Gysin sequences modulo 2 of this zero sphere bundle to obtain that H\Gk, I), Z,) is onto, %-g.k, if and only if W»+l=£0 in H*(G(k+l, I), Zt). It then follows that H*(G(k, I), Z,) is generated by Wt, • • •, Wk for dimensions ^ k ex­cept for the special cases already considered for integral coefficients. We thus obtain 5.3 for coefficients modulo 2.

REMARK 2. We have shown that the only invariant able to distinguish immersions of Mk in E**1 obtainable from the homomorphism t*: H*(G(k,l))^H*(M*)\B TV",, except for k odd and 1=1. Furthermore by duality Wt is determined mod 2 by the Stief el-Whitney classes of M. We have the problem, given a class r of H\M*), when is there an immersion of M* in E*+l with W' - r? If I - k, Theorem 3.3 says this is possible for all cohomology classes of H\M) not excluded by duality.

BIBLIOGRAPHY

1. A. BOREL, Selected topics in the homology theory of fibre bundles, University of Chicago, 1954. (mimeographed notes).

2. S. S. CHERN, Topics in differential geometry, Princeton, 1951. (mimeographed notes). 3. , " La geom6trie des son-variet^s d'un gspace euclidien a plusieurs dimensions "

in L'Enseignement Mathlmatique, March 1955, pp. 26-46. 4. , and E. SPANIER, A theorem, on orientable surfaces in four-dimensional space,

Comm. Math. Helv., 25 (1951), 205-209. 5. S. ElLENBERG and N. STEENROD, Foundations of algebraic topology, Princeton Uni­

versity Press, 1952. 6. P. HIRZEBRUCH, Neue topologische Methoden in der algebraischen Geomctrie, Sprin­

ger, Berlin, 1956. 7. M. KERVAIRB, Gowrbure integrate generalise et homotopie, Math. Ann., 131 (1956),

219-252. 8. J. MlLNOR, On the immersion of n-manifolds in (n+l)-spaee, Comm. Math. Helv.,

30 (1956) 275-284. 9. N. STEENROD, Topology of fiber bundles, Princeton University Press, 1951.

10. R. THOM, Qvelques proprHUs global** des varUUs diff&rentiables, Comm. Math. Helv., 28 (1954), 17-86.

104

ON THE IMMERSION OF MANIFOLDS 583 11. R. THOM, Espaces fibres en spheres et earres de Steenrod, Annales de L'Ecole

Normale Superieure, vol. LXIX fasc. 2 (1952) pp. 109-181. 12. H. WHITNEY, lite maps of an neomplex into an n-sphere, Duke Math. J. 3 (1937),

51-55. 13. , The self-intersections of a smooth it-manifold in 2n-space, Ann. of Math.,

45 (1944), 220-246. 14. WENTSUN WU, Sur les caracteYistiques des structures fibrees spheriques, Act.

Sci, Indus. No. 1183. Paris (1952). For more general results on self-intersections see:

H. WHITNEY, On the topology of differentiable manifolds, in Lectures in topology, Wilder and Ayres, Michigan, 1941.

R. K. LASHOF and S. SHALE, Self-intersections of immersed manifolds, J. Math, and Mech., to appear January 1959.

105

106

Self-intersections of Immersed Manifolds

R. K. LASHOF & S. SMALE

Communicated by E. SPANIER

Introduction. An immersion / : Af* —> Xk*' of one C manifold into a second is called n-normal if for each n-tuple of distinct points x, , • • • , x. of Af with /(xi) = • • • = /(a*,) the images MZi of the tangent spaces MIt under the differ­ential of / have the minimum possible intersection in X„ , y = /(x.)- Explicitly, it is required that dim f*Y_, Mx( = k — (n — l)r. We will prove that any immersion of a closed manifold can be C. approximated (any 5) by an n-normal immersion.

If /: Af —> X is n-normal and (/)": (Af)" -* X)" is the n-fold product map of /, then the restriction F of (/)" to the subspace of distinct n-tuples of the n-fold product space (Af)" is (-regular in the sense of THOM [7] on the diagonal A of (X)\ Then F-1(A) = 2 . is a manifold of dimension fc — (n — l)r which we call the n-self-intersection manifold of /. The reason for this terminology is as follows. Let ir, : (Af)" — Af be the projection onto the first factor. Then ir, restricted to 2 . is an immersion ir, : 2„ —> Af and ir, (2„) is the set of points of Af which are mapped n (or more) to one by /. That is,

Ti(2,) = x c Af; 3 *a , ••• , xn c Af, distinct, and /(x) = /(x<).

If Af is closed then 2 , is closed; if Af and X are orientable then so is 2 , . Assume now that Af, and hence also 2„ , is closed. Denote the image of the

orientation of 2 . under TM : Hk-iM-1)T(En) —* Hk-ln-l)T(M) by 2 t (use coefficients from Zt if 2B is non-orientable, otherwise use integer coefficients). We are able to compute this class as follows: Consider the composition

Ht(M) A Hk(X) ± Hr(X) A HTM)

where X is Poincare" duality (using cohomology with compact supports if X is not closed). Let Af denote the orientation of Af and r = /*X/t(Af). If Af is non-orientable then T is defined with coefficients from Z7 . Let W' eZT(Af) denote the rth Stiefel-Whitney class of the normal bundle of Af in X. I t is an integral class unless this normal bundle is non-orientable or r = 1, in which cases it is defined with coefficients from Zt . We will prove

143

Journal of Mathematics and Mechanic*, Vol. 8, No. 1 (19S9).

144 It. K. LASHOF & g. SMA.LR

Theorem Ix If f: Mk —* Xk*r is an n^normal immersion of a dosed, connected manifold M, then (FO X2* = ±(r - Try".

Here X is Poincare" duality and the product on the right is the cup product. The formula is meant with integer coefficients unless r = 1- or M or X is non-orientable in which cases coefficients are taken mod 2. (The sign on the right will be (_i)*<—»)<*+'> for o u r choice of orientation of 2m .)

WHITNEY in [10, p. 131] has given an early formulation of equation (F„) in the orientable case when n = 2 and X is closed. In place of our r he has the symbol AT, M) which he defines geometrically. In place of 2 2 , WHITNEY writes D(M), "the distant intersection of M with itself". He speaks of D(M) as a submanifold of M; however, D(M) may only be an immersed manifold in general.

In case Xk*T is Euclidean space Ek*T, it is an immediate consequence of its definition that T = 0, and we get, from Theorem I:

If Xk+T = E"*r, then (I*\) becomes

(FO X2t = ±(WT"-In particular, the set iri(22) is the set of all points of M which are not mapped

1-1 by /, and we have (FJ) X2*2 = ±W\ (For the case of (F2') where r = k, see also [4] and [11].)

Since (Wr)2 = P'r, r even, is the normal Pontrjagin class of the immersion (see [12]), we have (FO X2? = ± P J r , r even. In the case r = $A;, 2S consists of isolated points, i.e. triple points, and F2* t Hk(M) is determined by duality theorems from the Pontrjagin classes of M [A]. Thus

Corollary A. The number of triple points of an immersion f: Mk —> Eik of a closed manifold oriented with proper sign is independent of the immersion.

(The sign of the triple points is determined as in 1.2. I t is easy to obtain from this a geometric rule for attaching a + or — sign to an isolated triple point depending on how the pieces of M come together.)

This behavior of triple points contrasts with the number of double points of an immersion of Mk in Eik, see [4] or [11].

The following example shows that the preceding theorems have some content. M. HIBSCH [2] has proved that the real projective 6-space P* can be immersed in E7. From the Whitney duality theorem and equation (F£) follows

Corollary B. For every 7-normal immersion f of P* in E7, there exists y t E7

such that f~l(y) has at least 7 points. If /: Mk —* Xk*r is 2-normal and at most 2 to 1 e.g., if r > Jfc and / is 3-normal),

then 2S is imbedded in M. The immersion / restricted to Ti(2a) = 2, C M is a double covering. Let 2, be the image space /(2»). We will prove

107

108

SELF-INTERSECTIONS OF IMMERSED MANIFOLDS 145

Theorem II. If /: Mk -* Xk*' is Znormcd and at most 2 to 1, then 2» is a non-orientable manifold when r is odd and an orientable manifold when r is even.

In most cases it is very difficult to prove imbedding theorems from immersion theorems. However, from Theorem II it is very easy to prove the special theorem

Theorem III. If k is even, then near any immersion of Mk in E**'1 C E**, there is an imbedding of M in E**.

WHITNEY has proved that every manifold Mk can be immersed in E""1

(for a modern proof see HISSCH [2]). We would like to thank E. SPANIER for several helpful conversations during

the preparation of this paper.

1. Normal immersions. If M —* Mk is a manifold of dimension k, Mt will denote the tangent space of M at x. All manifolds considered will be C" with a countable base, but not necessarily closed or connected. The following definition of t-regviar has been used by THOM [7]. If N = 2V* is a submanifold of X = X%

and /: M — X is a C" map, / is said to be ^-regular on N if the composition Mt A JT. -> XJN,

is onto for all y z N and x t f~\y). If / is ^-regular on N, then Q = f~1(N) is a submanifold of M of the same codimension (n — q) as N. Moreover, / induces a map of a tubular neighborhood of Q onto a tubular neighborhood of N which can be considered as a bundle map of normal vectors over Q into the space of normal vectors over N. (For this purpose one uses a Riemannian metric and constructs the tubular neighborhood from geodesies orthogonal to the submanifold and hence corresponding to normal vectors.)

A C" map f: M —*X is an (r, t)-approximation of another C map f:M—*X if, roughly speaking, / ' and the first r derivatives of / ' are within « of / and the first r derivatives of / respectively. See WHITNEY [9] for a precise definition. WHITNEY calls such a map an (/, rj, M, ^-approximation.) An n-tuple, of C maps, f\ : Af — X, i = 1,2, • • • , n, is an (r, ^-approximation of a second n-tuple f : M —»X, if, for each t, f' is an (r, ^-approximation of f(.

The following theorem can be found in THOM [7]. See also WHITNEY [9] for the case of an immersion.

Theorem 1.1. Let f: M -* X be a C" map, N a submanifold of X and « > 0. Then there exists an (r, ^-approximation (any r) of f, say / ': M —* X, such that f is t-regular cm N and f agrees with f except possibly on f~\V,(N)), where V.(N) is an e neighborhood of N.

The basic class of a closed connected orientable manifold M — Mk is a choice of generator of Hk(M). We usually denote this class by M itself. The basic class of a closed connected non-orientable manifold M is the non-zero element of Hk(M, Zi). The basic class of a non-connected closed manifold Mh is the direct sum of the basic classes of the components, if each component is an orientable manifold; otherwise it is the direct sum of the generators of Hk(M, Zt).

146 R. K. LASHOF & S. 8MALE

Lemma 12. Let f: M -* X be C" and t-regidar on a manifold N C X, where M, X and N are oriented. Then the orientations of M, X and N determine an orientation of Q = f~\N).

Proof. The map f:M ->X induces / , : M, -* X, , q t Q, p = fq) t N. That / is ^-regular means the quotient map MJQt —* X9/N, is an isomorphism onto. From the orientations of X, and Np we get an orientation of XJN, and hence MJQ, . Then, since M, is oriented, an orientation of Qt is determined. ThiB in turn gives an orientation of Q. (To be precise, take an ordered basis for N, , the order determined by the orientation of N, and extend it to an ordered basis for X compatible with the orientation of X. The image of this basis in XJN9 is an ordered basis which determines the orientation of the factor space. An ordered basis for Q is then chosen so as to give the same result for MJQt .)

An orientation of a manifold determines a basic class in a natural fashion. Under the conditions of 1.2 we will always suppose Q is oriented as in this proof, and therefore the basic class of Q is determined.

The foEowing theorem is known.

Theorem 1.3: Let f: M" — X' be a proper C map, t-regular on a closed sub-manifold AT*"* C -X", with M and X connected. Let Q = f (N) and write N* and Q* for the images of the basic classes of N and Q in X and M respectively. Then

XQ* = f*\N*.

Here integer coefficients are meant in the case that M, X, N and hence Q (by 1.2) are oriented. Otherwise coefficients are to be in Zt. The symbol X always stands for Poincare" duality between singular homology and Cech cohomology with compact supports, in both M and X.

Proof. Let V0 and V'Q be tubular neighborhoods of Q and N respectively, and, using a Riemannian metric on M and X, we consider /: V0 —* V'0 to be a bundle map. Consider the following diagram (cohomology with compact supports and singular homology):

Hn_q(Q) < A - * H°(Q) <-l— H°(H) <e-2-> Vq(N)

I , I , I H ^ M ) +-*L+ H*(M) +J— H*(X) * — * _ , H p_q(X)

109

110

SELF-INTERSECTIONS OF IMMERSED MANIFOLDS 147

Here <j> and <£' are isomorphisms as in THOM [6], and the other vertical maps are all inclusions. The subscripts on X indicate duality in the different spaces. Various naturality theorems show that each smaller square commutes e.g. see [1] and [6]). The theorem is proved by commutativity of the large rectangle: Starting from the upper right hand corner, the basic class of N is mapped across the top row onto the basic class of M by 1.2 and the fact that f* takes the sum of the generators of H°(N) onto the sum of the generators H°(Q). These basic classes then map down into N* t H^^X) and Q* t Hm-,(M) respectively. Commuta­tivity then implies that f*\xN* = X*Q*.

An immersion f: Mk —* Xk*T is a C" map with Jacobian of rank k at every point. A property P of n-tuples of immersions fi:Mi—*X,i = 1, • • • , n, with M closed, will be called generic (after THOM [8]) if both the following conditions hold:

(1) Given an n-tuple /< : M< —* X, there exists an « > 0 such that every (2, «)-approxunation of (/< has property P.

(2) Given any n-tuple /<: Af < —♦ X and any e > 0, there exists an (r, «)-approx-mation of /, by an n-tuple with property P . (That the M( be closed is not necessary, it only simplifies matters.)

In the case n = 1, generic describes properties of immersions, and this case will be our main concern. We proceed to describe one such property; the proof that it is generic will be given later (1.5).

If /: M — X is an immersion of one manifold in a second, let Mx denote the image of M,, x t M, under the differential fm of /. Then an immersion /: Mk —* Xk~r

is called n-normal if, for any n distinct points x , , • ,x%zM with /(xi) = • • • = ffa),

n

dim C\Mxt = k-n- l)r.

Instead of 2-normal we may say simply normal. An immersion is completely normal if it is n-normal for every n.

Let (Af)" denote the n-fold Cartesian product of M with itself, A the diagonal of X)n and R = fa , • • , x*) t (M)* \ xt = x,- for some distinct t and ;'.

Lemma 1.4. The n-fold product map of an immersion f: M —* X, restricted to (M)n — R, is t-regular on A if and only if f is n-mormal.

Proof. Let F: (M)n -R^> (X)" be the restriction of (/)". The map / is n-normal if and only if dim (~\ Mwt = k — (n — l)r for all n-tuples Xi , • • • , x. of distinct points of M with ffa) = • • • = ffa). Let xx , • • • , x. be such an n-tuple with f(Xi) = y. Let V be the set of vectors Y of (M)U) , (x) = fa , • • , x„), with F.Y t A(f, , (y) = (y, y, • •_• , y), and let V = Fm(V). Then dim r\ M„ = k -(n — l)r if and only if dim V = k — (n — l)r. Since Fm is an immersion, dim V = dim V. Now F is t-regular on A at fa , • • • , xn) means that nk — dim V = n(k + r) — dim A = (n — l)(k + r) or dim V = k — (n — l)r. Putting these statements together we obtain 1.4.

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148 R. K. LASHOF & S. SMALE

If / is n-normal (using 1.4) we define 2 , = F~1(A), where F: (Af)* - 22 - (X)' is the restriction of the n-fold product map, to be the n-selfintersection manifold of /. It has the following properties:

(1) 2„ is a manifold of dimension k — (n — l)r. (2) If x: (Af)" —♦> ikf is projection onto the first factor then x restricted to

2„ is an immersion.

Proof. We will show that the differential x^ is 1-1. Let x: (Af)* — R — M and x': (X)" — X be projections onto the first factor. Then -K'F — fr. Now F(2«) C A and x ' restricted to A is 1-1. Since F+ is 1-1 this implies x„ is 1-1.

(3) x(2„) C M is the set of points of M which are mapped n or more to 1 by /. In particular, x(2a) is the set of points at which / is not 1-1.

(4) If r > k/n then dim 2n+i < 0, and hence / is at most n to 1. (5) Let p . : 2 , -> 2„_1, p , ( i , , • • • , x„) = (x,, • • • , x._x). Then p. is 2-normal,

if / is (n -f l)-normal. Denote by KK the self-intersection manifold of p» (.K, C 2 . X 2„). Then there exists adiffeomorphism an:Kn—> 2»+1 such that ira, = icy, where 7 is projection onto the first coordinate of 2„ X 2 . followed by x: 2 , —> Af. In particular, if 23 is empty then 22 is imbedded by x in Af.

Proof. The fact that p . is 2-normal is easily checked. To define a. , let [(x, , • • • , x„), yi , • , y»)] t K% . Then

a . [ (x , , ,xn),(Vi , •• , y»)] = (*i , • • • , * . , ! / - ) •

Note that since p.(x, , • • • , x.) = p,(y, , • • • , yn), x< = y4 , i = 1, • • • , n - 1. The details are left to the reader.

(6) n-normal is generic for closed Af. (See 1.5 for proof.) (7) 2,, is a closed subset of (Af )*, and hence 2„ is closed if Af is closed. (8) 2 , is orientable if Af and X are orientable. (This follows from 1.2.)

An n-tuple of immersions /< : Af J' —* X', i = 1, • • • , n, is called mutually normal if for each n-tuple of points xx , • • • , x„ , such that x< e Af < and /L(Xi) => • • • = f„(x„), dim 07-> U* (M.i) = Z.- * « - ( * - ««

Let /: Af — X be an immersion and i: N — X an imbedding of a submanifold in X. Then the following are equivalent:

(1) / is i-regular on N. (2) / and i are mutually normal.

The proof is very easy and left to the reader.

Theorem 1.5. The property n-normal is generic for immersions of closed namir folds.

Proof. We first show that n-normal satisfies property 1 of generic. If / ' is close to a given n-normal immersion f: M —* X then the n-fold product map of / ' will be close to the n-fold product of /, and hence F' will be close to F (the

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restrictions of the n-fold product mapa to (M)* — K). By 1.4, F is ^-regular on A, and it follows (see THOM [7]) that F' will be also, since any map sufficiently close to a f-regular map is again J-regular. Application of 1.4 again yields that / ' is n-normal if it is sufficiently close to /.

We next prove that 2-normal satisfies property 2 of generic. The proof will then be finished by induction. Let /: M —* X be a given immersion of a closed manifold M. For t = 1, • • • , m let Wt , Wi be open sets of M such that / is 1-1 on W,, Wt C W't, and VJ Wt = M. Given t > 0, choose e<, i - 1, • • • , m, such that £ r . i «. < « and «,_ < i d(X - f(WQ, f(Wt)), d being the distance function in X. Let fi :M — Wx —* X be given by Theorem 1.1 where we take «t for i, f(Wi) for N, and f\u-w, for the given map. Then fi = / except on rl(Y.x(J(Wi)); in particular, f[ = / on x e W[ \ d(f(z), f(Wx)) £ « and d(/(x), * - fWD) £ e,. Hence if we define /, = / on \x t Wi | d(/(x), X - f(Wfi) ^ t, and /i = /[ on the complement of this set, fx will be a C map of M into X. The immersion fx will have the property that f,^, and fx\u-», are mutually normal. For some «£ > (r» ^-approximations of fx will also have this property and will still be 1-1 on Wx . Define / 2 : M —* X in a fashion similar to fx using rain («2 , «£) instead of ei and Wt in place of Wx . Iteration finally yields an immersion fm:M—>X such that fm\wt and fm\U-«( are mutually normal and /„ is 1-1 on W , i = 1, • • • , m. I t follows that fm is an (r, «)-approximation of / which is 2-normal.

Proceeding by induction, let /: M —* X be an (n — l)-normal immersion. It is sufficient to find an (r, ^-approximation of / which is n-normal. For the proof we need the following lemma.

Lemma 1.6. Let g: V —> X be an immersion and g': N — X an imbedding. Then there exists an (r, ^-approximation gi: N — X of g' such that g and gi are mutually normal and g[ is given by hg' where h: X —* X is a diffeomorphism of X equal to the identity outside any prescribed tubular neighborhood of g'(N).

Proof. This is essentially in THOM [7]. (Compare the above lemma with 1.1.) In fact, THOM'B result gives a diffeomorphism hx : X —> X which is the identity outside a prescribed tubular neighborhood o£ g'(iV) and such that hxg: V —* X is ^-regular on g'(N). As noted above, this means that hxg and g' are mutually normal. But since h = hx* is a diffeomorphism, g: V — X and hg' : N — X are again mutually normal, and again h is the identity outside the prescribed tubular neighborhood of g'N).

Returning to the proof of the theorem, let W , W\ be as before and let S,_! be the (n — l)-self-intersection manifold of /. By the above lemma we can find an (r, e^-approximation fx to ft Wl by means of a diffeomorphism h: X —* X which is the identity outside of Vi,(/(TTl)) such that fx = A/, r , and fx: 2,_, -♦ X are mutually normal. Now hf = / on x t W'x \ d(/(x), f(W0) ^ tx and d(/(x), X — f(W[)) > ti. Hence we can extend fx to M by letting fx «= / on the complement of [x z Wi | d(/(x), f(Wx)) > «,. Then fx has the following property. Suppose x, , • • • , x„ are distinct points of M such that fx(xx) = • • . = •

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150 R. E. LASHOF & S. SMALE

fi(xn), and that, for some i, x4 tWx, at most one x t W[ since /»is 1-1 on W[. Hence (xx , • • • , £v, , • • • , x„) t 2._, , since fi(x,) = /(a:,), ; * t. But the mutual normality of fUWl and fx: 2«_I —* X implies that

dim H UM,,) = dim H KM.,) H U(MXI) J - l i*i

= k - (n- 2)r + k - (k +x) = k - (n - l)r.

This property is equivalent to saying that the map Fx , the cut down of the n-fold product map of /, to Wx X (M — Wx)n~l — R, is f-regular on the diagonal A of (X)*. But for sufficiently small 4 > (?, ^-approximations of fx will again have this property and will still be 1-1 on Wx . We now proceed by induction on i, obtaining a map /„ which will have the desired property for any n distinct points since U W< = M and /„ will be 1-1 on each W . This completes the proof of the theorem.

Theorem 1.7. Mutually normal is generic for n-iwples of immersions of a closed manifold.

Proof. Property 2 of generic is satisfied as follows. Let /; : M — X be an n-tuple of immersions where each Mf = M, i = 1, • • • , n. Let M* be the disjoint union of the M( and F: M* —> X be denned by F(z) = fi(x) where n l , - . By Theorem 1.5, there exists an (r, «)-approximation G: M* —* X of F which is n-normal. Let f't : Mi —* X be the restriction of G to M . It is immediate that f[\ is an (r, <)-approximation of /, and is mutually normal. Property 1 of generic is proved in a similar fashion.

2. The main theorem. The goal of this section is to prove Theorem I of the introduction. For simplicity, we will assume M and X are connected in the rest of this paper, also that M is closed. If /: Mk —* Xk*T is any continuous map of the manifold M into the manifold X, we denote the image of the basic class M under the composition

HkM) A Ht(X) -*♦ H'(X) A H\M) -^ H*_,(.li)

by y(j); i.e. y(j) t Hk-,(M). If M or X is non-orientable it is defined with coeffi­cients Z2 and otherwise with integer coefficients. We denote the dual of the cup product in the manifold M by °; i.e. Mx,yt H^(M), x ° y = X(Xx W \y). Theorem I may then be stated as follows:

Theorem 2.1. If f: Mh —* Xk*T is an n-normal immersion of a closed connected manifold M in a connected manifold X, then the n-self intersection class

2* = ±(7(/) - XTT)"-1,

where WT is the normal Stiefel-Whitney class, X is duality, y(J) is the class defined above, and the product is ° product.

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SELF-INTERSECTIONS OF IMMERSED MANIFOLDS 151

We start the proof of 2.1 by choosing immersions f: Af —»X, i = 1, • • • , n — 1, (r, ^-approximations of /, by 1.5 and 1.7 (we will choose the e later), such that /, f, ••• , f'1 are mutually normal and each is n-normal. Write f = /.

Let V0 be the normal bundle of vectors of magnitude < S. Then Va can be immersed as an (immersed) tubular neighborhood of M in X. Let /0 : M —* V0 be the zero-cross-section. Further, if j0 : V0 —* X is the above immersion, let /o : M — V0 , i = 1, • • , n — 1 be the cross-section uniquely defined by the condition

Jo/o = /'

(thia is well defined if / ' is sufficiently close to /). In order to use a unified notation, we set /; «= /, i = 0, • •• , n — 1, and Vx «= X. Then if we let a = (« t , • • • , «»_i) where.«, = 0 or 1 we define F. : (Af)" ->M XVm, where Va = Vtl X • • • X V...t by Fa = i X /J, X • • • X f £ , (i: Af - Af the identity). Further, we define 0, : (Af)" - * M X 7 . b y G . = t X f!, X • • • X £,_. .

Now let r „ C i f X 7 , be the image of Af under the composition

Af A (Af)»A- Af X F a ,

where d is the diagonal map.

Lemma 22. Fa is t-regular on Ta .

Proof. Let i X jm : Af X F„ - Af X (X)"-1, j . = ;., X • • • X ;'.._, , j0 : V0 - X, j , : X - X the identity. Then if (1) = (1, • • • , 1), r (1 ) = (i X ja) Tm and i^d) = (t X ja)Fa . Since t X ja is an immersion, it follows that Fa is ^regular on T. , for all a, if and only if F(U is i-regular on r ( l ) . Now if (x) = (io, • • • , *._,) t (Af)" and Fcx)(x) t r (1) then /°(x0) - /!(xi) - • • • - /"" ' (^- i ) . Let 7(.) be the subspace of (Af)"x) such that V(M) — F^] (r<i))<,) , (y) = (x0 , f(xi), ••• , /""l(x«_i)). We see immediately that dim FWVW = dim CX~-\ f(M.t), and hence

dim VM = dim f\ /'(Af,,) = ft - (n - l)r.

But then

dim (Af)u> - dim 7 t „ = ftn - (ft - (n - l)r) = (n - l)(ft + r),

and

dim (Af X (X)"_1)(,) - dim r (1 ) = ft + (n - l)(ft + r) - ft = (n - l(ft + r). Thus since Fm is an immersion, it is t-regular on r (1 ) .

At this point we stop to review some facts concerning the "slant operation" (see [5]). If X and Y are two topological spaces, then, using singular chains and cochains, let z t C"(X X Y), y e Cr(Y); then we define a cochain z/y t C~'(X) by the property

z/y(x) = z(x (g)y), i t C,_r(X).

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152 R. K. LASHOF & S. SMALE

Then

6(z/y) = &z/y + (-ly—Wdy;

and hence we get a pairing H'(X X Y) (x) Hr(Y) -+ H'-'(X), w (x) v - w/v, w z H'(X XY),vz HrY), w/v z F " " ( I ) , which again has the property

(A) w/v(u) = w(u®v), uz H„.r(X).

Here we use singular homology and cohomology with coefficients in any groups 0, (?, , Gt for X X Y, X, Y respectively; and then w/v will have coefficients in G ® G, and the Kronecker index in (A) will have values in G ® Gi ® G* -

Lemma 2.3. Let u t H'(X; Z) be an integral cohomology class of a space X. Then u is uniquely determined by its values (Kronecker index) on the homology classes x z HV(X; G), where G runs through Z„ (integers mod n), for all n, as well as Z.

This lemma is well known and we leave it to the reader to check.

Lemma 2.4. (Naturality Lemma.) Let f: X —* X', g: Y — 1"' be continuous maps. Then for u z H'(X' X Y'), y z Hr(Y)

f*(u/g*y) = [(/ X g)*(u)]/y.

This lemma follows immediately from the characterization (A) above and Lemma 2.3.

Lemma 2.5. Let X" and Y" be closed connected oriented manifolds and X be Poincare duality inX, Y or X X Y as the context indicates. Let /Y: H"(X X Y) — H*~m(X) and x t . : Hn+m.v(X X Y) —* Hnt.m.v(X), where wi. is projection onto the first factor and Y z Hm(Y) is the basic class. Then \-(/Y) = Tt.X.

Proof. Let u t H'(X X Y), x t H"+"-'(X; G). Then

\(u/Y)(x) = u/Y(\x) = w(Xi (x) Y) = \u(\(\x (x) K))

= Xu(x (X) 1) = Xw(ir (i)) = ir,.Xu(i).

Note that 1 e H°(Y) and hence the sign here is + 1 . By 2.3 our result follows. We now summarize some known facts (see CARTAN [1]) concerning duality

in a paracompact connected manifold X. The coefficients are in an arbitrary abelian group if the manifold is oriented and integers mod 2 if it is non-oriented.

Let H*(X) be locally finite singular homology supports in <t>, Hl(X) cohomology with supports in 4>.

(a) If 4> = C = all compact subsets of X, then HC,(X) = ordinary singular homology.

(b) If ^ = F = all closed subsets of X, then H;(X) = ordinary singular cohomology.

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SELF-INTERSECTIONS OF IMMERSED MANIFOLDS 153

(More generally, since X is H.L.C., Hl(X) is singular cohomology with supports in*.)

If X is n-dimensional, we have Poincare' duality:

(c) K:Hl(X)^Ht-p(X).

The fundamental class X of the manifold is the image by X, of a generator of H?(X), X t H"„(X) (integers or integers mod 2). From the cup -product

we get the cap product

(d) Hl(X) (g) H*t'(X) -+ ld'(X).

Taking <f>' = F, 0 H F = <t>, and X i H'n(X) we get from (d)

Hl(X)®X-+Ht-,(X).

This is precisely ^ [1, p. 88]. Further, the cup and cap products have the following properties: (e) Let u i HlX), v t H'r(X), x e H'P.,(X); then

(v VJ u)x) — u(v r\ x) (Kronecker index),

u(uH x) = (uUt>)(z).

(f) Then, for u t HrcX), v c Hy(X),

vn-\\cu") = (-l)pin-p)u'(\rvn-').

Proof.

v"-p(\cuv) = o"-'(u' H X) = (u' VJ t>-")(X) = ( - 1 ) ' ° — V " W *)(*)

= ( _1 ) "—> M V~ '^* ) = (-1),("-'V(X^""') Consequently, if z z H^,(X), y t H'P(X),

x;1!/ ,^-,) = (-D,tn-p)Xclx.-p(j/,).

(g) Let im : H%X) -> H',[X), i*: H;(X) -♦ ff;(X) be the natural maps; then

i^Xc = Xj , i * , i*Xc l = X ^ l i # .

Now let M be a closed connected manifold and /: M —* X a continuous map. Then if f% : HZ(X) -* tf \M) and /> : ff;(X) -> H'M) are the induced maps (with similar definitions for f° and /£),

(n) /?x* = n, iji = / ; . Finally, from (f), (g), (h) we get the following lemma:

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154 R. K. LASHOF & S. SMALE

Lemma 2.6. Let M and X be connected manifolds, M conpact. Letf,g:M-*X be continuous maps. Let z t H9(M), u> t Hn-,(M), n = dim X. Then (vrith nota­tions as in (f), (g), (b) above)

mtVA") - (-i)'("""<7SXc7X*). Proof.

n\~cgcAw) = \-cxgcMw) = #(x;7:w)(-i)'(—' = #(x;V»(- 1>' ("* )

= ^(t*Xc7^)(-D,("-" = *(tf»*x;,£w)(-i)'"-) = ^(<7?Xc7»(-D'(""'). lemma 2.7. T ,F? ( r . ) = 7 . - (-l)*("- ,)<** ,>7., o . . . o T,._, wtewy., =

7</.,). Proo/. By 1.2 and 2.2 we have

(1) F : ' ( r . ) = XF*x(ra), or (10 x.Wr.) = x.xFtxcrj. Consider

F* X ^(MXT*) «_*_> H ^ J M ^ X T . , ) J = _ ^ I ) < * « > ( ( H ) » ) -JL-*a^ (a.1)r(0:)B)

H(n'l)r(«) " ^ ( , , l ) r W

The above square commutes by Lemma 2.5. Hence (2) TjXFjxr. = x(F*xr./®r-1 Af). We will now show (3) X(F*Xr./<gr' Af) = 7.. o • • • o y

Combining (1'), (2), (3) we will have our lemma.

Proof of (3). Let x t H t._1)r(M). Then (F*Xra/<gTI M)x) = F*\l\(x (x) M ® Af ® • • • (x) Af)

= FJXG.^-MTx (g) JV/ (x) • - • (g) Af). By Lemma 2.6 this gives

(_l)*<-»«**'>firjXF..(s® Af ® ••• ® Af)(d,Af). Let a = (_i)*«»-»«♦'), Then since / j , is homotopic to /.< = /?< , and again since deg XAf = 0, the above becomes «r(Xx ® /.!X/.,.Af ® • • • ® /* _.X/,._,Jf)(rf#Af)

= <r(Xx U X 7 . , U - U X7.._.)(Af) = «r(Xr.. W • • - VJ Xy.. J ( x ) = <rX(7.. o . . . O Y . . , , ) ^ ) .

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This completes the proof. Le tK, = Gl'Tm). Then

R. = l(x0 , • • • , i»_,)e(A0"l/(^) = /(*o) all i; and if «,, = Othen x,-, = ar0.

Now partially order the a's by defining a < a' if X << < 2 e< an(* e< ^ «« f° r

each i. Let Q„...0 = iJo-o and Q. = Ra — Ua<aQa-. Then (as is easily given by induction)

Qa = (x0 , • , xn-x)t(M)n\1(x.) = /(*<>) all i, a;,-, = x0 tt «,, = 0, a;,., #= Xi if «,, = 1

and the Q„ are closed, disjoint and 2 , = &..., . Since the Q. are closed we can take open neighborhoods T« of the Q„ with

Ta r\ Ta- = 0 if a * a'. Let 7 be an open neighborhood of I\...i such that Gv..i(7) C ^ . r a . This can be done since G1...l((M)n - \JTa) is compact and disjoint from r,.. .!. By our construction of Fx.,.i we can assume that Ft...i and (?»..., agree outside of ^ . . . ( V ) . Let QL = Ta(~\ F;\.,(r,...,). Then Qi is a closed submanifold of (M)\ Further, Qf..., is isotopic to Qj..-i = Sm , since F,..., and (?!...! are ^-regular on IV.., , considered as maps of 7Y..j , at least if Fi...t is sufficiently close to G1...1 .

Now let R'„ = F ^ C r J . Then (A) Q'a = fl'. - U a - < a Q'„. . Further, we have seen that T ^ represents the class ya . Let sgn (a) = + 1 or — 1 according as a has an even or odd number of zeros.

Theorem 2.8. 7r,Q^ represents sgn (a) 2 a ' S a sgn (a')ya- and, in particular, TTIS, represents Sa sgn (a)ya .

This result follows purely algebraically from relation (A) and the proof is left to the reader.

Theorem 2.9. Z*H = <r(7, - -ya)"~l. This follows from 2.8 and a short algebraic argument which we again leave

to the reader. Then 2.1 follows immediately from 2.6 and the next lemma. Lemma 2.7. y0 = \W\ Proof. Consider the diagram

H'CT) 4 1 ^ ( 7 , 7 ) « - £ H. (7 )

t t ' 0

B*(M) < U ^ H°(M) ( * ^ ( M )

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156 R. E. LASHOF & S. SMALE

where V is the normal bundle of /: M —X of vectors with length ^ 5, for some S, and V its boundary. Each of these diagrams commutes according to THOM [6]. The proof of 2.7 is an immediate consequence of this picture.

3. An orientability theorem. We assume throughout this section that /: Mh — Xk*' is a 2-normal immersion without triple points and M is closed. If, for example, / is 3-normal and r > \k then from Section 1 it follows that 2 , is empty and hence we have the above situation. We will identify Ti(22) with 22 since T, is an imbedding (Section 1). Then / restricted to 22 , say /, is 2 to 1 and hence a double covering. Let 22 = /(22). Then if N is a component of 2, , f~\N) can have either one or two components. If f~\N) has two com­ponents, then / restricted to these components is a trivial double covering. This is a relatively simple case. We will now consider the second case.

Theorem 3.1. Let f: Mh —> Xk*T be a 2-normal immersion without triple points, M closed. Let N = / " ' ( # ) be connected where N is a component of 2a . Then (1) ij r is odd N is non-orientable and (2) ij r is even N is orientable.

Before we prove this we note a corollary.

Corollary 3.2. If k > 2 is even and f: Mk — E2k-1 is an immersion, then Mk

may be imbedded by a map h into E2k arbitrarily close to f: Mk -> E2k~1 C Eik.

Proof of corollary. By Theorem 1.5 we may assume / is 2-normal and without triple points. Then 22 is a disjoint union of circles. Since a circle is orientable and r = k — 1 is odd, by 3.1, each component of 22 is trivially doubly covered by /. Let 2V\ , AT, be the two circles of /~l(iV) where N is a component of 22 . Let g be a C" function which is not zero on 2V\ and zero outside a small tubular neighborhood of Nt . Let h: M —> Ek+r** be defined by having the same values as / on the first k + r coordinates but with last coordinate g. Then h has no intersections near 2V\ or N7, and will be close to / if g is chosen with small enough absolute value. Repeating this process gives the desired imbedding of the corol­lary.

We now prove 3.1. Let t: M X M — M X M be defined by tp, q) = (g, p). Then t(22) = 2 2 . We will write 2 for 2 , .

Lemma 3.3. If 2*,,,, is the positive orientation of the tangent space 2 ( , „> , then if r is odd t„(2*,,„) = 27„„) and if r is even ^(2*,,,, ) = 2f,,p) where t+ is the differential of t.

If 3.3 has been proved, then we show 3.1 as follows. Let <f>: M X M — X' where <t> = frt ; then <j>t — 4> and <f> restricted to iV is our double covering. If r is oddthen $**#iV*Pi„ = <f>0N*,,,) = <£*#(,,,) by the above diagram and 3.3. Hence N is orientable. On the other hand, if r is even <f>^N*,_t) = <£»AT7f.»> &n^ so N is non-orientable.

We now prove 3.3. Let t': X X X - X X X be denned by t'(x, y) = (y, x). Then (/ X f) ° t = f ° (/ X /).

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SELF-INTERSECTIONS OF IMMERSED MANIFOLDS 1 5 7

Suppose r is odd. Case L k is even. Then t' reverses the orientation o f l X l and t preserves

the orientation of M X M. Hence t reverses the orientation of 2. (See Section 1 for the definition of the orientation of 2.)

Case IL k is odd. if preserves the orientation of X X X while t reverses the orientation of M X M. Hence t reverses the orientation of 2.

Suppose now that r is even. Case I. k is even. Then t preserves the orientation of M X M and tf preserves

the orientation of X X X, so t preserves the orientation of 2. Case II. k is odd. Then t reverses the orientation of M X M and f reverses

the orientation of X X X; so t preserves the orientation of 2.

REFERENCES

[1] CARTAJT, K , Stmmaire de iopologie algebrique, E.N.S., Paris, 1950-1951. [2] HIBSCH, M., Thesit, University of Chicago, 1958. [3] HIRZBBBUCH, F., Neue topoiogiaehe methoden in der algebraitche geometry, Berlin, 1956. [4] LASHOF, R. & SMALE, S., Immersions of manifolds in Euclidean space, Ann. of Math.,

to appear. [5] STEENBOD, N., Homology groups of symmetric groups and reduced power operations,

Proc. Nat. Acad. Sri. USA., 39 (1953), 213-223. [6] THOM, R., Espaces fibres en spheres et carrfs de Steenrod, Ann. 3d. Eeol. Norm. Sup.,

69 (1952), 109-182. [7] THOK, R., Quelques proprietes globalee des varietea differentiable, Comm. Math. Helv.,

28 (1954), 17-86. [8] THOU, R., Leg singularites des applications differentiable, Ann. Inst. Fourier, 6 (1956),

43-87. [9] WHITNKT, H., Differentiable manifolds, Ann. of Math., 37 (1936), 645-680.

[10] WHrrNZT, H., On the topology of differentiable manifolds, Lecture* in topology (Ed. WILDER <fc ATRES), Michigan, 1941.

[11] WHTTNIT, H., The self-intersections of a smooth n-manifold in 2n-space, Ann. of Math., 45 (1944), 220-240.

[12] Wo-, W I N TSUN, SW lea caraderitliquta dea atruclurea fibreia apheriquea, Act. Sci. & Ind., No. 1183, Paris. 1952.

University of Chicago Chicago, Illinois

A CLASSIFICATION OF IMMERSIONS OF THE TWO-SPHERE

BY STEPHEN SMALE

An immersion of one C1 differentiable manifold in another is a regular map (a Cl map whose Jacobian is of maximum rank) of the first into the second. A homotopy of an immersion is called regular if at each stage it is regular and if the induced homotopy of the tangent bundle is continuous. Little is known about the general problem of classification of immersions under regular homotopy. -Whitney [5] has shown that two immersions of a ft-dimensional manifold in an n-dimensional manifold, n^2k+2, are regu­larly homotopic if and only if they are homotopic. The Whitney-Graustein Theorem [4] classifies immersions of the circle S1 in the plane E1. In my thesis [3] this theorem is extended to the case where £* is replaced by any C* manifold Mn, n > l . As far as I know, these are the only known results. In this paper we give a classification of immersions of the 2-sphere S1 in Euclidean n-space £", n > 2 , with respect to regular homotopy.

Let 7»,» be the Stiefel manifold of all 2-frames in £ \ If / and g are two immersions of 5* in En, an invariant Q(/, g)£ir»(V».») is denned.

THEOREM A. / / / and g are C* immersions of S* in E", they are regularly homotopic if and only if B(/, g) =0. Furthermore let Q»ETi(Vn,t) and let a C* immersion f: S1—*E" be given. Then there exists an immersion g: 5*-+£" such that Q(f, g) =»flo- Thus there is a 1-1 correspondence between elements of*\( V»,t) and regular homotopy classes of immersions of S* in En.

Since T»( Vi.t) = 0 , Theorem A implies:

THEOREM B. Any two C1 immersions of S* in E* are regularly homotopic.

That this should be so, is not obvious. For example, it is not trivial to see that a reflection of the unit sphere in £ ' is regularly homotopic to the identity on the unit sphere.

Since T»( V^t) = Z, there are an infinite number of regular homotopy classes of S* in E*. In fact we are able to obtain using results of [l ] ,

THEOREM C. Given yQH*(S*), y even, then there is an immersion of S* in £* such that the characteristic class of the normal bundle is y. Furthermore, any two such C* immersions are regularly homotopic. There is no immersion of S* in E* with odd normal class.

In [2] it is proved for say 5* in E* that the normal class of the immersion

Presented to the Society, February 23, 1957; received by the editors April 29,1957. 281

121

282 STEPHEN SMALE [February

is twice the algebraic intersection number of Whitney. Hence two C1 im­mersions of 5* in B* (for which the algebraic intersection number is denned) are regularly homotopic if and only if they have the same algebraic inter­section number. This answers a special case of a question of Whitney [6, p. 220].

Finally one can immediately obtain from Theorem A that two immersions of S* in EK, n>4, are always regularly homotopic.

M. Hirsch, using the theory of this paper, has obtained a regular homo­topy classification for closed 2-manifolds in E", n>2.

A slight extension of the methods in this paper yields a generalization of Theorem A to the case where £ • is replaced by any C* manifold M*, »>2 . We state the results as follows.

If M is a Cl manifold, Fj(Jlf) denotes the bundle of 2-frames of Af. Let N be a C* manifold of dimension greater than two, let xoEFiOS1) and let y»£.Ft(N). An immersion/: S*-*N is said to be based at y9 if /»(x0) =yo where /*: Fi(S*)—>Ft(N) is induced by /. A regular homotopy is based at yt if every stage of it is. If / and g are two immersions of S* in N based at y«, then an invariant Q(/, g)Gvt(Ft(N), y») is defined.

THEOREM D. / / / and g are as above, then they are regularly homotopic, with the homotopy based at yt, if and only if fl(/, g) = 0. Furthermore, let Oo E.*t(Fi(N), y9) and let a C1 immersion / : S*—*N based at yo be given. Then there exists an immersion g: S*-*N based at yt such that fl(/, g) =Ii0. Thus there is a 1-1 correspondence between elements of Tt(Ft(N), yi) and regular homotopy classes of C* immersions of 5* in N based at yo.

The methods used in this paper are extensions of methods of [3]. It is to be hoped that these methods can be used to solve further questions on regular homotopy classes of immersions.

§1 is on fiber spaces in the sense of Serre. In §2 a triple (£, p, B) of func­tion spaces is defined and shown to have the covering homotopy property (Theorem 2.1). To generalize 2.1 would be a big step in obtaining regular homotopy classification of higher dimensional spheres. In §3 Theorem 2.1 is applied to compute the homotopy groups of the fiber F, of (E, ir, B) (or at least to reduce the computation to a topological problem). Finding the 0th homotopy group of Tc is roughly the solution of the local problem in the theory of 2-dimensional regular homotopy. In §4, using the knowledge of To(I\), the main theorems stated in the Introduction (except Theorem D) are proved.

1. A triple (E, p, B) consists of topological spaces E and B with a map p from E into B (note P is not necessarily onto). A triple has the CHP if it has the covering homotopy property in the sense of Serre. In that case we call (£, p, B) or sometimes just E a fiber space. The following was proved in [3].

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1959] A CLASSIFICATION OF IMMERSIONS OF THE TWO-SPHERE 283

THEOREM 1.1. Suppose the triple (£, p, B) has the CHP locally; that is, for each point x£B, there exists a neighborhood V of x such that (£ - 1( V), p, V) has the CHP. Then (£, p, B) has the CHP.

A homomorphism h, h') from a fiber space (£, p, B) into a fiber space (£', p', B') is a pair of maps h: E—*E' and W: B-+B' such that the following diagram commutes.

h £ - £ '

B-+B' If B and B' are the same space and h' the identity we will speak of A as a

homomorphism. A map / : X—* Y is a weak homotopy equivalence if (1) its restriction to each

arcwise connected component of X induces an isomorphism of homotopy groups and (2) it induces a 1-1 correspondence between the arcwise connected components of X and Y.

LEMMA 1.2. If (£, p, B) is a fiber space then p(E) consists of a set of arc-wise connected components of B.

This follows immediately from the CHP.

LEMMA 1.3. Let (h, h') be a homomorphism from a fiber space (£, p, B) to a fiber space (£', p', B') such that h and W are both weak homotopy equivalences. Let XoG-B, yo=h'(x0), F^p-Xxt) and F=p'-l(y<l). Then the restriction of h to F, h: F—*F is a weak homotopy equivalence between F and F*.

This lemma is an immediate consequence of 1.2, the exact homotopy se­quence of a fiber space and the 5-lemma.

2. Let D be the disk in the plane £' , D=(x, y)\xi+y1^l] and t) = (x> y) I x,+y1 = 1. However, unless specifying otherwise, we will refer to the points of D with polar coordinates (r, 6) and use 6 as the coordinate for t). The points of Euclidean n-space £* will be considered to be vectors from some fixed origin.

A space £ is defined as the set of all regular maps of D into £" (we always assume n>2) which satisfy the following condition. If / £ £ then the first derivatives of / on the boundary of D, D are C1 functions of the boundary variable 0. The topology defined by the following metric is imposed on £ .

d(J, g) = max d'(Jp), g(p)), d'(Mp), g,(p)), d'<Jyp), gy(p)) \peD. Here d' is the ordinary metric on £" and f,(p),fvp), etc., denote the obvious partial derivatives. In general we will call such a topology on a function space, the C1 uniform topology.

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284 STEPHEN SMALE [February

Let 7„,» be the Stiefel manifold of 2-frames (not necessarily orthonormal, but ordered, and independent pairs) in £". Let V= Vn,tXE" be all 2-frames at all points of E". Let q be the projection of a frame onto its base point, q: V—»£", and gi, g», the projections of a frame into its first and second vector respectively.

Let B' be the space of maps of Z) into V with the compact open topology. Let B be the subspace of B' satisfying the condition: If /£2J, then

(1) a / W - / M i s regular, (2)/'(*)=ftf(0),and (3) / is C\ A map x: E—*B is defined as follows. If / £ £ , (1) *»(/)<*) =/(!.«). (2) »»(/)(»)-/,(1,«), (3)flWr(fW)-/t(l,«). The subscripts r and 0 as here always denote the respective partial de­

rivatives. From the definitions of the spaces, E and B, it follows that r is well defined and continuous. The purpose of this section is to prove the fol­lowing theorem.

THEOREM 2.1. The triple (£, T, B) has the CHP.

Let g£J3 be given. We choose a neighborhood U of g in B as follows: Let A be the minimum angle in radians between qig(0) and q*g8) as & ranges over Z). Let Z. be the minimum of the magnitudes of qig(6) and q*g(P) as $ ranges over t>. Choose U such that for A£ U,

(1) the angle between g,A(0) and gtf(0) is < J 4 / 1 0 0 , and (2) \qih(0)-q<g(6)\<AL/\00

for t = l, 2 and 0£Z>. By the topology of B, U can be chosen as above. By 1.1. it is sufficient for the proof of 2.1 to show that (r-^CO. *". U) has

the CHP. Let A.: P-»Z7 be a given homotopy, P a polyhedron, and A: P—£ a covering of Ao. We will construct a covering homotopy h~w: P—*E. We may assume that P is a cube (see, e.g. [3]).

A linear transformation of £", Q*(P)(0) is defined as follows. First, let V,(J>)(6) be the plane defined by qiht(p)le) and 2i*,(p)(0) (if it exists) and a,p)8) the angle from the first to the second of these vectors. Then Qt(P)(0) is to be the rotation of £* which takes V,(p)(B) through the angle a,(j>)(6) and leaves the orthogonal complement fixed (if V,(p)(d) does not exist then Q!(p)(6) is the identity e). Finally Q,(p)(6) is Q?(p)(6) multiplied by the scalar |»*.(*)(*)|/|**o(*)(*)|. Note that [**•(*)W]G.O)W=ji*.(*)W where (?,(/>) W is to be considered as acting on the right. Also Q*(P)(0) is C1

with respect to 0 and continuous with respect to v and p.

LEMMA 2.2. Let n>k>l,G».hbethe Grossman manifold of oriented k-planes in £" and 5"_1 the unit vectors of £". Let a map w: Q—*G».t, be given which is

124

1959] A CLASSIFICATION OF IMMERSIONS OF THE TWO-SPHERE 285

homotopic to a constant where Q is some polyhedron. Then there is a map u: Q-*Sn~1 suck that for all qG.Q, u(q) is perpendicular to the plane w(g).

The proof may be adapted from the proof of 4.1 of [3] substituting y»,t+i for V»,i in the proof.

Let u: PXIX Z^S"-1 be given by 2.2, taking for u»: PX JXZ?-*G»,i the map which sends (p, v, 9) into the plane of £* spanned by qihrp)(6) and qiht(P)(fi). Because ht(p) is covered by Jt, one can prove that w is homotopic to a constant map.

Now choose 5>0 so that if |t»—v'\ £5 then AL

| qk.(p) - qh..(p) | < iooor

where T = max u,(p, v, 0)\pGP, vGI, 0G7>. Choose f0, 0^r 0 <l , such that for p£P, 0£Z) and r£r9,

(i) I kr(pKr,e) - k\(p)(i,6) | < (A/ioo)| k\(p)(i, e) | , (2) | Ht)r, 9) - ht(p)(l, 6) i < (A/100) 1 J,(f)(1, 9) 1 , and

\jmi±jmrt<im/m\Mi,*>\ 1 — To

(4) \h(p)(r,6) -h(p)(l,0)\ < ^

where iV-max | Q*(P)(e)I I P€p> v^-r> 9^t>\. That r0 can be chosen satis­fying (3) follows from the definition of the derivative h\(p)(l, 0).

Let ri=-r0+(l—r0)/200 and choose a C* real function on J = [0, l ] , a(r), satisfying a(r)=0 for r^n, a(l) = l, a'(l)=0, |a(r)| £1 , and such that \c/(r)\ < 102/100(1 -f„). Let 0(f) be a real O function on I with /3(f) =0 for r^ro, /3(1)=0'(1)=O, |/3(f)| £200 and /3'(f)>100a'(f) for r^r^l. The proofs that functions a(r) and 0(f) exist as above are not difficult and will be omitted.

Let JfW = max f 1 «*.(#)(*> - qhtipM I 1 ? € P, 9 G />.

Let

where e is the identity transformation. The covering homotopy h\: P—*E is defined as follows for r^5.

h\(p)(r, e) - [*(#)(r, o) - Hp)(i, 8)]Q.(p)<r, e) + ar)[qh.(p)(e) - ?*,(»(«)] + 0(f) Jf («)«(>, t, 0) + *(*)(!, *)•

The following derivatives are easily computed.

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286 STEPHEN SMALE [February

PT = UpKr,6) = kp)r,6)^p)r,S) + [*(*)(',«) ~ * (» ( ! , * ) ]&r(» (M) + o ' W l l W W - qh*(P)(0)] + p'(r)Mv)up, v, 6),

P, - K.,(p)r, 0) = [h,(p)(r, 6) - h,(p)(l, 0)]Q,(p)(r, 0) + [K(p)r,e) - KP)(l,0)]&>(p)(r,6) + cc(r)[qth,(p)(e) - qtht(p)(e)) + p(r)M(v)u,(p, v, 6) + H,(p)(l, $),

Q-(p)(r,e) - a'(r)[Q.(p)(8) - e], Q.>(p)(r,8) = a(r)Q„(p)(0).

Then it can be checked that R,(p) has the following properties. (1) h.(p) is CK (2) h\(p) has derivatives with respect to 0 and r, Cl with respect to 9 on t>. (3) k*p)r,6)=hp)(r,6). (4) *.(/>)(l, 0) =qh.(p)(0)-(5) M/>)(l,0)=2iA.(/O(0). We will show (6) hw(j>) is regular. To show (6) it is sufficient to show that the derivatives PT and P# are

independent. From the various choices made it can be checked that P , is "close" to Kr(p)(h 0)+F(.r)M(v)u(J>, v, 6) while P, is "close" to t»(p)(l, 6). From this statement it follows that P , and P» are independent.

From (1), (2) and (6) it follows that h\(p) is really in B, from (3) that k, is a homotopy of It and from (4) and (5) that h~, covers A,. For 5±sp^28 define £, as before using h instead of h. Iteration yields a covering homotopy A", for all v £ J . This proves 2.1.

3. Let r 0 £ V, and let 5 0 be the subspace of B with the further condition that for/£2Jo,/(0) = Xo (see the previous section for notation). Let Ea=r_1(2?o) and let the restriction of T to EQ still be denoted by IT. Then from 2.1 we have the theorem.

THEOREM 3.1. The triple (E0, T, B0) has the CHP.

Let E' be the space of all maps of pairs (D, po) into (V, x0) with the com­pact open topology where pt is the point of D, (r, 6) = (1, 0). Let Bi be the subspace of B' (B' as in §2) with the condition if/£2?o' then / ( 0 )= ro . A map T ' : E'->Bi is defined by restriction to D, i.e., if / £ £ ' , (TT/)(0) = / ( l , 0).

LEMMA 3.2. Tfo <rt>fe (£'. r>, Bi) has the CHP and E' is contractible.

Proof. Let ht: P—*B& be a given homotopy and H: P—+E' cover h<, where P is some finite polyhedron. A covering homotopy A\:P—£' is defined as follows:

h\(p)(r, 0) = h(p)((v + l)r, 6) Q£r£ 1/(1 + v), Up)(r, 6) = hr^-tpKe) 1/(1 + .) S r £ 1.

126

1959) A CLASSIFICATION OF IMMERSIONS OF THE TWO-SPHERE 287

It remains to show that £ ' is contractible. Let ht: D—*D be a strong de­formation of D into the point Po£D. Then define H,: E'—*E' by

BtWV) -MO)) , /eE'.pED. It is easily seen that Ht contracts E' into the constant m a p / 0 : D—*x6. This

proves 3.2.

LEMMA 3.3. The homotopy groups of E0 vanish, i.e., xt(Eo) = 0 , fc^O.

Proof. L e t / : 5*—>£o be given, k^O. We will show t h a t / i s homotopic to a constant map. Let A be the angle between the two vectors of the frame *o£ V and let L be the minimum of their magnitudes. Then there exists a neighborhood U of pa of D such that for q and g' of 5* we have

(1) \Mq)(r,6)-Mq')(r,6)\ < AL/100,

(2) | f,(q)(r, * ) - / i ( ? ' ) ( r , *) | < AL/100

for all (r, 0) £ £/. The existence of such a £7 implies that there is a homotopy H,:Sh->E0, - l ^ v ^ O , with H-X(q)=f(q) and such that (1) and (2) above are satisfied with H9 replacing / and (r, 0) ranging over D. Let qtG.Sk and e=HQ(ql))€.E<>. Then define # , : S*-+E0, Ogt/^1 by

#.(<?) = (1 - »)#<>(?) + ve.

It follows from the choice of U that ff,(g) is regular. Then it is easily seen that H.(p) is well denned and that H\(q) —e. Thus / is homotopic to a con­stant and hence 3.3 is proved.

Let F be the space of maps of I1r = [0, 2ir] into V starting at x0, and let p': F'—*Vbe the map which sends a path onto its end point. I t is well-known that (F, p', V) has the CHP and that F is contractible. Note also that the fiber £'-1(*o) is the space Bi.

Denote by F the subspace of F which satisfies conditions (l)-(3) of B in §2. Let p' restricted to F be denoted by p. Then B0 = p-l(x0).

LEMMA 3.4. The triple (F, p, V) has the CHP.

Proof. Let hv:P—*V be a given homotopy where P is a cube and let h: P-+F be a covering of Ao- We will construct a covering homotopy h,: P—*F.

Let 0: V—>T be the map of V into T, the bundle of nontrivial tangent vectors of En, which is defined by dropping the first of the pair of vectors from each frame. Clearly V becomes a fiber bundle over T this way.

Let R be the space of regular curves of En starting with point and deriva­tive 0(xo) and with the C uniform topology. Let r : R—>T be defined by *•(/) = ( f ( l ) , / ( l ) ) . Then (R, r, T) has the CHP. For a detailed description of this theory, see [3]. In this paper, a regular curve differs slightly from those of [3], but the transition in the theory from one definition to the other may be made without trouble. If g,: P—*T is the composition 0fc„ the

127

288 STEPHEN SMALE [February

above remarks imply that there is a homotopy |»:P—*R such that (1) rf„=f. and (2) lo=<t>k.

Define a map H: PXluXl-^Tby H(p, 8, v) = (&•(?) (0), j . W W ) and let X be the bundle induced by H with respect to the bundle V over T. Let q: X—*V be the corresponding bundle map. Since P X i t » X / is contractible X will be trivial.

Let A—PXiuXl^PXluXO and define as follows a cross-section s: A->X thinking of XC VXPXIuXL

s(p, 0, v) = (*„, #, 0, v), s(p, 2x, t>) = (h,(p), p, IT, V),

s(p, e, o) =. (h(p)(fi), p, e, o). Now we can extend s to a cross-section, still denoted by s, of PXluXl

into -X" so it is O with respect to 8. Let K,(p)(0) =qs(p, 0,v). It can be easily checked that A. is our desired covering homotopy.

LEMMA 3.5. The homotopy groups TA(F) vanish, k£0. The proof of 3.5 is not difficult and will not be given. The method is a

slight extension of the proof of 6.2 of [3]. A map <f>: E»—>E' is defined by the following equations. For/G£o, let

q<t>(f)(*,y) - / ( * , y ) , ?!*(/)(*, y) =/.(*, y), 5**(/)(*,y)=/,(*,y).

Here obviously we are using rectangular coordinates for the disk D. The difficulty of using polar coordinates to define such a map 4> are apparent.

We define a map ^«: Bo—♦•Bo' in such a way that the following diagram commutes.

£,-!-» E'

2>o * SQ

We define first a map f: F-+F as follows. F o r / E P let

(4) 9 K . 0 M - / M , (5) ?,*(/) (*) = cos tf?1/(0) - sin *,/(*), (6) wK/) W - sin 0qif(8) + cos 8qtf(fi).

The map ^ is based on formulae for transformation from coordinates (r, $)

128

1959] A CLASSIFICATION OF IMMERSIONS OF THE TWO-SPHERE 289

to coordinates (x, y). In fact, (S) and (6) come from the following equations for the boundary of an immersion g: D—*En.

gm = cos 0g, — sin 8g,, g, = sin 6gr + cos Og,.

The map fa: B0—KBO' is the restriction of fa It can be checked now that (3) commutes.

LEMMA 3.6. The map fa: B0—>Bo' is a weak homotopy equivalence.

Consider the diagram,

(7) \P f e

V > V Here e is the identity. It can be easily checked that (7) commutes in spite of the fact that $ is not the ordinary inclusion. Then by 1.3, 3.4 and 3.S, fa: Bo—Bo is a weak homotopy equivalence.

Let cGBo, r« = T-'(c). and x'-'Of) =Qd, where d=fac). Let fa: r,-»fti be the restriction of <f>. Then from 1.3, 3.1, 3.2, 3.3 and 3.6 we obtain:

THEOREM 3.7. The map fa: I1,—Q* is a weak homotopy equivalence.

From the exact homotopy sequences of the triples (£', T', BO') and (F, p, V) (using 3.2) we obtain:

LEMMA 3.8. x*(ik) = T * + I ( 7 ) =T*+1(7».t).

Combining 3.7 and 3.8 we have

THEOREM 3.9. T*(r«) =**+!( V».i). In particular T»(T.) = T I ( 7 „ . I ) .

The last statement may be interpreted as giving the regular homotopy classes with fixed boundary conditions of a disk in En.

4. Let /and g be two C1 immersions of Si in £". We can assume without loss of generality that they agree on a closed neighborhood U of a point say SoGS*. An invariant fl(/, g)GTi(V»,i) is defined as follows. The space D = S* — int U is a topological disk, so we can assume there is a fixed field of 2-frame8 defined over it. From this field / and g induce maps of D into Vw,» which agree on the boundary D of D. Then by "reflecting" g we obtain a map of the 2-sphere 5* into Vn,t. The homotopy class of this map clearly does not depend on the choices made. We denote this class by Q(f, g)€*i(Vn,t) (we can ignore the base point of TI(V,,,J) because either Vn,t is simply con­nected or Tj(7»,t)=0).

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290 STEPHEN SMALE

Theorem A of the Introduction is an immediate consequence of the pre­ceding paragraph and 3.9.

Let Gn.i be the Grassman manifold of oriented 2-planes in £". An immer­sion/: S*—£" induces the tangential map T/\ S,—^Gm,t by translating a tan­gent plane at a point of f(S*) to the origin of £". We denote by T/ the homo-topy class of T>, thus T/ is an element of T»(G,,,I). Let p: Vn,t—*Gn,t be the map which sends a frame into the plane it spans, and pt: 7TJ(7»,J)—nr»(G»,») the induced homomorphism.

The next lemma follows from the definition of Q(J, g). LEMMA 4.1. / / / and g are two immersions of S* in £*, then pfiKJ, g) = T/

- T f . We now prove Theorem C of the Introduction. Let /and g be two immer­

sions of S* in £«. Let p*: H1(V4,i)-^Ht(Gi,1) and 7%, 7 * : Ht^-tH^GtA be the homomorphisms induced by p, T/, and T, respectively. Suppose that S* is oriented and that s is the fundamental class of Hi(S*). JThen by 4.1 and the Hurewicz theorem, />««(/, g) =Tf*(s)-T^(s), where fl(/, g)G.ffi(7«.j) corresponds to Q(/, g) under the Hurewicz isomorphism. Let W(J) and W(g) be the normal classes belonging to H*(S2) defined by the immersions / and g evaluated on s. Then by Chern-Spanier [l],

T,.(s) - T*(s) = 2~W(f) ~ W(g)b where 0 is a certain element of Ht(Gi,i). Thus

p*Q(f, g) = 2-»[TT(/) - W(g)]*

and since />« is 1-1, fi(/, g)=0 if and only if Wf)*~W(g). Then Theorem C follows from Theorem A.

Lastly we note an easy consequence of Theorem 2.1.

THEOREM 4.2. A regular map of t) into £", n>2 , can always be extended to a regular map of D into £".

REFERENCES 1. S. S. Chern and E. Spanier, A theorem on orientable surfaces in Jour-dimensional space,

Comment Math. Helv. vol. 25 (1951) pp. 1-5. 2. R. K. Laahof and S. Smale, On immersions of manifolds, to appear in Ann. of Math. 3. S. Smale, Regular curves on Riemannian manifolds, Trans. Amer. Math. Soc. vol. 87

(1958) pp. 492-512. 4. H. Whitney, On regular closed curves in the plane, Compositio Math. vol. 4 (1937) pp.

276-284. 5. , Differeniiable manifolds, Ann. of Math. vol. 37 (1936) pp. 645-680. 6. , The self-intersections of a smooth n-manifold in In-space, Ann. of Math. vol. 45

(1944) pp. 220-246.

UNIVBUITY OF CHICAGO, CHICAGO, I I I .

130

AN*AI,M aw MATIIHM.\T«I« Vol. 69, No. 2. March, 1959

f'tinted in Jaftoti

THE CLASSIFICATION OF IMMERSIONS OF SPHERES IN EUCLIDEAN SPACES

BY STEPHEN SMALE

(Received October 25, 1957)

Introduction

This paper continues the theory of [6] and [7]. See these papers for further information on the problem as well as Chern [1]. See also [8] for a partial summary of the results in this paper. For the most part, how­ever, we do not depend on our previous results.

An immersion of one differentiable manifold (all manifolds will be of class C") M* in a second V", n > k, is a regular map (a C1 map with Jacobian of rank k) of Jlf* into Vn. A homotopy/,: M" -* V" is a regular homotopy if at each stage it is regular and the induced homotopy of the tangent bundle is continuous. We are concerned with the problem of classifying immersions with respect to regular homotopy. Except for a few comments, we restrict ourselves to where M" is the fc-sphere S* and V is euclidean n-space E*. Then we are able to provide a solution to this classification problem, at least in terms of homotopy groups of Stiefel manifolds. The result may be stated as follows.

Let VHt be the Stiefel manifold of all fc-frames of E* (not necessarily orthonormal frames) and FKS") the bundle of all A;-frames of 5*. Then an immersion of / : S* -> £* induces a map /«, : Fk(SK) -* Vn,t x E*. Let Xa e Ft(S*) and ya e V„.* x E* be fixed. We shall say an immersion / : S* -* E* is a based immersion if/*(«„) = y0. A based regular homotopy is a regular homotopy which at each stage is a based immersion. For any two based immersions/and g, an invariant fl(f, g) e nk(Vn_„) is defined as follows. Given based immersions f,g:S"-*E*, by a small regular homotopy of g,f and g can be made to agree on a neighborhood U of g(x0) which is diffeomorphic to a closed A-disk. Here q : FkSk) -*• S* is the bundle projection. The space D — (SMnterior XT) is a topological A;-disk so we can assume there is a fixed field of A-frames defined over it. From this field/and g induce maps/* and g+ of D into Vn,k which agree on the boundary of D. Consider D as a hemisphere of the A;-sphere S* and reflect g+ to the opposite hemisphere to obtain a map of S* into V,.,. The homotopy class of this map is denoted by fi(/, g) e «»(V.*).

THEOREM A. Iffond g are C~ baaed immersions o/S" in E* they ore based regularly homotopic if and only if D(f, g) = 0. Furthermore, let

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328 STEPHEN SMALE

fij € nk( Vni) and let a based C" immersion f: S* -»• E* be given. Then there exists a based immersion g : S* -*• E* such that 0 ( / , g) = fi,. Thus there is a 1-1 correspondence between elements of nk(VnX) and based regular homotopy classes of immersions of S* in E*.

If n > k + 1, ^Kf, g) can be defined for non-based immersions and Theorem A is true omitting the word based wherever it occurB. Theorem A is a direct generalization of Theorem A of [7] where k = 2. The case of Theorem A for k = 1 is included in my thesis [6]. See these papers for implications of Theorem A when k = 1, or 2. Many of the groups ff*(y«.*) have been computed. See Paechter [5]. Since nt(Vn_t) — 0 for n > 2k + 1, Theorem A implies the following.

THEOREM B. TWO C" immersions of Sk in E* are regularly homotopic when n ^ 2k 4- 1.

Whitney in [9, p. 220] posed the question : Are two immersions of a manifold Af* in E1* regularly homotopic if they have the same intersection number I, ? We assume in this and the next two paragraphs that the immersions are nice enough to define an intersection number. (Whitney [9] defines I, to be number of self-intersections of an immersion/: M *-*• E1* counted properly when JXM*) intersects itself only in isolated points. If is an integer if k is even and an integer mod 2 if A; is odd, k > 1). We can obtain from Theorem A the following.

THEOREM C. TWO C" immersions of S" in E*, k>l, are regularly homotopic if and only if they have the same intersection number I,.

If k is even, then according to [3], Wk(f)=2I/ where Wk(f) is the IS* normal Stiefel-Whitney class with integer coefficients of the immersion/. Thus in this case Wt(f) characterizes the regular homotopy class of/.

The based regular homotopy classes of based immersions of S* in £*+' correspond to the elements of «»(#*+i) where i2» is the rotation group on •E"(since i?»4l = V,+li4). To study further the situation of immersions of S* in £*♦' recall Milnor's notion [4] of normal degree. If / : M* -* E**x is an immersion of closed oriented manifold let / : M* -* S* be the map obtained by translating the unit normal vector at a point of/(AP) in £*+' to the origin. The normal degree Nf o f / i s the degree off.

Milnor [4] asks the question : for what k can S* be immersed in E**1

with normal degree zero ? He proved that for this to be true S" must be parallelizable and he proved that S* could be immersed in E* with normal degree zero.

THEOREM D. There exists an immersion of S* in E*+l with normal degree zero if S* is parallelizable.

132

ON IMMERSIONS OF 8PHERES 329

The question of regular extensions is closely related to that of regular homotopy. In particular, we are interested in the following problem: Suppose / : S*_1 -* E* is an immersion where S*~l is the boundary of the fc-disk D*. When can/ be extended to an immersion of D* 1 From H. Whitney's work one obtains an affirmative answer whenever n ^ 2k. The following theorems give an answer to this question under the restric­tion n > k.

THEOREM E. Ifn>kan immersion/: Sk~l -* E" can be extended to an immersion of D* ifandonly if Q(f,e) = 0 where e: Sk~l-*-Ekc.EH is the standard unit sphere in a k-plane ofE*

In a certain sense the results of this paper are local in nature. M. Hirsch using these results together with obstruction theory has proved theorems on the regular homotopy classification of manifolds instead of spheres. He also obtains some sufficient conditions for manifolds to be immersible in euclidean space. For example, he proves every closed 3-manifold can be immersed in E* [2].

The above results suggest the following questions : (1) One problem is to replace E* in Theorem A by an arbitrary n-

manifold Af". I believe one would get a classification of immersions of S* in M* in terms of nt(Fk(M*)) where Fk(M") is the bundle of A-frames over M*. I don't think this will be very difficult to prove, following the proofs in this paper.

( 2) Find explicit representatives of regular homotopy classes. Whitney has essentially done this for the case n = 2k. What regular homotopy classes have an imbedding for a representative ?

(3) Develop an analogous theory for imbeddings. Presumably this will be quite hard. However, even partial results in this direction would be interesting.

1. The covering homotopy theorem A triple E, p, B) consists of topological spaces E and B with a map

p from E into B. A triple has the CHP if it has the covering homotopy property in the sense of Serre.

Let D" be the unit fc-disk in E*(k 1) with generalized polar coordi­nates. That is to say, points of Dk will be pairs (t, x) where t is the dis­tance from the origin 0 of £* and a; is a point of the boundary Z?* of !>*.

Let Ek<n = E be the set of all C immersions of D* in E*. n > k. The set E is given the Cl topology,1 i.e., is metrized by

> Added in proof: Assume all function spaces have the C topology instead of the C1

topology.

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330 STEPHEN SMALE

/>(/, g) = m&xjp(fv), g(y)), pXdUY),dgfY))\yaD", YczDS,\Y\ = l where f,geE,p is the metric on E*, E* being considered as its own tangent vector Bpace, and D* is the tangent space of D* at y e D*.

Let B»ifl = 5 be the set of all pairs (g, g") where g is a C" immersion of D* in E* and g' is a C cross-section in the bundle of transversal vectors of g(D*). Thus g' is a C map of & into J&" — 0 such that g'x) does not lie in D*(x) the tangent plane of g(Df) at flr(ar). The set B is given the following metric. For (g, g'), (h, h') e B let

P'[(g, 9'), (K h')] = max 0,(0, A), ^'(ar), A'(x))| x e & where p is as above and p, is defined as p above except that y is only allowed to range over Z)*.

A map «•: E-*B is defined as follows. For h e E let TT(A) = (5. flO where g is the restriction of h to ZJ* and flr'(a?) = A,(l, x) (the subscript t means differentiation with respect to t). The goal of this section is to prove that [E, n, B) is a fiber space in the sense of Serre.

THEOREM 1.1. The triple (E, n, B) has the CHP. PROOF. Some of the constructions in this proof are straightforward

generalizations of those of [7]. This proof is essentially independent, however, and somewhat more detail is given here than in [7].

In the article La classification dee immersions, Seminaire Bourbaki, December 1957, R. Thorn has an interesting exposition of the proof of 1.1.

A rough account of our proof is as follows. We are given a homotopy hi: P-+ B, hence AJ(p) for each p and v is an

immersion of a sphere hjjt) with a transversal vector field h'm(p). Fur­thermore, hl(p) for each p is covered, i.e., h^p) is the boundary of an immersed disk h\p) e E and h[(p) is a transversal field induced by the immersion of the disk. The problem is to follow the homotopy h%p) by an immersion of a diBk h\(p).

In our construction of h~.(p), Equation (17), the factors a(t), fi(t), and M(v) are introduced mainly so that various boundary conditions are met and the regularity of A,(p) is preserved.

The first and last terms of (17) roughly speaking are used to take care of the transversal field part of the homotopy. In particular, the trans­formation Q,(p) is the principle element here. This part of the covering homotopy could be taken care of directly by an isotopy of £". The latter, in fact, is what Thorn does.

The second term of (17), a(«)[A,(p)(aO - Ao(p)(a?)], is just what makes

ON IM MERSION8 OF SPHERES 331

the immersion of the disk \p) project onto the immersion of the sphere hJiv)- However, in general the introduction of thiB term will cause the map of D* to have critical points or points where the Jacobian has rank < k. To counteract this, the term fHt)M(v)u(v, p, x) is added. The effect of this term is, roughly, to blow up the immersion whenever it might have become critical.

The above construction is used in [6], but in simpler form. The reader might see the idea of this proof by looking, therefore, at that paper also.

Let hi: P-* B be a given homotopy where P is a cube (recall that it is sufficient to prove the CHP for cubes) and let h: P-+ E cover hi. We will construct a covering homotopy h\: P-+E.

We write h\j>) = (A,(p), h',(p)) (recall the definition of B). Let e^v, p, x) be the distance from K(p)(x) to the tangent plane of

A,(p)(Z?*) at x and let ^ = min ^(v, p, x) \ v, p, x. Let £, = min \Vvht(p)(x)\[x, v, p, V e Dt, \ V\ = 1

and take e = (1/10) min £„£„ 1. The symbol VvK(p)x) means the derivative of h,p) with respect to V at x.

We define a linear transformation of En, Q,(p)(x) for p e P,xeDk and sufficiently small v (we clarify this later) as follows. Let VJj>)x) be the 2-plane of En spanned by the vectors Kfcp)(x) and K(p)(x), if it exists, and let a,(p)(x) be the angle from the first to the second of these vectors. Let Q*(p): D* -»• Rn (the rotation group) be the rotation of £* which takes Vw(p)(x) through the angle ct.(p)(x) and leaves the orthogonal complement fixed (if V,(p)x) does not exist then Q?(p)(») is to be the identity rotation e). Finally, define Q,(p): D* -»• GL(n, R) to be the rotation Q,*(p) multi­plied by the scalar | K(p)(x) \ / \K(p)(x) \. We will consider Q»(p)(«) as act­ing on En on the right. It is immediate that Q,(p) is C" with respect to x and that (1) A&>)(*)Q.(P)(*) = Kp)x) .

See [7] for the following. LEMMA 1.2. Let n > k ^ 1, Gn,t the Grossman manifold of oriented k-

planes in En and S"'1 the unit vectors of E*. Let a map w: Q-*GKit be given which is homotopic to a constant where Q is some polyhedron. Then there is a map u:Q-*- S"'1 such that for all q e Q, u(q) is normal to the plane w(q),

Note that if w is C°° we may assume that u is also. Now apply 1.2 taking for Q, I x P x I>* and for w(v, p, x) the plane

spanned by h'.(p)(x) and the tangent plane of h.(p)(x). Because (A,(p)X(p))

135

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332 STEPHEN SMALK

18 in the image of n: E -* B, one can show w is homotopic to a constant. Thus one obtains a map u: I x P x D* -* S*-1.

Choose * > 0 so that for | v - v , | £d and allp 6 P, x e & and Ve &t,\ V\=1, the following conditions are satisfied. ( 2 ) The angle between AJ(PX») and Aj,(p) is less than 180° (this insures that Qj(pX*) i> well-defined). (3 ) |A:(pK*)~A;(pK*)l<(6/10)minl,|A:(pX*)l|u,p|a; . ( 4) | vAfeX*) - VrM*X*) I < «/10 .

(5) I *.(PX*) - \<*>X*) I < (c/100) - - i . ^ max | vMv, p, x)\[v, p, x)

(If quantity on right of inequality of (5) is undefined, omit (6)). It is clear that such a choice for d may be made and that our choice of

d depends only on hjjp) and not H(p). It is easy to check that (3) implies (3') I Q,(p)(ar) - Q.(p)(a) | < «/10 min 1, | #(?)(*> l b , * »&*>

We choose now t,, 1/2 < t, < 1, so that for all * e [*,, 1], v £ 8, p e P, V e Z>: and | V \ = 1, the following conditions hold.

(6) \Bm^^^\<\AMatmU,

(See Lemma 5.1 of [6]). ( 7 ) | fc(p)(*, x) - lSs>)(\, x) | < e/10 . ( 8 ) | Vrh(P)(t, x) - VT%P)(1, *) | < e/10 .

(9) |S(p)&*)~A(p)a.*)l<(«/10)-m**|VrQ.(pX*) II « , ? , * ' (If quantity on right of inequality of (9) is undefined, omit (9)). It is clear such a choice for t, may be made. Set ^ = «, + (1/3) (1 — «,).

Real C - functions on /, a(t) and £(t) are defined satisfying the follow­ing conditions; (10) o(t) = 0 0£t Ztt. (11) o(l) = 1 o/(l) = 0 . (12) l « ( * ) l £ l \c?(t)\<2l(l-tt). (13) /?(*) = 0 0 £ * £ * , . (14) ^(1) = ^(1) = 0 (15) |^(*)|>10|o'(*)| t,^t £ 1 . (16) I W ) l £ 2 0 .

ON IMMERSIONS OF SPHERES 883

Primes in this case denote the derivativeB. As in [7] we leave to the reader the task of constructing such functions.

Let Mv) = max \K(P)(X) - *,(*>)(*) 11 v e P, x e B> .

Then for v ^ 8 the desired covering homotopy K.(p) is defined as fol­lows.

A.(P)(*. *) = Wp)(t, x) - h\p)(l, *)] [e + a(t)(Q,(p)(x) - •)] + <*(t) [k,(p)(x) - hJip)(z)] + mM(v)u(v, P, x) + hip)(l, x) . We write down the following derivatives for reference.

h\t(j>)(t,x) = h,(p)(t, x)[€ + a(t)(Q.p)(x) - e)] (18) + WPW, x) - h(p)(l, *)K(*)(Q.(p)(x) - «)

+ a\t)\h.m*) - K(p)(x)] + nW(v)u(v, P, x). For Ve Dt

VA(P)«, *) = [VrMt, x) -_VTh(p)a, »)] [e + «(*)(Q.(p)(*) - «)] + WJ>W. *) ~ HP)(l, x)oc(t)VrQ,(p)(x)

+ at)[Vrh.(p)(x) - Vr*i(p)(*)] + KW(v)vMv, p, x) + Vrhtp)(l, X) .

We will prove that k\(p) has the following properties. (20) h\(p) is C- in * . (21) fUp)=h(j>). (22) ^(p)(l,*) = ft.(p)(*). (23) M P ) ( 1 , *) = WPHZ) .

(24) A".(p) is regular . First we show how 20-24 imply Theorem 1.1. Properties (20) and (24)

yield that the homotopy A,: P-* E is well-defined, (21) says that h\ is a homotopy of A and (22), (23) imply that h. covers hi. Thus it only re­mains to prove (20-24).

Property (20) follows from the fact that all the functions used to de­fine h\(p) are C in x (17).

One can check (21) immediately from (17). One can obtain (22) from (17) noting (11), (14) and that hJip)x) =

X&0<1.*). Property (23) follows from (18) using (11), (1), (14) and the fact

£(p)(l, *) = *(*>)(*).

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384 STEPHEN SMALE

To prove the regularity of h\(p) it is sufficient to show v^w(p)(t»x)^0 where V e £>?,,„. Then V can be written V = V, + V, where Va is the projection of V into Z?* and V, is the projection of V into the vector space orthogonal to Di in £>?,,,). Then (25) Vyh\(p)(t, x) = pthwt(p)(t, x) + pxVwkP)t, x) where W is Vx normalized and p„ px are appropriate scalars. Either pt*0orpx* 0.

LEMMA 1.3. There is a vector b' of En, 16' | < e (see beginning of proof of 1.1 for definition of e) such that

Vwh.(p)(t, x) = Vwh\p)(l, x) + b' . PROOF. From (12), (3*) and (8) it follows that

| \.vwh\p) it, x) - vJiPHh x)][e + a(t)(Qt(p)(x) -e)]\< 2s/10 . By (12) and (9)

| lh\p)(t, x) - h\p)(l, z)]ct(t)VwQ.(pHx) I < e/10 . By (12) and (4)

I a(t) [Vwh.(p)(x) - VWUPHX)] I < s/10 , and finally by (16) and (5)

| P(t)Mv)vwu(v, p,x)\< 2£/10 . Then by (19) and these four inequalities we have 1.3.

LEMMA 1.4. There exist vectors in En, b, u, scalars A, A' where \b\ <e, A > 10 A' and \ u | = 1, and u = u(v, p, x) such thai

KXv)(t, x) = A,(P)(1, x) + b + Au + A'w .

PROOF. We can easily obtain from (18), h.,p)t, x) = htp)l, x) - Hp)(\, x)a(t)[Q0(p)x) - Q,(p)(z)]

- [X,(p)(l, x) -ht(,p) (t, *)] [e + a(t)(Q,(p)(x) - e)] + [h(PHt, x) - h(p)(l, x)]ct'(t)mP)(x) - e] + ct\t)M(vyu + p(t)M(v)u(v, P, x)

where w is a unit vector. Then from (12), (3') and (7) it follows that

I [A.(P)(1. *) - Hp)(t, ar)][6 + a(t)(Q,(p)(x) - e)] | < 2«/10 . By (12) and (3')

138

ON IMMERSIONS OF SPHERES 335

IA,(p)(l, x)a(t)mP)(z) -e]\< e/10 and by (12), (6) and (3') we have

I ct'(tMp)(t, x) - k\p)l, x)] mP)(x) - *] l < 4e/10 . By (15) and noting that K(p)(x) = ht(p)(l, x) and Q„(p)(x) = e the above inequalities yield 1.4.

LEMMA 1.5. Let a, b, u, u, a' and b' be vectors in En, A, A' scalar* with the following properties

16 |, 16' | < (1/lOHo, sa') aU reals 16' |< (1/10) | a'|, | u | = l, l u | = l, A > 1 0 A '

and u normal to both a and a'. Then a + b + Au + A'u and a' f 6' are linearly independent.

PROOF. If the lemma is false then there is a scalar v such that v(a + b + Au + A'u) = a' + b'

or (26) Av(u + (A'/A)«) = o' + b' + v(a + b) .

Since | (A'/A)zi | < (1/10) \u\,u + (A'/A)u has angle less than 25° from u and thus since u is normal to o and o' the term on the left of Equation 26 is at an angle of greater than 65° from the a — a' plane (the case A = 0 offers no difficulty). On the other hand a + b has an angle less than 25° from o, and a' + b' has an angle less than 25° from a'. This implies that the term on the right of Equation 26 has an angle from the a — a' plane less than 50°. Thus Equation 26 is false, and hence 1.5 is proved.

From the last three lemmas we are now able to prove the regularity of h,(p). By (25) it is sufficient to show that hvi(p)t, x) and VwKp)t, x) are linearly independent. This fact follows from 1.5 making the substitutions from 1.3 and 1.4, o = A,(p)(l, x), and a' = Vwh,(p)(h x). One also uses the definitions of e and u to check the hypotheses of 1.5. Thus we have proved (24).

The above construction may be repeated if 8 < 1 using h,(p) in place of h. This yields a covering homotopy for v ^ 28. Iteration yields a cover­ing homotopy h, for all v e I. This proves 1.1.

2. The weak homotopy equivalence theorem

Let/ 0 : D* -+ En be a C immersion, n > k. Denote by r=r». . ( / , ) the space with the C topology of all C" immersions/of D* in E* such that / agrees with /„ on D* and df with df0 on the restriction of the tangent

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386 STEPHEN SMALE

bundle of £>* to D*. Let x = (xu • • •, xk) be rectangular coordinates on E* Z) I? and if / : D* -*• E* is an immersion denote by/l4(x) the deriva­tive of /t«) along the curve xt. Let /» "• I* -* Vn.k (the Stiefel manifold) be the mapj;(») = (fXl(x), • ••,/,,(»)) and I" = r;,B(7i) the space with the compact open topology of all maps of D* into Vn.* which agree with/0 on Z>. Let <t> be the map typ)x) = (/,,(»), • • • ,fZt(x)). The proof of the following theorem is the goal of this section.

THEOREM 2.1. / / / „ : D*-+En is the standard immersion of SK in ak+1 plane of En then * : Tk,,(/0) -* r'ktn(ft) is a weak homotopy equivalence.

A map / : X-* Y is a weak homotopy equivalence if its restriction to each arcwise connected component of X induces an isomorphism of homo­topy groups and it induces a 1—1 correspondence between arcwise con­nected components of X and Y.

One could probably prove 2.1 without much trouble even if /«>: LP-+E* is an arbitrary immersion.

Let xm be the South pole of Z)**1. Then impose a coordinate system x = (x„ • ••,**), x e LT+^-x., on I>+ ,-x.. such that r(x) = C*^) , /* = 1 is the equator Q of Z)**1. From now until the end of this section we identify x and x, x, and xt; identify D* with the upper hemisphere of of Z>+1 and Z> with Q. Let

A = (*, x) 6 Z>*+11 * £ 1/2, r(x) ^ 1 or x = x. and let &: Z>*+1 -* ZC" be the standard inclusion into a k + 1 plane of En.

A subspace B = Bk*t. • (ffo) of i?»+1. „ (flr0) is defined as follows. An ele­ment (g, flr') of B belongs to B if g restricted to LP*1 n A is gt, and 0' restricted to Z)**1 n A is 0M (the derivative of gt with respect to t).

Let A' = (*, x) € Z)**11 x * x- and define jj,: A' — V,.,*, by g(y) = (9t^(y)t • •. ft.,(v). &.(!/)). 1/ e A'.

Let # = 5i*i,«( i,) be the space with the compact open topology of all maps of D* D A' into V».*+i which agree with £„ on Z>*+1 n A' n A.

A map 9:B-* ff is defined by *(0. flO(*) = (?,,(*), . . . . «rIt(x)F flf'(x)), (AT, g>) e B, x e A' n ZW1.

Let/o: Z>* -»• Z£" be the restriction of gt.

THEOREM 2.2. If 4>: I\.„(/0) - r'kt%(fk) is a weak homotopy equivalence, then "9 : Bk+Un(gt) -* 2&+i,a(9o) is also a weak homotopy equivalence (n > k).

PROOF OP 2.2. Let r* be the subspace of r = r,,(/0) of those immer­sions which have all derivatives (of all orders) agreeing with the

140

ON IMMERSIONS OF SPHERES 337

derivatives of/, on JD*. The restriction of * to r* will still be denoted by 4>.

LEMMA 2.3. The inclusion i: F* -* r ia a weak homotopy equivalence. The proof of this lemma is not difficult and will be left to the reader.

The idea of the proof is that any compact subset S of r can be deformed so that elements of S agree with g0 on a neighborhood of the boundary of

Define maps p : B -* T* and p': B -* V as follows. For (g, g') e B let p(c, flO be g restricted to Z>* = upper hemisphere of D**1. For / 6 B'.fiD'*1 n A'^Vn.„uM=(fito). '•-,Mv),f*+i(v))tet p'(/)(*) = (fi(x), •••,/*(«)) for xeB*. It is easily checked that the p is well-defined, continuous and that the following diagram commutes.

B-^B-\p \P'

♦

We claim that (1) (B, p, r*) has the CHP, (2) B, p', r ) has the CHP, and (3) Y restricted to a fiber is a homeomorphism between corresponding fibers.

PROOF OF (1). Let k,: P-+ r* be a given homotopy and h": P-+B cover h" where P is a polyhedron. We will construct a covering homotopy

K-.P-+B. Let qx: V»,t+1 -> V,.* be defined by dropping the last vector of a frame

and qt: Vn.t -* Gy* be the map which sends a fc-frame into the oriented fc-plane it spans. Then q = q^ has the CHP.

Define kf: P x D* -> Gnt so that &*(&, x) is the tangent plane of h,(b) at x translated to the origin. Let h* : P x D" -+ F„,*+i be the map

h*(b, *)_= (*,,(&)(*), • • •, hXk(b)(x), h'(b)(x)), x e Z>» where h\b)(x) = (h(b)(x), h'(b)(x)). Then since ti> covers h, it follows that q~K* = A»*. By the CHP of (V»,*+1, q, Gy») we obtain a covering homotopy h? • P x Z>* -»• V,.»+J from h* and A,*.

Define K • P -* B as follows. Let £(&X*) = (KMx), £(&X*))

where *.(&)(*) = *,(&)(*) x e Z>*

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838 STEPHEN SMALE

h.(b)(x) = 9<£x) IT e D»+1flA h\b)(x) = (k + l)-component of Ht(b, x) x e Z>* ft(&)(«) = 0Uz) x 6 />♦' D A .

Now it can be checked that hl(b)(x) is well-defined and is the desired covering homotopy. This proves (1). One proves (2) the same way.

Lastly, (3) can be seen as follows. If g e r*, g: D* -* E" is regular, and then p'\g) is the space of all g1: Z>* -* E* such that for each XBU1, g"x) is transversal to the tangent plane of g at x and g1 obeys a boundary condition. The fiber over <bg) e I* is the same while the restriction of ¥ is a homeomorphism between these fibers. To finish the proof of 2.2 we note that Y maps the exact homotopy sequence of (B, p, r*) into the exact homotopy sequence of (B1, p', IT"). By the five lemma the theorem follows using (3) above, the given condition on <t> and 2.3. This proves 2.2.

Let gv:D*-> E* be the standard inclusion of Z?* into a fc-plane of E*, and/0 be as in 2.2.

LEMMA 2.4. If 4>: l\.n(ff»)-*r'kt%(g9)*» a weak homotopy equivalence then 8o«*:r„(/,)^r;.(7.).

The proof of this lemma offers no trouble and we leave the proof of it to the reader. One can use for example a diffeomorphism of E*.

THEOREM 2.5. Jf "9 : BK,n(gt) -* B'ti%(gt) is a weak homotopy equivalence then aois<P: rkn(g0) -* r*.,^)-

Before we prove 2.5 we note that 2.1 follows from 2.2, 2.4 and 2.5 by induction on k keeping n fixed. The first step, that ¥ : Bl,n(ga)-*B[,nQt) is a weak homotopy equivalence, is trivially checked. In fact, roughly speaking, my thesis [6] contains the second step and [7] is the third step in this induction.

PROOF OF 2.5. An outline of the proof is contained in the following diagram. The spaces and maps will be defined as the proof proceeds.

nt(jS) - 5 - nt(E, F) x-±+ nUD - ^ ^ ( r ) [vt A \vt B bjt C \*t

*AP) *^- *t (£*, F') -*U nUF') - ^ *,_,<!") Let Xt be a (k—l)-frame of Sk'1=Dk whose base point is a «„, the south

pole of S*~l. The gjx„) is a (k — l)-frame say yt of En (with base point) &nd gtt(x„) is a vector say yt transversal to the plane of yt. Let Bt be the

142

ON IMMERSIONS OF SPHERES 339

subspace of B of elements ( / , / ' ) where ffa) = y, and / '(a-) = yt. Let E0 - n-^B,) c E. Then by 1.1 (4) (Et, n, B,) has the CHP. (We sometimes denote the restriction of a map by the same Bymbol as the original map).

We let E be the subspace of Et of immersions/: D* -»• En which agree with gtt on A. Note that B c. Bt and let n: E-* B be the restriction of jr. Let F = ff"'^^). Then we will prove ( 5) For all i, 5r,: (E, F) -* nt(B, n(g<>)) is an isomorphism onto.

For the proof of (5) consider

E Et

!% I-B-?-* Ba

where j and / are inclusions. Then it is sufficient for (5) to show for all i £ 0 ( 6) j , : nt(E, F) -* 7TtE„, F) is 1-1 onto, (7) rrt: 7Zt(Et, F) -> n,(B0) is 1—1 onto, and (8) j ' t : ^(B) - 7rt(B„) is 1-1 onto. The truth of (7) follows from (4).

By the exact homotopy sequence of the pairs (E, F) and (Et, F) for (6) it is sufficient to show irt(Et) = nt(E) = 0 for all i ^ 0.

Let g : S -* E be given. It is sufficient to show that g is homotopic to a point. For every e > 0 there is a differentiable strong deformation re­traction H, of D* into N, where N, is diffeomorphic to jy, N,Z)A and for every y e N„ d(y, A) < e. Then for each such s, there is a homotopy gt of flr denned by gtp)(y) = 9(p)(Ht(v)), P e S3 and y e Z>*. On the other hand, for each e > 0 we have the homotopy ht: S' -* E between gx and the fixed map /„ defined by kt(p)(y) = (1 - t)g,(p)(y) + t/M- We leave it to the reader to show that ht(p) will be regular (hence h, will be well-defined) if £ has been chosen small enough. Thus g iB homotopic to a point. This proves n(E) = 0.

In a similar fashion one proves n3(Et) = 0. PROOF OF (8). We wish to show i:B-*Bt is a weak homotopy

equivalence. L e t / : P - + 5 , where P is a polyhedron. It is sufficient for the proof to show there is a homotopy Fk: P -»• Ba such that (a) Fx = / , (b) if J\p) e B then FK(p) = AP), and (c) Fs(p) e B.

143

340 STEPHEN SMALE

The homotopy Fx IB defined in stages with some of the details omitted. First, it can be Bhown that there is a neighborhood N of x. and a homo­topy Fk: P-*Bt, 0 ^ X ^ 1 satisfying (a) and (b) and Buch that .F,(p) agrees with g„ on N.

There is a number e > 0 such that it g e Bt and satisfies p+(g, g0) < e where

P*(9, gj=maxf(g(y), g0(y)), piVrffiv), Vr9»(v)) 11/6 Z>* n A, VcD*,, \ V\=l

then Gk(y) = Xg(y)+(1 - X)gt(y) is regular with 0 £ J £ 1, and ye D*f)A. Furthermore, if e is small enough one can use the map GK to define a homotopy Hk e B, where 0^-1^1 such that Hl=g, H=g<, and if g=ga, Hk=Ht. To define HK from Gk one uses a ribbon around the equator of D*. Everything in this paragraph is valid on a compact set of such g all satisfying p+(g, gQ) < e.

Taking eas in the last paragraph one can define Fk: P-*-B01 ^ X ^.2 such that FJO?) is the previously defined F^p), pm(Ftp), gB) < e, and if Fi(j>) e B, Fk(p) = F,(p). Here F is taken as a stretching of E* moving Fi[p) except in a neighborhood of a%. Also Fk> 1^ X ^ 2 involves a simple re-parameterization of Z>*. Finally, Fk for 2 ^ X ^ 3 is defined directly by the HK of the last paragraph. This proves (8).

Let A and A' be as before and using the fixed map gt, define another fixed map g,: A' -* F».» by

g,(v) = (s^iv), • • •, 9**k_y), g,(v)), v e A'. Let E'k,n(gt) = E' be the space with the compact open topology of all

maps of A' into Vn.„ which agree with gt on A D A'. Let #,„(&) = Pt be the space with the compact open topology of all maps of D* (1 A' into V,,, which agree with gt on & n A' n A. Define n': E1-* ff by restrict­ing a map to fy n A'. Let K..(i.) = F* = *'" Vfo)) . (9 ) The triple (£', *\ # ) has the CHP.

To prove (9), let ft,: P -*• S be a given homotopy where P is a polyhed­ron and let H: P-» E' cover h,. A covering homotopy H,: P-> E1 is defined by

A»(*. *) = MjP)(t, x) 0 ^ * ^ 1 / 2

fe.(p)(t,g)=A<p)(t + r _ - 1> a») l /2£*<Jr

and if r =£ 1

144

ON IMMERSIONS OF SPHERES 341

where T = T(VtX)=2 + v(l+\x\)

2(1 + v)

1 = Xv, x, t) = *T ~ *> , r * l . r — 1

and

It can be checked without much trouble that this is a good covering homo-topy. We show now, (10) KKE) = 0, i ^ 0

PROOF. Let H,be& strong deformation retraction of Z>* into A, i.e., a homotopy H,: V-* D* such that H9 is the identity, Hx(x) e A for x e Z)* and if a; e A, #.(«) = * f° r aH '• Then define a homotopy H,:E'-*E>

by H,(f)(x) = f(Ht(x)). It is easily checked that J?, is a strong deforma­tion retraction of E into the point/^H, of E'.

A map y>: E-* P is defined as follows. >Fg)(v) = ( .,(1/). • • •, 0.,-,(v), fl'id/)), jr e ^ , 1/ e A'.

Then the following diagram commutes.

E-^E'

1". 1 K'

V

Let ^ : F-+ F' be the restriction of <p. It is easy to check that <p is continuous and that diagrams A and B at

the beginning of the proof of 2.5 commute. Then we have (11) <p,:*<_, (F) -* rt^F') is 1 - 1 onto.

For (11) first note that <pt: nt (E, F) -* nt(E', F') is an isomorphism onto since 5?, is, by (5), and it' is by (9). Then A and A' are isomorphisms onto because Jt,E) = 0 (proof of (6)) and ^ (£0=0 by (10). This proves (U). (12) There are maps ax: F -* V and a,: F-* V which are weak homo­topy equivalences and such that the following diagram commutes.

I' 1-

145

342 STEPHEN SHALE

PROOF. Let a : N-*D* be a diffeomorphiBm of a small closed neighbor­hood N of //-interior A into IP which sends the t,xu • • •, a?t_i) coordi­nates which were introduced after the statement of 2.1 (in one dimension higher) into the (xlf ••,»*) coordinates of D* which were introduced at the beginning of Section 2. If a has been chosen properly, it induces weak homotopy equivalences a, and a, as above with the above diagram commuting.

Theorem 2.5 now follows from (11) and (12).

3. Applications The aim of this section is to prove the theorems stated in the Introduc­

tion from the theory developed in Sections 1 and 2. Thus we prove Theorems A, C, D and E.

We first prove Theorem A. Let «, e Fk(S") with qx, the South Pole of S*, (q: Fk(Sk) -> Sk, the

bundle map) and let fif0: S* -♦ E**1 c E* be the standard inclusions. Let g0+(xt) = yt e Vn,k x En. Denote by A the Bpace of C immersions of S* based with respect to x„ and y0, with the C1 topology. Let r0 be the subspace of immersions of A which agree with gu on the lower hemisphere of 5*

THEOREM 3.1. The inclusion i : r„ -»• A is a weak homotopy equivalence. The proof of 3.1 follows from the argument used in the proof of 2.5,

statement (8). It is immediate that r„ and V* are naturally homeomorphic where T*

is as in Section 2. Thus we have by 2.1, 2.3, 2.4 and 3.1 that JT0(A) and Tr r") are in a 1 — 1 correspondence where r" = ri.»(^) is defined as in Section 2. The arcwise connected components of r ' correspond to the elements of nt(Vn,k) and the arcwise connected components of A are based regular homotopy classes of based immersions of Sk in En. The reader may check that the correspondence is that given by Theo­rem A. This proves Theorem A.

Theorem C is proved as follows. If k = 1, see [6]. Now suppose k is odd, k > 1. Then there exist two regular homotopy classeB of S* in E* by Theorem A since *»( Vlttk) = Z, if k is odd, k > 1. On the other hand, Whitney [9] showed that there exist immersions of S* i n £ u with arbitrary intersection number / „ I, defined to be an integer mod 2. Whitney further showed that / , is invariant under regular homotopy. These facts together prove Theorem C for k odd, k > 1.

Now suppose k is even. Let (?,»,» be the Grassman manifold of oriented fc-planes in 2£u and v'. V»*,* -* Gu.* be the map which sends a A-frame

146

ON IMMERSIONS OF SPHERES 343

onto the plane it spans. Then we have the following commutative dia­gram.

nt(S>) -r^-> nk(G,k,k) — 5 nk(Vik<k)

1 * 7.-. lhi 1 * i ^ S * ) - 7 - ^ — Hk(Gu,k) < ™—Hk(Vtk,k) .

Here the vertical maps are Hurewicz maps ; / and gr are immersions of S* in Eu;f,g:S*-* Glkik are induced by/ , g by translating tangent planes to the origin of Eu. ThenJ^ and/«, are induced byj\

Suppose now Jf = I,. We wish to prove that / and g are regularly homotopic. For this we use the result from [3] which says that for an immersion / : M"-+E'k (with M" compact oriented and I, defined) Wk(f) = 21f where Wkf) is the k* Stief el-Whitney class with integer coefficients of the normal sphere-bundle over M*. Thus in our case Wk(f) = Wk(g). But by [3] this implies that/„ = pv

It is easily checked that pfrif, g) = A(S,)—g^St) where St is a gener­ator of n*(S*). Then by the previous diagram hiP,Cl(f, g)=JZ(ht(St))— g*(k,S,))=0 since/, = g,. Then by the diagram p+hfl(f, g)=0. By [3] pm is 1 — 1 and since A, is 1 — 1 this implies n(f, g) = 0. Thus / and g are regularly homotopic by Theorem A. This proves Theorem C.

Theorem D is proved as follows. Let p : Vk+ltk -* Gk+itk = S* send a k-frame into the Ar-plane it spans. Since p is a fiber map, we have the following exact sequence.

*»(r»M.») - ^ T*(S*) - ^ - »r».,(Fiber = Rk) . The map A is zero since S" is parallelizable. Hence p, is onto. Therefore by Theorem A, there exist immersions/, g Sk-*E"*1 such that pfl(f, g) is a generator of ir,(S»). Since pfl(f, g) = J7(S,) - y,(S,) either/ or g has even normal degree, and there exists an immersion of Sk in E"*1 with normal degree zero [4]. This proves Theorem D.

Lastly we prove Theorem E. If n ( / , e) = 0 then / is regularly homo-topic to e. Furthermore, this regular homotopy can be covered by a transversal vector field/' by the argument of Theorem 2.2 (1). By 1.1, ( / , / ' ) is in the image of n ; hence the desired extension exists.

Conversely, in order that / have an extension it must lie in the image of n (with some/*). But it follows from the proof of 2.5, Statement 6 that E is arcwise connected. This implies f2(/f e) = 0.

147

344 STEPHEN BMALE

Addenda Here we note that the solution of another problem posed by Milnor

follows from our work. On page 284 of [4] he asks : Let n be a dimension for which S" is not parallelizable. Can some

parallelizable n-mamfold be immersed in E"*1 with odd degree ? Can some (necessarily parallelizable) n-manifold be immersed in E**1 both with odd and with even degree ?

The answer to the first and hence to the second question is seen to be no, as follows.

As in the proof of Theorem D, Section 3, consider the horaomorphism P% ' T«(^«*I,II) -*• "n(Sn). It is sufficient to consider the case of n odd with S" not parallelizable. By Theorem A, the image of p, consists of even elements of JT„(S") since only odd degrees of immersions of Sn in En*1

are possible [4]. Now if M" is parallelizable and/: M* -* E**1 is an im­mersion, then/ induces a map F: M*-+Vn^,n withpF = / . Then/must have even normal degree, proving our assertion.

UNIVERSITY or CHICAGO

REFERENCES

1. S. S. CBERN, La giamitriB das sous-varUUs d'wn espaoe euelidean a plusiers dimen­sions, L'Enseignement Matbematique, 40 (1955), 26-46.

2. M. W. HlRSCH, On immersions and regular homotopies of differential)!* manifolds, Abstract, Amer. Math. Soc., 539-19, Notices Amer. Math. Soc., 5, No. 1 (1958), 62.

3. R. LASHOF AND S. SHALE, On the immersion ofmanifolds in euelidean space, Ann. of Math., 68 (1958). 562-583.

4. J. MlLNOB, On the immersion of n-manifolds in(n+ \)-spaee, Comment. Math. Helv., 30 (1956), 275-284.

5. G. F. PAECHTER, The groups «,< V».«)(I). Quart. J. Math., 7 (1956), 249-268. 6. S. SHALE, Regular curves on Riemannian manifolds, Trans. Amer. Math. Soc., 81

(1958), 492-512. 7. , A classification of immersions of the two-sphere. Trans. Amer. Math. Soc., to

appear. 8. , On classifying immersions of S» in euelidean space, Summer Conference on

Algebraic Topology, Chicago, 1957, mimeographed notes. 9. H. WHITNEY, The self-intersections of a smooth n-manifold in Zn-spaee, Ann. of

Math., 45 (1944), 220-246.

148

DIFFEOMORPHISMS OF THE 2-SPHERE STEPHEN SHALE1

The object of this paper is to prove the theorem. THEOREM A. The space Q of all orientation preserving C diffeo-

morpkisms of S* has as a strong deformation retract the rotation group 50(3).

Here 5* is the unit sphere in Euclidean 3-space, the topology on fi is the O topology °° £ r > l (see [4]) and a diffeomorphism is a differentiable homeomorphism with differentiable inverse.

The method of proof uses THEOREM B. The space IF (C topology) of C diffeomorphisms of the

unit square which are the identity in some neighborhood of the boundary is contractible to a point.

The analogue of Theorem A for the topological case was proved by H. Kneser [2]. The problem in his case seems to be of a different nature from the differentiable case. J. Munkres [3] has proved that Q is arcwise connected.

Conversations with R. Palais have been helpful in the preparation of this paper.

Let 7' be the square in the Euclidean plane E* with coordinate (t, x) such that (*, x)£7» if 0£t£l and 0 £ * £ l . Let l: 7*-»7» denote the identity diffeomorphism and 7, the space of diffeomorphisms, with the C topology, of 7* onto 7* which agree with I on some neighborhood of /*, the boundary of 7*. The C topology is such that two maps are close with respect to it if they are close and their first r derivatives are close. See R. Thorn [4] for details. We assume that r is fixed in this paper, » £ r > l , and that all function spaces con­sidered possess the O topology. We further assume that all diffeo­morphisms are C".

Let 71C71 denote the subset (t, x)G7*|/= 1, df, be the differen­tial of a diffeomorphism / at p, and u0 be the vector (1,0) in E* con­sidered as its own tangent vector space. Then denote by 8 the space of diffeomorphisms of 7* onto 7* such that if / £ S , then (a) f—l on some neighborhood of t*—I\, and (b) d/,(«0)=M0 for a l l - i n some neighborhood of 7j.

Received by the editon September 29,195& 1 The work on this paper was supported in part by a National Science Foundation

Postdoctoral Fellowship.

621

149

622 STEPHEN SMALE (Aagutt

Let iy. P—*u9 be the constant map and define 8 to be the space with the compact open topology of maps of P into S which agree with i0 in some neighborhood of 1*, where S—E1—(0, 0).

A map «£: 6—♦* is defined as follows: *(/)(', *) - dfrHt..)M, f e e.

LEMMA 1. There is a homotopy ^.: Z-*i such that for eachf^S, (a) 4>,f)(t, x)isCin (v, x, t), (b) *.C/) = «., (c) <tn-4>, and (d) * . («) - «.. PROOF. Let p: R—*S be the covering map where R is the universal

covering space of S, and let fl6/»-,(«o). Let Tw: R—*R be a differenti-able contraction of R to fl.

Define now a homotopy A,: 8-»8 by *.(/)(*) "PT^Qx)) where / is the unique lifting of / taking / ' into 0. Then it is easily checked that £,—A, o £ may be taken as our desired homtopy.

LBMMA 2. 7*Aer« M O homotopy Ht: S-+S swcA *Ao//or each /G6, (a) # , ( / ) w C in (v, x, t), (b) Ht(f)-t, (c) Hx(f)-f,and (d) # ,(*)-«. PROOF. Let $,(/)(*, x) be considered as a vector field on P as given

by Lemma 1, for each / £ S and » £ / , J the unit interval. Let P,(J)(u, U, xt) be the integral curve of 4>,(j) with the initial condition -P.C/)(0, t», x,) = (<0, xt).

Define Q.(f)(t, x)=P.C0«, 0, x). Now suppose there were an integral curve P,(j)(u, h, xe) starting

at (<o, xe) * (0, x) which did not leave /*. If this were so then it would approach asymptotically some simple closed curve in P. In the in­terior of this curve the vector field #,(/) would have to have a singu­larity. But this is impossible. Thus we conclude that there is a t, say l, with Q*f)(t, x) meeting Ji. This is exactly the part of the proof of Theorem A which does not extend to the case of S*.

Denote the above t by i(v,f, x). Then l(v,f, x) is C" in (r, x) con­tinuous (»,/, x) and positive.

We need the following lemma which can be found for example in [l .p.172].

LEMMA 3. Let gbea real lower semi-continuous positive function on a paracompact space X. Then there is a real continuous function h on X such that for all x £ X , 0 <A(x) <f (x).

150

19591 DIFFE0M0RPH1SMS OF THE o-SPHERE 623

Let g be the function on 8 defined by

U - <(»,/, x) )

Then let ij be the function on S given by Lemma 3. Let 7 be a real function on 6X2?, C* in t such that y(J, t) =0 for *

in some neighborhood of 0, fif, t)=*l for t in some neighborhood of 1 and

o<fr^<l+,OT. at

We leave to the reader the task of showing that such a function exists. Now define if,: 8—»S by

*.(/)(', *) = &(/)(< + y(J, <)(<(*,/, x) - l), *).

We prove now that Hm(J): I*—*I* is regular (has Jacobian of rank 2). Note that !?,(/) can be written as the composition fog where

g: t, x) -> (/ + y(j, <)(/(*,/, *) - 1), x) = f, x1), +-f,x')^Q.f)V,*?)-

From the choice of 77 it follows that dtf /dty*0, and hence g is regular. Now we prove that ^ is regular. Let

?(«, *, x) = P.(f)(u, t, x) and *<(«, t, *), * = 1, 2

be the coordinates of <p. Also let X(t, x) denote the vector field #•(/) with

Xt,x) = Xi(t,x)- + Xx(ttx)-. di dx

Then yp(t, x) =tp(t, 0, x). It is sufficient to prove _1 is differentiable. Let rt, X) be the unique « such that <pl(u, t, x) =0. Then ^_,(<t *) = ( -T(* . x), vlrt, x), t, x), ?*(T(*, X), t, x)). The map *~ l is differ­entiable if r is and T is differentiable if

V(«,',*)

But d<pK«,t,x)

*0. r(t,m)

du = A'1(v»(r(<,x),<,x)-^(0,^-1.

151

624 STEPHEN SMALE [Aqgott

It is easy now to check that H9 is our required homotopy. The following amplifies Theorem B. THEOREM 4. The space S is contractible. In fact there is a homotopy

G>: 1F-+S such that for eachf&r, (a) G,(/)(x, t)isC*in (», x, t), (b) G.Cf)-«, (c) G*(f)-f,and (d) G.(*)-*. PROOF. Let JFi be the space of diffeomorphisms of I into I which

agree with the identity in some neighborhood of the boundary /. Let K.iSi-tf! be defined by X.( / ) (0- / ( f )v+f( l -v) . Then K,(J)(t) is C" in (*, v), K9(J)(t) =/, Ki(f) - / , and if e is the identity X,(e) =e.

Let H, be as in Lemma 2, and define h,<*H,(h) for each AG6. Let h, denote A, restricted to h. Let /9(f) be a C" function of t such that P(t) «=0 in a neighborhood of 0, 0 ( 0 - 0 in a neighborhood of 1.

The desired homotopy G%: S—*5 is defined as follows. G.(h)(t, x) - (f, [JT,(,,(fc-1)](*'))

where <' and x7 are the t and x components respectively of h,(t, x). The map ((, x)—*<, x*) is a diffeomorphism because A. is. It is easily checked that

is a diffeomorphism. Hence the composition G,(h) is also a diffeo­morphism. One can further check that Gw(h) satisfies all the proper­ties demanded by the theorem.

Let S* be the unit sphere in B? with Cartesian coordinates (x, y, s) and xe the South Pole. Let eu et be unit tangent vectors of S* at x( in the directions of the x and y axes fespectively. Let Qo be the space of diffeomorphisms of 5* such that if /£G 0 then/(x0) —x0 and dfm,(e) - « « , * - 1 , 2,

THEOREM 5. The space Q0 « contractible in the sense of Theorem 4. PROOF. Let E be the open southern hemisphere of 5*. Then £*

induces a natural Euclidean coordinate system on E and the tangent bundle T of £. We will use these coordinates to add points on E and T.

Let g: Qt-*R be denned by

«(/) - «wp q % 11 for all X € 2 W * ) ) , I WXi - X\ < l where T(Nt(x9)) is the unit tangent bundle of Nt(x»). Then f is

152

•9591 DIFFEOMORPHISMS OF THE l-SPHERE 625

lower semi-continuous. Hence by Lemma 3 let e be a positive con­tinuous function on fio such that \df(X)-X\ <1 for XGT(N,w(x»)) and /GQ0.

Let 7 be a function on OoX-S*, with y(J, x) C for each / and such that y(f, x) = 1 for x£iV.(/)/*(xo) and y(f, x) — 0 for x in the comple­ment of N,(/)(xo).

Define 5,: Q0-+Q» by &(/)(*) - (1 - ')/(*) + •[?(/. *)* + (1 " 7(7,* ))/(x)].

Then St(f) =/and 5i(/) agrees with the identity on iV«(/)/»(xo). Let p: S*—Xr-*E* be stereographic projection using x« as pole with

E* being the x—y plane of £*. Let J* denote the square in E* with center (0,0) and with sides of length i f which are parallel to the axes. Let DM: IM—*^I be the obvious canonical diffeomorphism onto.

Let M be a positive continuous function on Q« such that £_l(exterior /*</)) CW.c/)/»(xo). Let IF be the space of diffeomorphisms of E* which are the identity in a neighborhood of t\ and outside if-Then let G,:V—»ff be a9 in Theorem 4. Let a homotopy F,:fl»—»Oo be defined by

*(/)(*) = ^[Gtffh-^bix), b = DMp, x £ N.wtix,), *".(/)(*)=/(*) x E N.(f)t»(xoh

Let /3(v) be a C" function which is 0 in a neighborhood of 0 and 1 in a neighborhood of 1/2. Let y(v) be a C" function which is 0 in a neighborhood of 1/2 and 1 in a neighborhood of 1. Then we define our desired contraction Tt: 0(—>Oo by

T. = S,<„ 0*v± 1/2, T. = FyM 1/2 S v < 1.

The following amplifies Theorem A. THEOREM 6. The space ft of all orientation preserving diffeomorphisms

of S* has as a deformation retract the rotation group 50(3). In fact there is a homotopy Hw: ft—»Q such that for eacfc/GQ

(a) H,f)*) « C** t'n («, x), (b) ff.(/)-/, (c) Hif) is a rotation of S», and (d)i//GSO(3), # . ( / )= / . PROOF. Define 0 as the subspace of ft with the property that for

/GO, e1U)~df,t(ei) and «*(/)« d/.,(«t) are orthonormal. We first show that ft is a deformation retract of 0.

If /GO, let v( be the unit vector perpendicular to el(J) in S/<« the

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626 STEPHEN SMALE

tangent space of S* at/(*<>)» and Ut be e\f) normalized. Then define

«!(/) - (1 - t)e(f) + /««,

«!(/) = (1 - <)«fy) + too Let gt be the linear transformation which sends («'(/), e*(J)) into

('(/), «?(/))• Let />: 51—»5>(. be the natural projection. By Lemma 3, choose a positive continuous function < on fl with <(/)<l, such that for a unit tangent vector JT in the tangent space of Ntif)(J(x»))> d(p~lgtp)(X) and X are independent.

Let 7 be a function on 0X5* such that for each / £ 8 , y(J, x) is C, is zero outside NtV)(J(xt)) and is 1 on iV,(/)/i(/(xo)). Let p induce an affine structure on the hemisphere of S* with center /(xo). Define G,: Q-*Q by

c(/)(*) = 7fy, *)rlbK*) + a - T(/, *»*• Then G, retracts fl onto Q. _

We now define a retraction of ft onto 50(3). For each / £ Q let <*(f)ES03) be the rotation sending (/(x0), dffa), i/(e*)) into (xo, eu ei). Then define X,: 8->Q by X,(/) - r , ( /o aJ))af)'1 where r, is as in Theorem 5.

The desired homotopy H,: Q—»B is obtained by composing G, and if, as in the proof of the previous theorem.

REFERENCES 1. J. L. Kelley, General topology, New York, 1955. 2. H. Kneser, Die DeformationssOtu der einfack ttuammenkdngenden Flacker,

Math. Z. voL 25 (1926) pp. 362-372. 3. J. Munkres, Differentiable isotopies on the too-spkere, Abstract 548-137, Notice*

Amer. Math. Soc vol. 5 (1958) p. 582. 4. R. Thorn, Let singularites das applications differentiables, Ann. I nit Fourier,

Grenoble vol. 6 (1956) pp. 43-47.

INSTITUTB FOB ADVANCED STUDY

154

ON INVOLUTIONS OF THE 3-SPHERE.*

By MORRIS W. HIRSCH and STEPHEN SHALE.1

1. Introduction. If T is an orientation reversing involution of the 3-sphere Ss, it follows from the Smith theory [15] that the fixed point set F of T is either a 2-sphere or two points. If F is a 2-sphere, it may be wildly embedded in 3s [2], in which case T will not be equivalent to a linear involu­tion. (If 7\ and T2 are involutions on spaces Xx and X2 respectively, an equivalence between 7\ and T2 is a homeomorphism h: XX—*X2 such that hTi — TJi.) On the other hand if F —= S2 is tamely embedded, it is easy to show that T is equivalent to a reflection through an equatorial 2-sphere of S3. The main purpose of this paper is to study the case where F is two points. We will prove

THEOREM 1.1. / / T: S*-*S* is an involution with fixed point set F consisting of two points, then T is equivalent to the linear involution: L: S8-»SS , Lxlrx2,x,,xt) =— (2,, — x2) — x3, — xt).

Here we suppose that Ss is the unit sphere in Euclidean 4-space with coordinates (x,, x2, x3, x,). If T is differentiable, then one can obtain actually a differentiable equivalence by our methods.

Theorem 1.1 answers a question of Floyd [6, p. 92].

The problems concerning an orientation-preserving involution on S* remain largely unsolved.2

As a by-product of the proof of 1.1, the following two theorems are proved:

THEOREM 1.2. Let M be a non-orientable triangulated 3-manifold with an element B of TT1(M) such that /2 = 1 and B reverses orientation. Then the protective plane P can be embedded in M piecewise linearly.

THEOREM 1.3. Let T be an involution on a topological 3-manifold M

* Received January 26, 1959. 1 Both authors have been supported in this research by the National Science

Foundation. * For the 2-sphere these problems have been solved by Eilenberg and Kerekjarto;

see S. Eilenberg, Fund. Math. XXII (1934), pp. 28-41. 893

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894 1I0KRI8 W. HIB8CH AND STEPHEN 8MALE.

with an isolated fixed point ya not in the boundary of M. Then there is an invariant Euclidean neighborhood V of y0 such that T restricted to V is equivalent to a linear involution. Furthermore, suppose V is a Euclidean neighborhood of y0 and h: dV ~* 8* C E* is on equivalence between T \ dV and 8 \ 8*, the reflection through the origin of E*. Then h can be extended to an equivalence between T and 8.

The proofs of the above theorems depend strongly on the methods of Papakyriakopoulos [ I I ] , especially in the version of A. Shapiro and J. H. C. Whitehead [13].

2. Proof of 1.2. Let a be the generator of »! (P) ~ Zt. An orientation reversing embedding f: P -* M is a one-one piecewise linear map such that /#(a) reverses orientation in M. The proof of 1.2 is similar to the proof of Dehn's lemma by Shapiro and Whitehead [13]. We shall prove the following statement, which implies 1.2:

2.1. There exists an orientation reversing embedding f:P-+M if and only if there exists B£*i (M) which reverses orientation, and such that j82 — 1.

It is clear that given an orientation reversing embedding / : P-*M, then fi — ft(<x) reverses orientation (by definition) and p* — 1 because a* — 1. It remains to find /, given B. It should be remarked that we do not prove that there exists an embedding f:P-*M such that ft a) —/3. The proof of 2.1 is broken up into several lemmas.

A map / : P-* M is canonical if it is piecewise linear, maps each 2-simplex in a one-one fashion, and f(P) has double curves and triple points (see [11, § 2]) as singularities.

To obtain canonical maps, it is convenient to use their differentiable analogues, normal immersions. We can assume M has a differentiable struc­ture [5] , which by [14] is unique up to diffeomorphism. A map f:P-+M is an immersion if it is differentiable and has Jacobian of rank 2 everywhere. We say an immersion / : P-*M is normal if whenever i „ - • -,z» are points of P such that y — /(a;,)—- • • — f(xn), then the intersection of the n tangent planes to f(F) at y has minimal dimension. It follows that n ^ 3 , and that f(P) has only double curves and triple points as singularities.

It is shown in [7] that if A and B are manifolds, dim A < dim 2?, and A can be immersed in B, then any map A—*B can be approximated by an

156

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INVOLUTIONS OF THK 3-8PHEHE. 895

immersion.3 Since P can be immersed in Euclidean 3-space (Boy's surface) P can be immersed in any 3-manifold M. In [8] it is Bhown that any immer­sion can be approximated by a normal immersion. We leave it to the reader to prove that a normal immersion can be approximated by a canonical map. (Compare this statement with [4], for example). Using these approximations and the fact that every 3-manifold has a triangulation, unique up to sub­division [9], we have proved:

LEMMA 2.2. Any map f: P-*M can be approximated (and is thus homotopic to) a canonical map.

LEMMA 2.3. Let fit-r^M) be such that fi2 — l. There exists a canonical map f:P->M such that /#(<*) — fi.

Proof. Let D be the unit disc in the complex plane and S1 its boundary. Let A: &ll-+M be a map representing fi. Let /*: S1->8i be the double covering e<s—»e2W. Then A/*: Sl-*M represents >32 — 1, and is therefore extendable to a map h: J)->M. Since h sends antipodal pointB of S1 into the same point, h defines a map g: P—>M, if we consider P to be obtained from D by identifying antipodal boundary points. I t is clear that g*(a) — J8. By 2. 2 there is a canonical map / : P-* M, homotopic to g. Thus /#(«) = fi, and 2.3 is proved.

Let f:P-*M be a canonical map, and put P0 — f(P). By a regular neighborhood of P0 we mean a 3-manifold V (with boundary) such that V is a sub-complex of M containing P0, and admitting P0 as a deformation retract. It is easy to construct regular neighborhoods (cf. [13]).

Given / and V as above, we say a covering space p: V—> V is proper if it is two sheeted, and f*(<x) € P#0™"i(^'))-

LEMMA 2.4. Let V be a compact non-orientable 3-manifold with non­empty boundary dV. If Hi(V) is finite, then at least one component of 8V is a projective plane.

Proof. We first show that not every component of dY is the boundary of a 3-manifold. This is because a compact 2-manifold N which bounds some 3-manifold bounds a (possibly non-orientable) henkelkorper Q, in the sense of [12]. The embedding i: N^*Q has the property that t» : H*(N) -*EX(Q) is onto. Now suppose that each component 2V« of dV bounds a henkelkorper Q+ Then the union of V and the Qt forms a compact non-orient-

* The map A-* B must be homotopic to an immersion. Using the results in [7 ] , it is easy to show that any map P-» M has this property.

7

896 MORRIS W. HIR8CH AND STEPHEN SHALE.

able 3-manifold W without boundary, and because im: H1(Ni)-*H1(Qi) is onto, and fli(V) is finite, it follows that 2?i(W) is finite, in contradiction to [12, Satz IV, p. 206].

Thus some component of dV is not a boundary, and therefore is non-orientable. Let there be s 2r 1 non-orientable components Xt of dV, whose genera (number of cross-caps) are ku- • •,&„ and r ^ O orientable com­ponents Yj whose genera are hlt • •, h„ if r > 0. We must show that some fc« — 1.

The Betti numbers bn of V satisfy b0—l, &i=»0, 6» — 0, and 6 » ^ r ; the last relation follows from the fact that H*(V,dV) — 0, since V is non-orientable. Let x stand for Euler characteristic; then [12, p. 223] x ( ^ ) — 2X (dV). Substituting in this the values x(Xi) — 2 — h, x(Y f) — 2 — 2h,, and the relations among the bn above, we obtain, after simplifying, 5 ^ 2irky + i 2i**« + 1- Since each h, ^ 0 and each fc4 1, it follows that some h — 1. This prove 2.4.

LBMMA 2.5. Let f:P-*M be a canonical map and V a non-orientable regular neighborhood of f(P). If V has no proper cover, then there is an orientation reversing embedding of P in V.

Proof. The boundary dV of V consists of a finite number of compact 2-manifold8. We shall show that some component of dV is a protective plane. Let f: TTI(V) —»27,(F) be the Hurewicz homomorphism. If 27,(F) were infinite, we could map E\(V) onto Zz such that tfiff (a) would be in the kernel K; the covering of V corresponding to ^(K) would then be proper, contrary to hypothesis. Therefore H: (V) is finite. By 2.4, dV contains a protective plane Pf which is piecewise linearly embedded in M. I t is clear that the embedding P" —» V reverses orientation in V, and therefore also in M, and 2.5 is proved.

LEMMA 2.6. Let f: P —* V be an orientation reversing embedding, and let p: V'—*V be a double covering. There exists an orientation reversing embedding P—*V.

Proof. Let g=>pf:P-*V. By a slight deformation of /, keeping it an embedding, we may assume that g is a canonical map. Clearly, g*(tx) reverses orientation in V, since /#(«) does so in V. Since / is one-one and p is two-one, g(P) has no triple points, but only double curves as self-inter­sections and these consist of mutually disjoint simple closed curves, piecewise linearly embedded in g(P). Let 0 be such a double curve. Then g~x(C) consists of a pair (Dot necessarily distinct) of simple closed curves C, C" contained in P. There are several cases to consider:

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INVOLUTIONS OP THK 3-8PHEBB. 897

i) c — C", C homotopically trivial ii) C' — C", C not homotopically trivial

iii) G'^=C", both homotopically trivial iv) C'j^C", one (Bay C") not homotopically trivial.

Case i) cannot occur, for both C and C, being null-homoptopic, preserve orientation, but it is easily seen that C must reverse orientation if C" =- C (cf. [8, 3.1]) . Case ii) is impossible, for C — C" means the map g: C->C is a double covering, hence the homotopy class g*(a) can be represented as /?a, where /? is the homotopy class of C (with some orientation). But </#(<*) reverses orientation, while the square of any homotopy class preserves orien­tation. Case iii) In this situation C and C" bound discs and the original " c u t s " (umschaltungen) of Dehn can be used, as described e.g. in [11]. It is easy to check that after making the cuts, the map gx: P -> V is canonical, has no triple points, fewer double curves than g, and <7i#(a) reverses orien­tation. Case iv) is impossible, for gf (a) j ^ 1 (because it reverses orientation), and if C represents a, C" must aho; but on a protective plane any two non-null-homotopic curves must intersect, contradicting the result that C and C" are disjoint. Thus all pairs C", C" are as in Case iii), and by making a finite number of cute, we obtain an orientation reversing embedding of P in V.

Proof of 2.1. By 2.3, there is a canonical map g: P-*M, such that gf (a) — p, p reversing orientation. Let V be a regular neighborhood of gP). If V has no proper cover, then 2. 5 implies 2.1. Otherwise let p: V~* V be a proper cover. Since ##(<*) €/)#•»-! (F ' ) by the definition of proper cover, there is a map g': P-*V auch that pg"=-g. It is clear that cf is canonical and g't(ct) reverses orientation. Moreover, it is easily seen, as in [11, 9.1], that g'(P) has fewer curves than g(P). If ^is an embedding, 2.1 follows from 2.6. If not, we take a regular neighborhood V" of P', a proper cover of V" (if possible; if not, 2.1 follows from 2.5 and 2.6), and proceed inductively. Thus 2.1 is proved by 2.5, 2.6, and induction on the number of double curves of gP)-

Remarks. Concerning the problem of embeddings f:P—*M such that /#(a) preserves orientation, we have the following results, which are not used in the rest of the paper.

2. 7. If there exists an embedding f:P-*M such that ft (a) preserves orientation, then Z2 is a free factor of ^(M).

Proof. Let V be a regular neighborhood of f(P), where / is as in 2.6. Then x(V) — x(f(p)) = !» s mce f(P) is a deformation retract of V. Since

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898 MOHBI8 W. HIHSCH AND 8TKPHKN SHALE.

2X(F) — x ( a 7 ) , we have x(&v) "~2- S i n c e e a c t component of dV is orient-able, dV is a 2-Bphere 3, which separates JW into two parts, whose closures are V and Vi. (say), with F n 7 , - I It follows that ^M) is the free product of wl(V1) and in(7)S2Z, .

In the case where / is assumed to be a differentiable embedding, we can choose for V a "tubular" neighborhood of f(V). In this case it is easily seen, by considering the normal bundle of f(P), that V is homeomorphic to the mapping cylinder of the double covering S2 -* P. This proves:

2.8. A necessary and sufficient condition that there exist a differentiable embedding f:P-*M such that /#(<*) preserves orientation, is that M be the sum (in the sense of [12, p. 218]) of protective 3-space and another 3-manifold.

8. Proof of 1.3. We will use the following lemma.

LBHMA 3.1. Let R: S*Xl-*S*Xl oe an involution without fixed points such that for i — 0,l,R(82Xi)—82Xi and R restricted to 82X* is equivalent to the antipodal map A on S2. Then R is equivalent to i X « : 8zXl^*8IXl> where « : / - > / is the identity. Furthermore, an equivalence between R and A X « on 8* X 0 can be extended to an equivalence on 82XI.

Proof. The orbit space S2Xl\R — X is triangulable by Moise [9] , and hence there is an induced triangulation on S2 XI in which T is simplicial. In the rest of this section objects (maps, embeddings, etc.) will be considered from the piecewise linear point of view. Let p: 8* XI -* X be the orbit map and let Xt — p(82 X *)> * — 0,1. It is well known * that an involution on S2

without fixed points is equivalent to the antipodal map. Hence the Xi are homeomorphic to the protective plane.

We define a map of the cylinder J — 81Xl into X as follows: Let fa: S^Xi-tXt, t = 0,l , be embeddings which represent the non-trivial elements of iri(X<). By consideration of the covering space maps (8* X *» P> Xt) -* (82 Xl>P,X), i -=0 ,1 , it is easily seen that f0 and ft are homotopic in X. Thus we obtain a map F: J-+X which is an extension of f„ + ft.

Since the boundary ft/ of J is embedded in dX, we can deform F slightly to a map F": J-+X such that if </« — *"(«/) and dJa — F'(dJ), then there are no self-intersections in a neighborhood of ft/0 and ft/0 C dX. Then J0

is a Dehn surface of type (0,2) in the terminology of [13]. Application of [13, 1.1] yields an embedding 0: J-+X which agrees with F' on a neighbor­hood of ft/. (Note that 1.1 of [13] first yields a non-singular surface of type (°> ?)> 0 < <ltk 2, but in our case q must equal 2 since our boundary circles are essential.)

INVOLUTIONS OP THE 3-8PHEBE. 899

Denote QJ) by Jz and p_1(^i) by Q. Since 0 is essential in X, Q will be a cylinder in S* X I which doubly covers <7, and 0Q C d(S2 X I ) • Further­more Q remains invariant under 22.

Let h: fl2 X 0 -» 5* X 0 be an equivalence between 22 and i X « , both restricted to S* X 0, and let K — A(C n S* X 0). Let

( ? 0 = ( x , y ) € 5 f ' X / | a : e i r .

Then @0 is invariant under A X «• By the well known analogue of 3.1 in dimension 2, A can be extended to an equivalence of R and A X « on Q and @0-Thus we have h: (S2 X 0)U Q-+ (S* X 0)U (),,. Let the components of # 2 X 2 — Q be denoted by J. and B, the components of S2Xl — Qo by ^o and 2?0. Then it follows from Newman's [10] and Alexander's [1] theorems that A, B, A0, and B0 are 3-cells. Extend A to a homeomorphism between A and A. Finally define h on B to be (AX e)hR. This h is our desired equiva­lence between R and A X «•

We now start the proof of 1.3. As before we can assume without loss of generality that T restricted to the complement of the fixed point set is piecewise linear. The following lemma is where 1.2 is used.

LEMMA 3.2. In every neighborhood of yt there is an invariant 2-sphere.

Proof of 3.2. Let U be a given Euclidean neighborhood of y0 in M. Choose 7 to be an open connected neighborhood of y0 such that V C V and TV C U. Let T : M -> M/T be the orbit may, 7* — •»-( V), and 7 0 — 7*—ir(y,). We claim that 7 0 satisfies the conditions of 1.2. From [3] it follows that T is orientation reversing, so 7 0 is non-orientable. Let x 6 7 , x^y0, and let 6(<) be an arc joining x to Ta; in ( 7 U T 7 ) —y0. Then it is easy to see that B — *•& (<) satisfies the conditions of 1.2. Hence there is an embedding / : P—> 7 0 of the projective plane in 70 . The inverse image ir'l(j(P)) C U will be the desired 2-sphere. This proves 3.2.

Returning to the proof of 1.3, let U be as in 3.2. If N is an embedded 2-sphere in U, denote by C(N) the closure of the component of M—N containing yp. Then by 3.2 choose disjoint invariant 2-sphere Vu 7 „ 7 „ • • • such that lim(diameter 7<) =-0, U D Vx and for each i, C(7«) D VUi-

Let W be the sphere about the origin in E* of radius \/i and let T0 be the reflection through the origin of E*. Let h: Vl-^Wlbe some equivalence between the restrictions of T and T„. By 3.1 extend h to

CiVi) —interiorC(V t) -i-G(W1) — interiorC(W t).

Here we again use Newman's and Alexander's theorems to show that

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900 MORBI8 W. HIE8CH AND STEPHEN 8MALE.

(7(F!)—interiorC(V2) is homeomorphic to S*Xl- Repeated applications of 3.1 yield the desired equivalence of 1.3.

4. Proof of 1.1. Let the fixed points of T be denoted by yr and yt. By 3.2 let 7 i and V2 be disjoint invariant 2-spheres about y t and yt respec­tively. In Sl let Wr = x 6 S* \ x1 — £ and W2 -= x € S* \ x1 = — £. Let h: V t-> Wi be an equivalence between the restrictions of T and L. By 3.1, extend A to be an equivalence on the domain of 8s — Vx — V2 not containing the fixed points. Then by 1.3 h can be extended to an equivalence on all of S\

REFERENCES.

[1] J. W. Alexander, " On the subdivision of 3-space by a polyhedron," Proceeding* of the National Academy of Sciences V. 8. A., vol. 9 (1923), pp. 6-8.

[2] R. H. Bing, "A homeomorphism between the 3-aphere and the sum of two solid horned spheres," Annals of Mo thematic*, vol. 56 (1952), pp. 354-362.

[3] G. Bredon, " Orientation in generalized manifolds and application to the theory of transformation groups," to appear.

[4] S. S. Cairns, " Polyhedral approximations to regular loci," Annals of Mathematics, vol. 37 (1936), pp. 409-415.

[5] , " Homeomorphism between topological and analytical manifolds," ibid., vol. 41 (1940), pp. 796-808.

[6] E. E. Floyd, Summary of Lectures and Seminars of Summer Institute on Set Theoretic Topology, Madison, Wisconsin, 1955, pp. 92-96.

[7] M. W. Hirsch, Immersions of manifolds, Thesis, University of Chicago, 1968, to appear in Transactions of the American Mathematical Society.

[8] R. K. Lashof and S. Smale, " Self-intersections of an immersed manifold," to appear in Journal of Mathematics and Mechanics.

[9] E. E. Moise, "Affinc structures in 3-manifolds, V. The triangulation theorem and hauptvermutung," Annals of Mathematics, vol. 56 (1952), pp. 96-114.

[10] If. H. A. Newman, " On the foundations of Combinatory Analysis Situs," Pro­ceedings of the Royal Academy of Amsterdam, vol. 29, pp. 610 641.

[11] C. D. Fapakyriakopoulos, "On Dehn's lemma and the asphericity of knots," Annals of Mathematics, vol. 66 (1957), pp. 1-26.

[12] H. Seifert and W. Trelfall, Lehrbuoh der Topologie, New York, 1947. [13] A. Shapiro and J. H. C. Whitehead, "A proof and extension of Dehn's Lemma,"

Bulletin of the American Mathematical Society, vol. 64 (1958), pp. 174-178. [14] 8. Smale, " On diffeomorphisms of the 3-sphere," to appear. [15] P. A. Smith, Fixed points of periodic transformations, Appendix B in Lefschetz,

Algebraic Topology, New York.

162

THE GENERALIZED POINCARE CONJECTURE IN HIGHER DIMENSIONS

BY STEPHEN SMALE1

Communicated by Edwin Moise, May 20, 1960

The Poincare conjecture says that every simply connected closed 3-manifold is homeomorphic to the 3-sphere 5 ' . This has never been proved or disproved. The problem of showing whether every closed simply connected n-manifold which has the homology groups of S", or equivalently is a homotopy sphere, is homeomorphic to S*. has been called the generalized Poincare conjecture.

We prove the following theorem.

THEOREM A. / / M" is a closed differenliable (C") manifold which is a homotopy sphere, and if n?*3, 4, then M" is homeomorphic to S".

We would expect that our methods will yield Theorem A for com­binatorial manifolds as well, but this has not been done.

The complete proof will be given elsewhere. Here we give an out­line of the proof and mention other related and more general results.

The first step in the proof is the construction of a nice cellular type structure on any closed C"» manifold M. More precisely, define a real va lued/ on M to be a nice function if it possesses only nonde-generate critical points and for each critical point 0, /09)=M/3), t n e

index of /3.

THEOREM B. On every closed C" manifold there exist nice functions.

The proof of Theorem B is begun in our article [3]. In the termi­nology of [3], it is proved that a gradient system can be Cx approxi­mated by a system with stable and unstable manifolds having normal intersection with each other. This is the announced Theorem 1.2 of [3]. From this approximation we are then able to construct the func­tion of Theorem B.

The stable manifolds of the critical points of a nice function can be thought of as cells of a complex while the unstable manifolds arc the duals. This structure has the advantage over previous structures that both the cells and the duals are differentiably imbedded in M. We believe in fact that nice functions will replace much of the use of Cl triangulations and combinatorial methods in differential topology.

1 Supported by a National Science Foundation Postdoctoral fellowship.

373

163

164

374 STEPHEN SMALE [S*p««mbcr

In our work here wc do not actually use the cells themselves as much as the neighborhoods of the skeletons of this structure. More precisely, i f / i s a nice function on Af, let A'»=/_ l[0, k+\/2\.

Then each A"» is a compact O manifold with boundary and one can think of A"» as being obtained from A"»_( by attaching a number of 4-cells and "thickening" them.

If Af is a manifold with boundary, D a disk. / : dD—*dM a differ-entiable imbedding with a trivial normal bundle B and y a homotopy class of cross-sections dD—»B, then a new manifold Af\JD = M\J/,yD is denned. We do not define AfKJD here but only remark that Af\JD is roughly Af with D attached by / , thickened according to y. and smoothed.

Then it can be shown that A"* - AViWDVJ . . . \JD*. where dim D\ « * (D, Dk, etc. always denote disks).

The handlebodies are defined to be all manifolds of the form D'KJDIKJ • • KJD't. Fixing n. k, s, the set of all such manifolds is denoted by 3C(n, *, s).

For our main theorems we prove that under homotopy assump­tions, the Xk are handlebodies. The following is a major step.

THEOREM C (THE HANDLEBODY THEOREM). Let n £ m a x ( 2 ; + 2 , 5), H£X(n,k, s),K = IfUDxKJ • • \JD„ dim D . - 5 + 1 and x . ( / O « 0 . Also assume r,(H U D, U • • • U £>,_») = 1 if s = 1. Then KC=X(n,r-k.s+\).

We shall not try to summarize the proof of Theorem C. Using Theorems B and C and the previous consideration one can

obtain the following theorem.

THEOREM D. Let Af' be a closed C* manifold which is (m —1)-connected and «^max(2m, 5). Then there is a nice function f on M with type numbers satisfying

Aft «= Mn I and A/< = 0 for 0 < »' < m and n — m < i < n.

Special cases of Theorem D are first Theorem E if n — 2m+ 1, and Theorem F if n =» 2m.

THEOREM E. Let M be a closed (m - \)-connecled closed C~ (2m + 1 ) -manifold. Then M = HKJH' where H, H'£X(2m + l, *. m).

This generalizes the Heegard decomposition of a 3-manifold.

THEOREM F. JM M be a closed (m — 1)-connected O 2m-manifold, m^l. Then there is a nice function on M whose type numbers equal the Betti numbers of Af.

i9«ol POINCARfe CONJECTURE IN HIGHER DIMENSIONS 375

Theorem F implies the even dimensional part of Theorem A. According to Thorn (see [2]) two closed oriented C* manifolds Mt

and Mi are J -equivalent if there exists a C manifold X with boundary Mi —Mt and each Mi is a deformation retract of X. Using methods similar to the preceding ones we are able to prove the following:

THEOREM G. / / two homotopy spheres Mi and Mt of dimension 2m — 1, my* 2 are J-equivalent, then they are diffeomorphic.

Using Mazur's Theorem [ l ] , the part of Theorem A for odd di­mensional manifolds follows from Theorem G.

THEOREM H. There exists a triangulated manifold with no differenti-able structure at all.

This follows from Theorem G and work of Milnor [2j. Take his manifold of W9 of Theorem 4.1 of [2] for fe = 3 and attach a 4A cell. (In our context W* can be viewed as a certain handlebody in 3C(4ifc, 8, 2*).) The following also follows from Milnor [2] and Theo­rem G:

THEOREM I. The groups r'",+l are finite, for all m.

For example, for r* = 0, there are precisely 28 differentiable struc­tures on S7, etc.

REFERENCES

1. B. Mazur, On embedding of spheres, Bull. Amer. Math. Soc. vol. 65 (1959) pp. 59-65.

2. J. Milnor, DifferenliahU manifolds which are homotopy spheres, (mimeographed) Princeton University, 1959.

3. S. Smale, Morse inequalities for a dynamical system. Bull. Amer. Math. Soc. vol. 66 (1960) pp. 43-49.

INSTITUTE FOB ADVANCED STUDY AND INSTITUTO DE MATEMATICA, RIO DE JANEIRO, BRAZIL

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A N N A L * Or KfATMBMAYlCB Vol. 74, No. 1. July, lttl

Prinltd in J*pm

ON GRADIENT DYNAMICAL SYSTEMS BY STEPHEN SHALE*

(Received August 29, 1960) (Revised November 28,1960)

We consider in this paper a C" vector field A" on a C" compact manifold M" (dM, the boundary of M, may be empty or not) satisfying the follow­ing conditions:

(1) At each singular point £ of X, there is a cell neighborhood N and a C" function f on N such that X is the gradient of / on N in some riemannian structure on N. Furthermore /9 is a non-degenerate critical point of / . Let fflt • • •, /9« denote these singularities.

(2) If x € dM, X at * is transversal (not tangent) to dM. Hence X is not zero on dM.

(3) If x € M let <pt(x) denote the orbit of X (solution curve) through x satisfying <P,(x) = x. Then for each x e M, the limit set of <Pt(x) as t —+ ± oo is contained in the union of the y9«.

(4) The stable and unstable manifolds of the /8< have normal intersec­tion with each other. This has the following meaning. The stable manifold Wt* of /8t is the set of all x e M such that limit, «y>,(x) = /9«. The unstable manifold Wt of /3« is the set of all x e M such that limit,..-.. <pt(x) = fit. It follows from conditions (1), (2) and a local theorem in [1, p. 330], that if /?, is a critical point of index X, then W, is the image of a 1-1, C" map <p\ U-* M, where Uc if"x has the property if x e U, tx e U, 0 ^ t ^ 1 and <p< has rank n — X everywhere (see [4] for more details). A similar state­ment holds for Wt* with the UczR\ Now for x e W% (or W<*) let Wlx (or W*x) be the tangent space of Wt (or Wt*) at x. Then for each i, j , if a; e Wt n W,*, condition (4) means that

dim Wt + dim W? - n = dim Wtz n W£) . Here Wu and W,* are considered as subspaces of the tangent space to M at x.

For closed manifolds, these vector fields are a special case of those considered in [4].

THEOREM A. Let f be a C~ function on a compact C" manifold Mn

with non-degenerate critical points. Suppose M is provided with a * Supported in part by a National Science Foundation Postdoctoral Fellowship, and in

part by National Science Foundation Contract G-11594. 199

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200 STEPHEN SMALE

riemannian metric and that grad / is transversal to dM. Then grad / can be C1 approximated by a vector field satisfying conditions (1) to (4).

THEOREM B. Let Xbe a C" vector field on a compact C" manifold Mn

satisfying (l)-(4). Denote by V^ those points of dM at which X is ori­ented in, and Vt those points ofdM at which X is oriented out. Then there is a C" function f on M which has these properties:

(a) The critical points of f coincide with the singular points of X and f coincides with the function of condition (1) plus a constant in some neighborhood of each critical point.

(b) If X is not zero atxeM, then it is transversal to the level hyper-surface of fat x.

(c) If/3eMisa critical point of f, then f(fi) is \fi), where \(J3) is the index of /9.

(d) / has value — \ on Vt and n + J on Vt. REMARK. It is easily proved from (a)-(d) that there is a riemannian

metric on M such that grad f=X. The next theorem follows easily from Theorems A and B. THEOREM C. Let M" be a compact C~ manifold with dM equal to the

disjoint union of V, and Vit each Vt closed in dM. Then there exists a C~ function f on M with non-degenerate critical points, regular on dM, /(V,) = — i, /(V,) = n + i and at a critical point /3of f,f(fi) — index /S.

For some motivation of these theorems see [4], [5], and [6]. In [4] Theorem A was announced for the case dM = 0 , while Theorem C was announced in [5] for the case dM = 0 . These theorems have implications in differential equations on one hand and topology on the other, both of which we will pursue in future papers.

As this paper was finished, an article by A. H. Wallace [7] appeared and seems to bear some relationship to this paper.

1. Proof of Theorem A.

First it is easily shown that there exist C1 approximations / ' of / such that / ' is C~ and has distinct values at distinct critical points. Thus in proving Theorem A we can assume / has these properties.

LEMMA 1.1. Let fbeaC" function on a compact riemannian manifold with non-degenerate critical points and X = gr&dfis transversal to dM. Then a sufficiently close C1 approximation X' of X with X' = X in a neighborhood of the singular points, satisfies condition (3) above. (One does not need such strong hypotheses on X'.)

PROOF. One can assume that X and X' have the property that, except

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DYNAMICAL SYSTEMS 201

at singular points, dfX and dfX' are positive. Then an orbit <p,(x) of Xor X' is either a singular point or has the property that /<p,(x) increases as t increases. Property (3) then follows. This fact that /<p,(x) increases as t increases is used in the rest of the paper without mentioning it again. It implies, for example, that there are no recurrent orbits of X and X1.

By 1.1 it is sufficient for the proof of Theorem A to show: LEMMA 1.2. Iff isaC" function on a compact C riemannian mani­

fold M, with non-degenerate critical points, distinct critical points having distinct values and X = grad / transversal to 9M, then there exist Cl approximations Yof Xsatisfying condition (4) and X= Yon some neighborhood of the critical points.

Index the critical points /9, of/of 1.2 so that/(£«) > /C9«-i), t = 1, • • •, r. Thus ft is the minimum of / . Denote by Wf and W* respectively the unstable and stable manifolds associated to £«. Let /?, = /(/9J, each t.

LEMMA 1.3. Given sufficiently small ^ > 0, j , there is a Cl approxima­tion X' of X such that X' = X outside off~l 0, + e„ &, + 3e,) and in the X' system W, and W* have normal intersection, each i. (" Wj in the X' system" has the obvious meaning.)

PROOF. Assume fifi,) + Se, < &SVi. Let dim W = n — k and Q be the submanifold f-\&, + &,) n W, of M. Let P=x = (x\ •••,**) 111 x || ^ 1 be the fc-disk and Im- z\ —m £ z £ m, m > 0. Then for small enough m there is a diffeomorphism h of /„ x P x Q onto a neighborhood U of Q sending identically 0 x 0 x Q onto Q and such that X = dldz' on U where / = h(z x 0 x 0) and U<zf-l(P, + e, JH, + 3e). We will identify points under h so that points of Uwill be represented by (z, x,y),\z\ ^ m, ||a;|| ^ land y e Q.

The proofs of the following two lemmas will be left to the reader. LEMMA 1.4. Let Im = [-m, m] and e > 0. Then there isaS> 0such

that ifv~<$, there is a C~ function (3(z) on Im, zero in a neighborhood of dlm, 0 £ £(z) ^ e, I £'(«) \^&,and

m/3(z)dz = ± V . 0

LEMMA 1.5. Let P be the k-disk as above. There is a C function y on P which is zero in a neighborhood of 8P, 0 ^ 7 ^ 1, | (frrldx1) | ^ 2 and 7(x) = l /or || x || £1/3.

With e arbitrary, let 5, be the minimum of the £ in 1.4 and 1/100, and let g be the restriction of xr: Im x P x Q — 0 x P x 0 = P to L C (0 x P x Q) n FFi*. Now by Sard's theorem [3] choose v e P such that || v|| = v < 8j and + 2v is a regular value of g. We can assume,

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using an orthogonal change of coordinates in P, that v = (x1, •••,**) = v,0, . . . , 0 ) .

Let X' be the vector field on M which equals X outside U and on U is given by

j r - . i + flWtfoJL. where /9 and 7 are chosen by 1.4 and 1.5. We claim that X' satisfies 1.3 if the eof 1.4 has been chosen small enough.

To see that X' is well defined it is sufficient to note that the second term vanishes in a neighborhood of dU. It is easy to check that X' can be made arbitrarily close in the C1 sense to X by choosing the e of 1.4 small enough.

It remains to prove that Wt and W,* have normal intersection in the X' system for each i. So fix t in what follows.

Let i?, be the orbit in the X' system through x with ^(x) = x and denote by W(*' and W, respectively W* and W, in the X' system. It is sufficient to prove Wt*' and Wj have normal intersection in U since any point q e Wj n W*' is of the form ^,(p), pe U and V, preserves the property of normal intersection.

Let V = (z, x, y) eU\\\x\\g 1/3. On V,

X' = - ? - + /3(z)— dz v 'dx1

and integrating the corresponding system of differential equations, we get z(t) = t + K„, x'(t) = I &(t)dt + Ku with the other coordinates con­stant. Then as long as we are in V,

*«(0, x, y) = (t, x1 + \'p(t)dt, x\ • • •, x*, j/) .

Using the main property of £(«) in 1.4, i/r,(0, x, y) stays in V for \t\ £ m, || as || £ 1/6, and ^±m(0, x, y) = (±m, x ± v, y) for || x || £ 1/6.

Let V, and V, denote respectively J7t*' n V and W/ D V where V = (0, x, y) e tfl || x || g 1/6. Then it is sufficient to show that Vt and V, have normal intersection in 0 x P x Q.

Since W,n0x PxQ= (0, 0,y)eU\ye Q, and )P, = Wj when restricted to ( -w , x, y) e t/ and also ^zl

m(—m, x, 1/) = (0, x + v, 1/) for || s || £ 1/6, we obtain V, = (0, +v, y) e [/. Hence if *,: IT— P is the previously defined projection, TT,(VJ) = +«• If ff is the restriction of it, to Vu then ff'l(+v) = V, n V€.

Since the intersection of JF<* and W,*' with (+m, x, y) € U are the same and V- . (+»i . x, 1/) = (0, x - v, y) we have

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DYNAMICAL SYSTEMS 203

V% = (0, x-v,y)\ (0, x, y) e W? n V, \\ z - v \\ S 1/6 . This implies that since g has a regular value at +2v, g has a regular value at +v. Hence dim V = dim P+ dim (Vt n F,) and since dim P= k, Vt and Vj have normal intersection i n O x P x Q . This proves 1.3.

We show that 1.2 follows from 1.3 by induction on the following hy­pothesis:

M(q)'. There is a C1 approximation Xq of X (of 1.2) such that Xq = X in a neighborhood of the /9„ Wr_, and Wj* have normal intersection in the X, system for all p ^ q and all i.

Then JC(0) is trivial and JC(r) implies 1.2. We will now show that M(q — 1) implies JC(q). Given .X ., by JCq — 1) we will construct X,. We can suppose that df(Xq^ = 0 only on the Bt. Let«! = l/4(/8«+l — #«> and apply 1.3 to obtain an approximation Xq of X,^ with df(Xq) = 0 only on the /9«, JT, = -ATj., on a neighborhood of the 8if and in the X, system, W,* and WT-q having normal intersection for all i. But also W and Wf will still have normal intersection in the Xq system for j > r — q and all i since this is true in the -X,-, system, Xq ss Xq-^ on f~l([&q+u £,]) and W, n W af-'d^,, &.]). This finishes the proof of 1.2.

2. Proof of Theorem B. LEMMA 2.1. Let X be a C vector field on a compact C" manifold Af*

satisfying (l)-(4) with Vx and V, the subsets of dM described in Theorem B. Then there exists a set of disjoint closed (n — l)-dimensional sub-manifolds Bt of M, i = —1, 0,1, • • •, n with the following properties:

( i ) B.X=V„ Bn=Vt. (ii) Each Bt is transversal everywhere to X. (iii) Each Bk, k^ — 1, n, divides Minto two regions whose closures we

denote by G* and Hk, with Gk D Gk-lt Hk D Hk+l and Gk containing exactly those singular points of index ^ k. For completeness we let G_, = B-lt H-t = M,Gn = Mand Hn = Bn. Hence, for k= - 1 , 0 , -•fn,GknHk = Bk and G, U Hk = M.

(iv) On Bk, X is oriented into Hk. The proof goes by induction on k. Roughly having constructed Bk-ir

we augment G»_, by tubular neighborhoods of the stable manifolds cor­responding to singular points of index k to obtain G» (and hence Bk).

PROOF. Take JB_, = Vx and assume we have constructed £»_, with M= G»-j U Ht-i, G»_, n Ht-i = £*_„ G,_, containing those singular points of index ^ k — 1, and on Bk-lt X is oriented into fl»_,. We will now con­struct Bk.

Let B*_, x [—1,1] be a product neighborhood of Bk-X (in case k = 0,

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take B„ x [0,1]) with £»_, = 5*^ x 0, Bx., x [0,1] c i?*_s and Bt_s x t transversal to X for each t.

Denote by 74, i = 1, • • •, r, the singularities of X of index k, and changing notation let VF* = W*k and Wt = PT""* denote the stable and unstable manifolds respectively of 7t, i — 1, •••,r. Then if x e W*, the orbit of x passes through V = 5»_, x 1 by Lemma 3.1 of [4] at least once and hence exactly once (the proof of 3.1 in [5] is for closed manifolds but applies equally well to our case; this easy lemma is the only use we make of [4]).

Let 7 be one of the iit W = Wt, W* = W*. One chooses from condi­tion (1) an open neighborhood N of 7, / on N and 8 > 0 such that the (n - ft)-disk bounded by f~\8) Q W = W is in N. Let E, be the normal bundle of W in M restricted to FT of vectors with magnitude ^ e. Denote by S, the image of E, under the exponential map. Assume e > 0 is so small that S, is transversal to X.

If e > 0 is sufficiently small one can define an imbedding T: S, — W—+ V, by sending x e S, — W into the point of the orbit through x meeting V,. Assume s is this Bmall and denote the image of T with 7 = y( by Kt, for each i = 1, • • •, r. We assume that e is small enough so that these Kt, are mutually disjoint.

Now define a C" imbedding F: dS, x [-1,1] — M by sending (p, —1) into p, (p, 1) into Tp) and (p, t) into the orbit joining p and Tp), the distance from p proportional to t. Then extend F to C" imbedding of dS, x [—2, 2], which sends p x [—2, 2] into a single orbit, each p.

Next in the construction of Gk and Bk we modify F slightly to a new C" imbedding. Fixing some riemannian metric on M, let v(j>, t) be the unit normal vector field on the image of F whose orientation is determined by the vectors on dS, oriented away from W. For r], a small positive con­stant, let F„(p, t) be the point at distance rjl from F(p, t) along the geodesic determined by v(p, t).

Choose y] so small that image F, = im F„ is disjoint from the K4», im F, is transversal to X everywhere, and im F„ n S„ im F, n V, are diffeo-morphic respectively to im F n S„ im F n t i.

Repeating this construction for each singular point 74 we obtain a hypersurface (singular) B't in M made up of the following pieces:

(a) The part of S, bounded by im F, ("1 S„ one corresponding to each 7<; (b) Vi minus pieces bounded by im F, n Vl and containing W* n V„

one such piece corresponding to each 7t; and (c) the part of im F, bounded by im F, n St and im F, D V,, one for

each 7t.

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DYNAMICAL SYSTEMS 205

Then B't has the property that, on each piece, it is transversal to X, M — B't = Gi U Hi, with G't containing G»_j and all the singular points of index k. In fact G'» only fails to satisfy G» of 2.1 in that dG't = Bi is not a differentiate submanifold, but has corners along im F, n Vx and im Fn n S, for each singular point. This is easily modified however to obtain the desired G» and Bt by the device of "straightening the angle" (see [2] for some discussion), the details of which we leave to the reader. This finishes the proof of 2.1.

LEMMA 2.2. Let XbeaC vector field on a manifold M" satisfying conditions (1), (2) and (3) with only singular points of index k. Let V, and V, be as in Theorem B. Then there is a C" function on M which satisfies conditions (a), (b), and (c) of Theorem B and has value k — on Vu value k + J on V,.

PROOF. Let yit • • •, 7, denote the singular points of X, Wt and W,*, their respective unstable and stable manifolds. We will first define the desired function in a neighborhood of U,„ (^t U Wf). Let Nt and / , be neighborhoods and functions of condition (1) but suppose also N( is as in the proof of 2.1. Furthermore assume /4(74) = k by adding appropriate constants.

Take y = yt, some i,f = /„ W = Wit W = Wf, and N= Nt. Then let f~l(k + 8) n N = R, f~\k - 8) = R~, with 8 chosen as in previous lemma, R, = (x, y) e A11| y || £ e), and Rr = (as, y) e R~ \ || x \\ £ e.

Fix a riemannian metric on M and take e = 1/10. For x € R„ re-define / o n <pt(x), t ^ 0, so that /(^>,(x)) = k + 8,f(y) = k + where y is the point of <p,(x) meeting V„ and on the points between <p„(x) and y on <p,(x), f is defined proportionally to arc length. Thus we have obtained an / on a neighborhood of W satisfying the right boundary conditions, but is not differentiable on/_1(8). By a smoothing process similar to the one dis­cussed by Milnor 8.1, 8.2 of [2], /can be made C" on /_1(S).

In the same way using R^, one gets / defined on a neighborhood Q of W* as well as a neighborhood of W which satisfies the condition f(Q flV)= k — I. This, by iteration, yields a function / defined on disjoint open neighborhoods P, of Wt U W* which agrees with the / , on some neigh­borhoods of the ytf of /(P t r\ VJ = k - i , / (P , D V.) = k + i, and / has only critical points at the yt. Furthermore / satisfies condition (b) of Theorem B. We can assume without loss of generality that the closures of the P, are disjoint and if x e P«, all of <p,(x) lies in Pt. We will now extend / to all of M.

Choose t7« c Vi n Pi. to be a compact neighborhood of W n V„ i = 1, • • •, r. Then let X be a real C" function on V, satisfying 0 £ X £ 1,

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206 STEPHEN SMALE

\ = loneach r7„\ = 0on Vi -LK^P t n V1.Porx€Jlf-U;.1(WiU W?) let l(x) he the length of the orbit through x, vx) be the distance from ?>»(«) n V1 to * along q>t(x) and g(x) = fc — J + (v(x))l(lx)). One can now show that the function \f + (1 — X)y on M has the desired properties of the function of 2.2, where \(x) = \(<pt(x) (~l V,) or 1 if <pt(x) does not meet V,.

Finally we prove Theorem B. Take / on the closure of Gt — G»_, of 2.1 to be the function of 2.2, fc = 0,1, • • •, n. One obtains a well defined function and by smoothing this in a neighborhood of Bt, • • •, 5 , - , as in the proof of 2.2, the desired function of Theorem B is obtained.

UNIVERSITY or CALIFORNIA, BERKELEY

BIBLIOGRAPHY

1. E. A. CODDINGTON and N. LEVMSON, Theory of Ordinary Differential Equations, Mc­Graw-Hill, New York, 1955.

2. J. MILNOR, Differentiable manifolds which are homotopy spheres, (mimeographed) Princeton, 1959.

3. A. SARD, Ths msarur* of th* critical values of differsntiablt map*, Bull. Amer. Math. Soc., 48 (1940), 883-890.

4. S. SHALE, Morse inequalities for a dynamical system, Bull. Amer. Math. Soc., 66 (1960), 43-49.

5. , Generalized Poineari conjecture in higher dimensions, Bull. Amer. Math. Soc., 66 (1960), 373-375.

6. , On dynamical systems, to appear. 7. A. H. WALLACE, Modification* and eobounding manifolds, Canad. J. Math., XII (1960),

503-528.

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AMKAXJB O » M t A i m M A T i c a Vol. 74. No. 2, September. 1961

Printtd in Japan

GENERALIZED POINCARE'S CONJECTURE IN DIMENSIONS GREATER THAN FOUR

BY STEPHEN SMALE*

(Received October 11, 1960) (Revised March 27, 1961)

Poincare has posed the problem as to whether every simply connected closed 3-manifold (triangulated) is homeomorphic to the 3-sphere, see [18] for example. This problem, still open, is usually called Poincare's conjec­ture. The generalized Poincare conjecture (see [11] or [28] for example) says that every closed n-manifold which has the homotopy type of the n-sphere S" is homeomorphic to the n-sphere. One object of this paper is to prove that this is indeed the case if n 5 (for differentiable manifolds in the following theorem and combinatorial manifolds in Theorem B).

THEOREM A. Let M*be a dosed C" manifold which has the homotopy type of S", n ^ 5. Then M" is homeomorphic to S".

Theorem A and many of the other theorems of this paper were announ-ed in [20], This work is written from the point of view of differential topology, but we are also able to obtain the combinatorial version of Theo­rem A.

THEOREM B. Let Mn be a combinatorial manifold which has the homo­topy of S", n 5. Then M" is homeomorphic to S".

J. Stallings has obtained a proof of Theorem B (and hence Theorem A) for n ;> 7 using different methods (Polyhedral homotopy-spheres, Bull. Amer. Math. Soc., 66 (1960), 485-488).

The basic theorems of this paper, Theorems C and I below, are much stronger than Theorem A.

A nice function/on a closed C" manifold is a C" function with non-degenerate critical points and, at each critical point £,/(£) equals the in­dex of 0. These functions were studied in [21].

THEOREM C. Let M* be a closed C~ manifold which is m — ^-con­nected, and n 2m, n, m) =£ (4, 2). Then there is a nice function f on M with type numbers satisfying M0 = Mn = 1 and Mt = 0/or 0<i<m, n — m < » < n.

Theorem C can be interpreted as stating that a cellular structure can be imposed on M" with one 0-cell, one »-cell and no cells in the range 0<i<m, n — m<i<n. We will give some implications of Theorem C.

♦The author is an Alfred P. Sloan Fellow. 391

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392 STEPHEN SMALE

First, by letting m --- 1 in Theorem C, we obtain a recent theorem cf M. Morse [13].

THEOREM D. Let M" be a closed connected C" manifold. There exists a (nice) non-degenerate function on M with just one local maximum and one local minimum.

In § 1, the handlebodies, elements of SC(n, k, s) are defined. Roughly speaking if HeJ(n, k, s), then H i s defined by attaching s-disks, k in number, to the n-disk and "thickening" them. By taking n = 2m + 1 in Theorem C, we will prove the following theorem, which in the case of 3-dimensional manifolds gives the well known Heegard decomposition.

THEOREM F. Let Mbe a closed C~ (2m + \)-manifold which is (m—1)-connected. Then M = HUH', HC)H' = dH = dH' where H, H' e M(2m + 1, k, m) are handlebodies (dV means the boundary of the mani­fold V).

By taking n = 2m in Theorem C, we will get the following.

THEOREM G. Let Mm be a closed (m—l)-connected C" manifold, m^2. Then there is a nice function on M whose type numbers equal the cor­responding Betti numbers of M. Furthermore M, with the interior of a 2m-disk deleted, is a handlebody, an element of JC(2m, k, m) where k is the mtb Betti number of M.

Note that the first part of Theorem G is an immediate consequence of the Morse relation that the Euler characteristic is the alternating sum of the type numbers [12], and Theorem C.

The following is a special case of Theorem G.

THEOREM H. Let Af"" be a closed C°° manifold m =t 2 of the homotopy type of S1". Then there exists on Ma non-degenerate function with one maximum, one minimum, and no other critical point. Thus M is the union of two 2m-diska whose intersection is a submanifold of M, diffeo­morphic to Slm~l.

Theorem H implies the part of Theorem A for even dimensional homo­topy spheres.

Two closed C~ oriented n-dimensional manifolds Mt and Mt are J-equiv­alent (according to Thom, see [25] or [10]) if there exists an oriented manifold V with dV diffeomorphic to the disjoint union of Af, and — M„ and each Mt is a deformation retract of V.

THEOREM I. Let Mx and M, be (m — l)-connected oriented closed C" (2m+l)-dimensional manifolds which are J-equivalent, m =£1. Then Af, and Af, are diffeomorphic.

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We obtain an orientation preserving diffeomorphism. If one takes M^ and M, /-equivalent disregarding orientation, one finds that M, and M, are diffeomorphic.

In studying manifolds under the relation of /-equivalence, one can use the methods of cobordism and homotopy theory, both of which are fairly well developed. The importance of Theorem I is that it reduces diffeo­morphism problems to /-equivalence problems for a certain class of mani­folds. It is an open question as to whether arbitrary /-equivalent manifolds are diffeomorphic (see [10, Problem 5])(Since this was written, Milnor has found a counter-example).

A short argument of Milnor [10, p. 33] using Mazur's theorem [7] ap­plied to Theorem I yields the odd dimensional part of Theorem A. In fact it implies that, if M*"+l is a homotopy sphere, m =£ 1, then MM+l minus a point is diffeomorphic to euclidean (2m+l)-space (see also [9, p. 440]).

Milnor [10] has denned a group JC" of C" homotopy n-spheres under the relation of /-equivalence. From Theorems A and I, and the work of Milnor [10] and Kervaire [5], the following is an immediate consequence.

THEOREM J. If n is odd, » =£ 3, JC" is the group of classes of all dif-ferentiable structures on S" under the equivalence of diffeomorphism. For n odd there are a finite number of differentiable structures on S". For example:

n

Number of Differentiable Structures on S"

3

0

5

0

7

28

9

8

11

992

13

3

15

16256

Previously it was known that there are a countable number of differen­tiable structures on 5" for all n (Thorn), see also [9, p. 442]; and unique structures on S" for n ^ 3 (e.g., Munkres [14]). Milnor [8] has also es­tablished lower bounds for the number of differentiable structures on S" for several values of n.

A group rB has been defined by Thom [24] (see also Munkres [14] and Milnor [9]). This is the group of all diffeomorphisms of S"_1 modulo those which can be extended to the n-disk. A group A* has been studied by Milnor as those structures on the n-sphere which, minus a point, are dif­feomorphic to euclidean space [9]. The group r* can be interpreted (by Thom [22] or Munkres [14]) as the group of differentiable structures on S" which admit a C" function with the non-degenerate critical points, and hence one has the inclusion map i: V — A" defined. Also, by taking / -equivalence classes, one gets a map p: A* —♦ M".

176

394 STEPHEN Slf ALE

THEOREM K. With notation as in the preceding paragraph, the follow­ing sequences are exact:

(a) A" - ^ JC > 0 , n =* 3, 4

(b) r" - ^ A" 0 , n even *= 4

(c) 0 > A" - ^ .#" , n odd * 3 .

Hence, if n is even, » =f- 4, r n = A" and, if n is odd + 3, A" = « # \ Here (a) follows from Theorem A, (b) from Theorem H, and (c) from

Theorem I. Kervaire [4] has also obtained the following result. THEOREM L. There exists a manifold with no differentiable structure

at all. Take the manifold W„ of Theorem 4.1 of Milnor [10] for k = 3. Milnor

shows dW„ is a homotopy sphere. By Theorem A, d W„ is homeomorphic to S". We can attach a 12-disk to W0 by a homeomorphism of the bound­ary onto dWtU> obtain a closed 12 dimensional manifold Af. Starting with a triangulation of W„, one can easily obtain a triangulation of Af. If Af possessed a differentiate structure it would be almost parallelizable, since the obstruction to almost parallelizability lies in H'(M, Kt(SO (12))) — 0. But the index of Af is 8 and hence by Lemma 3.7 of [10] Af cannot possess any differentiate structure. Using Bott's results on the homotopy groups of Lie groups [1], one can similarly obtain manifolds of arbitrarily high di­mension without a differentiate structure.

THEOREM M. Let C " be a contractible manifold, m^2, whose bound­ary is simply connected. Then C** is diffeomorphic to the 2m-diak. This implies that differentiate structures on disks of dimension 2m, m^2, are unique. Also the closure of the bounded component C of a C" im­bedded (2m — l)-sphere in euclidean 2m-space, m ^ 2, is diffeomorphic to a disk.

For these dimensions, the last statement of Theorem M is a strong version of the Schoenflies problem for the differentiable case. Mazur's theorem [7] had already implied C was homeomorphic to the 2m-disk.

Theorem M is proved as follows from Theorems C and I. By Poincare duality and the homology sequence of the pair (C, dC), it follows that dC is a homotopy sphere and J-equivalent to zero since it bounds C. By Theorem I, then, dC is diffeomorphic to Sn. Now attach to C " a 2m-disk by a diffeomorphism of the boundary to obtain a differentiable manifold V. One shows easily that V is a homotopy sphere and, hence by Theorem H, V is the union of two 2m-disks. Since any two 2m sub-disks of Vare

177

178

POINCARfi CONJECTURE 395

equivalent under a diff eomorphism of V(for example see Palais [17]), the original Clm c V must already have been diffeomorphic to the standard 2m-disk.

To prove Theorem B, note that V = (M with the interior of a simplex deleted) is a contractible manifold, and hence possesses a differentiable structure [Munkres 15]. The double Wof Vis a differentiable manifold which has the homotopy type of a sphere. Hence by Theorem A, W is a topological sphere. Then according to Mazur [7], dV, being a differentiable submanifold and a topological sphere, divides PTinto two topological cells. Thus V is topologically a cell and M a topological sphere.

THEOREM N. Let C"", m + 2, be a contractible combinatorial manifold whose boundary is simply connected. Then C^is combinatorially equiv­alent to a simplex. Hence the Hauptvermutung (see [11]) holds for combinatorial manifolds which are closed cells in these dimensions.

To prove Theorem N, one first applies a recent result of M. W. Hirsch [3] to obtain a compatible differentiable structure on C*\ By Theorem M, this differentiable structure is diffeomorphic to the 2/n-disk #**. Since the standard 2m-simplex ff**isaC triangulation of £>**, Whitehead's theorem [27] applies to yield that C*" must be combinatorially equivalent to a**.

Milnor first pointed out that the following theorem was a consequence of this theory.

THEOREM 0 . Let M*M, m^2,bea combinatorial manifold which has the same homotopy type asS**. Then M*~ is combinatorially equivalent to S*". Hence, in these dimensions, the Hauptvermutung holds for spheres.

For even dimensions greater than four, Theorems N and 0 improve recent results of Gluck [2],

Theorem O is proved by applying Theorem N to the complement of the interior of a simplex of M lm.

Our program is the following. We introduce handlebodies, and then prove "the handlebody theorem" and a variant. These are used together with a theorem on the existence of "nice functions" from [21] to prove Theorems C and I, the basic theorems of the paper. After that, it re­mains only to finish the proof of Theorems F and G of the Introduction.

The proofs of Theorems C and I are similar. Although they use a fair amount of the technique of differential topology, they are, in a certain sense, elementary. It is in their application that we use many recent re­sults.

396 STEPHEN SMALE

A slightly different version of this work was mimeographed in May 1960. In this paper J. Stallings pointed out a gap in the proof of the handlebody theorem (for the case 8 = 1). This gap happened not to affect our main theorems.

Everything will be considered from the C°* point of view. All imbed-dings will be C°°. A differentiable isotopy is a homotopy of imbeddings with continuous differential.

E" = x = (x„ • • •, *„), II x || = (£ . " .*?)'" , D" = x e En 11| x || £ 1, dDn = S"-1 = x e En \ || x || = 1 ;

Z)* etc. are copies of D". A. Wallace's recent article [26] is related to some of this paper.

1. Let M" be a compact manifold, Q a component of dM and

fi:dD\xDr'-^Q,i= 1, • • - , * imbeddings with disjoint images, s ^ 0, n ^ s. We define a new compact C°° manifold V — x(M, Q;fu ••,/*;*) as follows. The underlying topo-logical space of Vis obtained from M, and the D;xX>*"' by identifying points which correspond under some/ ( . The manifold thus defined has a natural differentiable structure except along corners dD\ x dD*~' for each i. The differentiable structure we put on V is obtained by the process of "straightening the angle" along these corners. This is carried out in Milnor [10] for the case of the product of manifolds Wt and W, with a corner along dWx xdW,. Since the local situation for the two cases is es­sentially the same, his construction applies to give a differentiable struc­ture on V. He shows that this structure is well-defined up to diffeomor-phism.

If Q = dM we omit it from the notation x(Af, Q; / „ • • •, /*; s), and we sometimes also omit the s. We can consider the "handle" D'x I)*~' c V as differentiably imbedded.

The next lemma is a consequence of the definition.

(1.1) LEMMA. Let U dD\x D* - — Q and ft. dD\x D'< - — Q, T=-1,- * - ,/c be two sets of imbeddings each with disjoint images, Q, M as above. Then *(M, Q;fu---,/»; s) and %(M, Q; f[,• • •, f'„; s) are diffeomorphic if

(a) there is a diffeomorphism h: M —» M such that /, ' = hfit i — 1, • • •, k; or

(b) there exist diffeomorphisms h: D' x £>•*'-» D* x Dnt such that f'i=fihi,i = 1, '-',k;or

(c) the f\ are permutations of the / , . If Vis the manifold x(M, Q;/„- • • , /»; s), we say a = (M, Q;/„« • •,/*; *)

179

POINCARfi CONJECTURE 397

is a presentation of V. A handlebody is a manifold which has a presentation of the form

(D*; / „ • • •, /»; »). Fixing n, k, 8 the set of all handlebodies is denoted by J(n, k, e). For example, JC(n, k, 0) consists of one element, the disjoint union of (k+1) n-disks; and one can show JC(2,1,1) consists of S1 x I and the Mobius strip, and M(3, k, 1) consists of the classical handlebodies [19; Henkelkbrper], orientable and non-orientable, or at least differentiable analogues of them. The following is one of the main theorems used in the proof of Theorem C. An analogue in § 5 is used for Theorem I.

(1.2) HANDLEBODY THEOREM. Let n ^ 2* + 2 and, if s 1, n ^ 5; let H e SCn, k, a), V = x(H; / „ • • •, / r ; « + 1), and n,( V) = 0. Also, if 8 = 1, assume KX(XH; / „ • • ' , fr-k; 2)) = 1. Then VeJC(n, r - k, a + 1). (We do not know if the special assumption for 8 = 1 is necessary.)

The next three sections are devoted to a proof of (1.2).

2. Let Gr = Gr8) be the free group on r generators Dlt • ♦ •, Dr if 8 = 1, and the free abelian group on r generators Dlt • • •, Dr if 8 > 1. If o = (Af, Q;flt '••, fT; 8 + 1) is a presentation of a manifold V, define a homo-morphism f„: Gr —> n,(Q) by /„( A) = 9><» where <p( e 7r,(Q) is the homotopy class of /<: 9Z>*+1 x 0- >Q, the restriction of ft. To take care of base points in case 7r,(<2) ^ 1, we will fix x0 e 9Z)'+1 x 0, y0 e Q, Let U be some cell neighborhood of y„ in Q, and assume f^x0) e /. We say that the homomor-phism / , is induced by the presentation a.

Suppose now that F: Gr — :r,(Q) is a homomorphism where Q is a com­ponent of the boundary of a compact n-manifold Af. Then we say that a manifold V realizes F if some presentation of V induces F. Manifolds realizing a given homomorphism are not necessarily unique.

The following theorem is the goal of this section.

(2.1) THEOREM. Let n ^ 2s + 2, and if 8 = 1, n ^ 5; let a = (Af, Q; /IT m",fr',s + 1) be a presentation of a manifold V, and assume TT^Q) = 1 if n = 2s + 2. Tften / o r ouy automorphism a:Gr—*Gr, V realizes f*«.

Our proof of (2.1) is valid for 8 = 1, but we have application for the theorem only for s > 1. For the proof we will need some lemmas.

(2.2) LEMMA. Let Q be a component of the boundary of a compact manifold Af" and/x: dD'xD%~'-Qo» imbedding. Let/,: dD'xO — Q be an imbedding, differentially isotopic in Q to the restriction fx of / , to dD' x 0. Then there exists an imbedding / , : dD' x Dn~'—*Q extending / , and a diffeomorphiam h: M—> Msuch that hft = / , .

180

398 STEPHEN SHALE

PROOF. Let / , : 9D* x 0 — Q, 1 ^ £ <; 2, b e a differentiable isotopy be­tween / , and / , . Then by the covering homotopy property for spaces of differentiable imbeddings (see Thorn [23] and R. Palais, Comment. Math. Helv. 34 (I960)), there is a differentiable isotopy F^dD'xD' ' —Q, 1 ^ t ^ 2, with F, = / , and Ft restricted to 3D* x 0 = / , . Now by apply­ing this theorem again, we obtain a differentiable isotopy G,:M—>M, 1 ^ t ^ 2, with G, equal the identity, and Gt restricted to image F, equal FtF^. Then taking h - - Gf\ F, satisfies the requirements of / , of (2.2); i.e., hft = G?Ft = FJSFt = / , .

(2.3) THEOREM (H. Whitney, W.T. Wu). Let n ^ max (2k+ 1,4) and f, fir: Mk — X" fee two imbeddings, M closed, M connected and X simply connected if n = 2k + 1. Then, iff and g are homotopic, they are dif­ferentially isotopic.

Whitney [29] proved (2.3) for the case n ^ 2k + 2. W.T. Wu [30] (us­ing methods of Whitney) proved it where X* was euclidean space, n = 2k + 1. His proof also yields (2.3) as stated.

(2.4) LEMMA. Let Q be a component of the boundary of a compact manifold Mn, n ^ 2a -\ 2 and if s = 1, n ^ 5, and 7T,(Q) = 1 if n = 2s + 2. Let / , : dD'+I x I)*--1 ^Q be an imbedding, and / , : dD'+J x 0—Q an imbedding homotopic in Q tof, the restriction of'/, to dD**1 x 0. !Tfte» Mere exists an imbedding / , : 9D'+1 x D""*-1 — Q extending ft such that X(M, Q: /,) ia diffeomorphic to x(M, Q; / , ) .

PROOF. By (2.3), there exists a differentiable isotopy between / i and / , . Apply (2.2) to get/r.SD'+'xD—*-1 —Q extending/,, and a diffeo-morphism h: M — M with fc/, = / , . Application of (1.1) yields the de­sired conclusion.

See [16] for the following.

(2.5) LEMMA (Nielson). Let G be a free group on r-generators D;, • « •, Dr\, and Jl the group of automorphisms of G. Then Jl is gen­erated by the following automorphisms:

R: A - D , \ D. - A » > 1 2 V A — A . A — A , D,->D, j * l , j * i , i = 2t»-,r

S:A —AA, A - A , » > 1. The same is true for the free abelian case (well-known).

It is sufficient to prove (2.1) with a replaced by the generators of Jl of (2.5).

First take a = R. Let h: D'*lxDu-"' - D,+lxDn—1 be defined by

181

POINCARE CONJECTURE 399

h(x, V) = (T, x, y) where r: D'+1 — Z),+1 is a reflection through an equa­torial »-plane. Then l e t / ; =fh. If a' = (AT, Q;/,',/„ • . . , / r ; s + 1), X^) is diffeomorphic to Fby (1.1). On the other hand xtf) realizes /„. = f„a.

The case a = Tt follows immediately from (1.1). So now we proceed with the proof of (2.1) with a = S.

Define 7, to be the manifold x(M, Q; / „ • • •, / , ; s + 1) and let QiddVt beQ1 = 8V1 — (dM—Q). Let 9\ e i:,(Q), i = 1, • • •, r denote the homotopy class of /«: 8DJ+1 x 0 -* Q, the restriction of /«. Let 7: 7r,(Q n QO — *»(£) and /9:7r,(Q n Qi) — n.(Qi) be the homomorphisms induced by the respec­tive inclusions.

(2.6) LEMMA. With notations and conditions as above, <ptey Ker /9. PROOF. Let q € dDi~'~l and ^: 9D;+1 x q — Q n Qi be the restriction of

/ , . Denote by -f e s,(Q n &) the homotopy class of ijr. Since $• and / , are homotopic in Q, 7$ = <pt. On the other hand ffi = 0, thus proving (2.6).

By (2.6), let + e n,Q n Qi) with 7^ = «P, and /3^ = 0. Let 0 = 1/ + ^ (or !/•?• in case * = 1; our terminology assumes *>1) where y e x,(Q 0Q0 is the homotopy class of / ,: dDl+> x 0 -»Q n Q,. Let p: M?,+1 x O - Q n Q , be an imbedding realizing g (see [29]).

If n = 2s + 2, then from the fact that JT,(Q) = 1, it follows that also ffi(Qi) = 1. Then since g and f are homotopic in Q„ i.e., ySg — y9y, (2.4) applies to yield an imbedding e: dD'+1 x Dn~*_1—Q, extending g such that Z( i» Qu e) an(* Z(Vi» Qi!/i) a r e diffeomorphic.

On one hand V= Xi.V,Q;f,"-,fr) = X(Vi, Qi?/i) and, on the other hand, x(V, Q; « , /„ ••',/,) = X(vi> Q» e)> so by the preceding statement, Vand x(V, Q; e,/„« • •,/,.) are diffeomorphic. Since 70=£r1+g„/„a(A)= / , ( A + A) = ft + 0>./^A) = ffA = 0. + fir,,/.a = / , . , where ^ = (V, Q; e, / „ . . . , / r ) . This proves (2.1).

3. The goal of this section is to prove the following theorem. (3.1) THEOREM. Let n g 2s + 2 and, i / « = 1, n ^ 5. Suppose H e

.#(«., A, 8). TTien given r ^ fe, t/t«re exists ow epimorphism g\ GT — ;r,(if) sweft that every realization of g is in JC(n, r — k, s + 1).

For the proof of 3.1, we need some lemmas. (3.2) LEMMA. / / M(n, k, s) then n,(H) is (a) a set of k + 1 elements if a = 0, (b) a free group on k generators if s = 1, (c) o free abelian group on k generators if s > 1.

Furthermore if n ^ 2s + 2, then n&dH) — *<(#) is an isomorphism for i ^ 8.

PROOF. We can assume 8 > 0 since, if s = 0, H is a set of n-disks k+1

182

400 STEPHEN SMALE

in number. Then H has as a deformation retract in an obvious way the wedge of k s-spheres. Thus (b) and (c) are true. For the last statement of (3.2), from the exact homotopy sequence of the pair (H, dH), it is suf­ficient to show that n^H, dH) = 0, i <; 8 + 1.

Thus let / : (D\ dD*) — (H, dH) be a given continuous map with t ^ 8 + 1. We want to construct a homotopy fT: (D\ dD*) — (H, dH) with / , = / a n d / , ( ! ) • ) cSff.

Let / , : (D*, dD*)—>(H, dH) be a differentiate approximation t o / . Then by a radial projection from a point in D* not in the image of / „ / , is homo-topic to a differentiate map/ , : (D\ dD*) —• (H, dH) with the image of/, not intersecting the interior of DnczH. Now for dimensional reasons / , can be approximated by a differentiate map/ , : (D*, dD*)-*(H, dH) with the image of/, not intersecting any D'xO<zH. Then by other projections, one for each i,f, is homotopic to a map / , : (D*, dD1) —»(H, dH) which sends all of D* into dH. This shows 7T<(if, dH) = 0 , i g s + l, and proves (3.2).

If /3 e 7T,_,(0(TC — s)), let Hfi be the (n — s)-cell bundle over S' deter­mined by /3.

(3.3) LEMMA.. Suppose V = x(i/„; / ; s + 1) where 0 e n^Oin - s)), n ^ 2s + 2, or t / s = 1, n ^ 5. Let aiso 7T,( V") — 0. TAe» V is diffeo-morphic to D".

PROOF. The zero-cross-section a: S' — Hf is homotopic to zero, since 7r,(V) = 0, and so is regularly homotopic in Kto a standard 8-sphere S'0 contained in a cell neighborhood by dimensional reasons [29]. Since a reg­ular homotopy preserves the normal bundle structure, a(S') has a trivial normal bundle and thus # = 0. Hence Hfi is diffeomorphic to the product of S' and Dn-.

Let a,: S' — dHf be a differentiable cross section and / : dD'*1 x 0—>dHfi the restriction of / : dD'*1 x Z)n_,_1 —< dHp. Then ^ and / are homotopic in dHp (perhaps after changing / by a diffeomorphism of D'+1 x D%~'~1

which reverses orientation of 3D*+I xO) since TT,V) = 0, and hence dif-ferentiably isotopic. Thus we can assume / a n d s, are the same.

Let / , be the restriction of / to dD'+1xDK,"1 where D;-'-1 denotes the disk x 3 £>—-111| x | | ^e , and e>0. Then the imbedding g,: dD'+l x D*-'-1 — dHp is differentially isotopic t o / where g,(x, y) = / fr,(x, y) and r,(x, y) = (x, ey). Define k,: dD,n x £>—»-» — dH, by p ^ x , y) where px: flr«(* x D"''~l) — Fx is projection into the fibre Fx of dHp over <7~'sr«(x, 0). If e is small enough, kt is well-defined and an imbedding. In fact if £ is small enough, we can even suppose that for each x, kt maps x x Dn~'~l

linearly onto image k, n Fx where image k, n F, has a linear structure

183

POINCARE CONJECTURE 401

induced from F„. It can be proved A;, and g, are differentiably isotopic. (The referee has

remarked that there is a theorem, Milnor's "tubular neighborhood theorem", which is useful in this connection and can indeed be used to make this proof clearer in general.)

We finish the proof of (3.3) as follows. Suppose V is as in (3.3) and V = x(Ht>;f; s + 1), ntV) = 0. It is sufficient to prove Vand V are diffeomorphic since it is clear that one can obtain D* by choosing / ' prop­erly and using the fact that Hf is a product of S' and Dn~'. From the previous paragraph, we can replace / and / ' by k, and k', with those prop­erties listed. We can also suppose without loss of generality that the images of k, and k't coincide. It is now sufficient to find a diffeomorphism h of Hft with hf=f. For each x, define h on image/n F, to be the linear map which has this property. One can now easily extend h to all of Hfi and thus we have finished the proof of (3.3).

Suppose now M* and Af J are compact manifolds and /<: Dni xi—> dMt are imbeddings for i -= 1 and 2. Then %(M, U Af,;/, U/,; 1) is a well de­fined manifold, where/, U/,: a.D,x.D"-1^9Af1U0Af, is defined by/ , and/,, the set of which, as the / vary, we denote by Af, + Af,. (If we pay at­tention to orientation, we can restrict Af, + Af, to have but one element.) The following lemma is easily proved.

(3.4) LEMMA. The set Af" + D* consists of one element, namely Af".

(3.5) LEMMA. Suppose an imbedding / : dD' x £>"-*—>8Mn is null-homotopic where M is a compact manifold, n ^ 2« + 2 and, if s = 1, n ^ 5. Then z(Af; / ) e Af + H$ for some 0 e ^-,(0(n - s)).

PROOF OF (3.5). Let / : 8D' x ? - > dM be the restriction of / where q is a fixed point in dD"''. Then by dimensional reasons [29],/can be ex­tended to an imbedding <p: D' —• 8Af where the image of <p intersects the image of / only on / . Next let T be a tubular neighborhood of <p(D') in Af. This can be done so that T is a cell, T u' [D' x £>"-*) is of the form H, and Ve M + Hf. We leave the details to the reader.

To prove (3.1), let H = 2(Z>";/„ • • • , / * ; * ) . Then / defines a class % 6 n,(H, Dn). Let 7t e 7z,(dH) be the image of 7\ under the inverse of the composition of the isomorphisms izJ@H) — n,(H) —* TZ,(H, D*) (using (3.2)). Define g of (3.1) by gD( =yiti^ k, and gD( = 0, i > k. That g satisfies (3.1) follows by induction from the following lemma.

(3.6) LEMMA. X(H>9»8 + l)e«#(n, A; — l,s)ifthe restriction of gi to dD'+l x 0 has homotopy class <y, e x,(dH).

Now (3.6) follows from (3.3), (3.4) and (3.5), and the fact that g, is dif-

184

402 STEPHEN SMALE

ferentiably isotopic to g\ whose image is in dHB T\ dH, where Hf is defined by (3.5) and / , .

4. We prove here (1.2). First suppose 8 = 0. Then H e M(n, k, 0) is the disjoint union of n-disks, k+\ in number, and V=x(H)fi, • • - , / r ; 1)-Since n0V) = 1, there exists a permutation of 1, • • •, r , iu • • •, i r such that Y — x(H>filt • ",fit; 1) is connected. By (3.4), Y is diffeomorphic to D\ Hence V = x( Y; /«,+1. • • •, fir; 1) is in SCn, r - k, 1).

Now consider the case s = 1. Choose, by (3.1), g: G*—7r,(9iJ) such that every manifold derived from g is diffeomorphic to D*. Let Y = x(H; fu • • • , / r _ t ) . Then TT,( Y) = -l and by the argument of (3.2), 7r,(d Y) = 1. Let gi:dDixO-^dHbe disjoint imbeddings realizing the classes flr(A)c Xi(dH) which are disjoint from the images of all .A, i = 1,- • •, k. Then by (2.4) there exist imbeddings glt • • •, gk: dB* x Dn~' —- dH extending the gt such that V = %(Y; /r_»+„ '••,fr) and z( Y; gu • • •, flf») are diffeomorphic. But

XY,g>, -~,gt) = XH;gu'",gk,f.,'",fr-*) = XiDn, fw", fr-t) e ^fl[n, r - k,2) .

Hence so does V. For the case s > 1, we use an algebraic lemma. (4.1) LEMMA. If f, g:G —- G' are epimorphisms where G and G' are

finitely generated free abelian groups, then there exists an automorphism a:G—*G such that fa -- g.

PROOF. LetG"be a free abelian group of rank equal to rankG — rankG', and let p: G' + G"— G' be the projection. Then, identifying elements of G and G' + G" under some isomorphism, it is sufficient to prove the existence of a for g = p. Since the groups are free, the following exact sequence splits

0 »/- l(0) G -^-> G' 0 .

Let h: G —»/-1(0) be the corresponding projection and let k: /_1(0) — G" be some isomorphism. Then a:G—>G' + G" defined by / + kh satisfies the requirements of (4.1).

REMARK. Using Grusko's Theorem [6], one can also prove (4.1) when G and G' are free groups.

Now take a = (IT; / „ • •, fr; s + 1) of (1.2) and g: Gr — K,(dH) of (3.1). Since it,V) = 0, and s > 1, / , : Gr — n,(dH) is an epimorphism. By (8.2) and (4.1) there is an automorphism a: Gr — Gr such that fjx. = g. Then (2.1) implies that Vis in -BC(n, r—k, «+ l ) using the main property of g.

5. The goal of this section is to prove the following analogue of (1.2).

185

POINCARE CONJECTURE 403

(5.1) THEOREM. Let n ^ 2s + 2, or if s = 1, n ^ 5, M"_1 be a simply connected, (s — l)-connected closed manifold and JCM(n, k, s) the set of all manifolds having presentations of the form (Mx [0,1], M x 1; / , • • •, /»; «). Now let He JCM(n, k, s), Q = dH- Mx0, V = *(#, Q;ff»'~, 0/, s + 1) and suppose n,(Mx 0)—»;r,( V) is an isomorphism. Also suppose if

8 = 1, that 7ti(x(H, Q; gu • • •, gr-t; 2)) = 1. Then V e JCMn, r -k,s +1). One can easily obtain (1.2) from (5.1) by taking for M, the (n — 1)-

sphere. The following lemma is easy, following (3.2).

(5.2) LEMMA. With definitions and conditions as in (5.1), x,(Q) — Gt if s = l, and ifs>l, 7r,(Q) = K,(M) + Gk.

Let p,: TT,(Q) —- K,(M), p,: n,(Q) — Gk be the respective projections.

(5.3) LEMMA. With definitions and conditions as in (5.1), there exists a homomorphism g: Gr—>K,(Q) such that ptg is trivial, p,g is an epimor-phism, and every realization of g is in Mu(n, r — k, s + 1), each r ^ k.

The proof follows (3.1) closely. We now prove (5.1). The cases 8 = 0 and s = 1 are proved similarly to

these cases in the proof of (1.2). Suppose s > 1. From the fact that x,Mx 0)—>K,( V) is an isomorphism, it follows that P i / , is trivial and p , / , is an epimorphism where a = (H, Q; glt • • •, gr, s + 1). Then apply (4.1) to obtain an automorphism a: Gr —»Gr such that Pifjx = pjj where g is as in (5.3). Then f„a = g, hence using (2.1), we obtain (5.1).

6. The goal of this section is to prove the following two theorems.

(6.1) THEOREM. Suppose f is a C" function on a compact manifold W with no critical points onf~l[—e, e] = Nexcept k non-degenerate ones on /" '(0), all of index X, and N f) dW = 0 . Then f~l[— °o, e| has a pre­sentation of the form (/"'[— °°, —s],f~l(—e);fu •••, /*; X).

(6.2) THEOREM. Let (M, Q;fu -••, fk; s) be a presentation of a mani­fold V, and g be a C°° function on M, regular, in a neighborhood of Q, and constant with its maximum value on Q. Then there exists a C" function G on V which agrees with g outside a neighborhood of Q, is constant and regular on dV — (dM — Q), and has exactly k new critical points, all non-degenerate, with the same value and with index s.

SKETCH OF PROOF OF (6.1). Let & denote the critical points of / at level zero, i = 1, • • •, k with disjoint neighborhoods Vt. By a theorem of Morse [13] we can assume V has a coordinate system x=(*„ •*,£») such that for || x || £ 8, some S > 0,f(x) = - E ^ x J + SXx + l t f . Let E, be the (*„ • • •, xk) plane of V and Et the (*x+1, • • •, xn) plane. Then for s, >0 sufficiently small # , n / _ 1 [ - e i . « , ] is diffeomorphic to D \ A sufficiently

186

404 STEPHEN SHALE

small tubular neighborhood T of Ev will have the property that T' = T 0 / - I [ - e„ £,] is diffeomorphic to Dx x Dn~K with T n / _ 1 ( -£ i ) corres­

ponding to dDK x Dn'K. As we pass from / " ' [—°° , —ej t o / _ 1 [ — ° ° . £i]» it happens that one

such T' is added for each i, together with a tubular neighborhood of f ' \ — £j) so that / _ 1 [ - co, £,] is diffeomorphic to a manifold of the form Z( / - 1 [ - °°» - £ iL / _ 1 ( - £1): / 1 . • • •. /*: M- Since there are no critical points. between —£ and —e„ £x and E, £, can be replaced by £ in the preceding statement thus proving (6.1).

Theorem (6.2) is roughly a converse of (6.1) and a sketch of the proof can be constructed similarly.

7. In this section we prove Theorems C and I of the Introduction. The following theorem was proved in [21].

(7.1) THEOREM. Let Vn be a C°° compact manifold with dV the dis­joint union of V and Vit each Vt closed in dV. Then there exists a C" function f on V with non-degenerate critical points, regular on QV, fVx) = -(1/2), /(V,) = n + (1/2) and at a critical point 0 off, / (£ ) = index /9.

Functions described in (7.1) are called nice functions. Suppose now M" is a closed C°° manifold and / is the function of (7.1).

Let X, = / - ' [0 , s + (1/2)], s = 0, • • -, n.

(7.2) LEMMA. For each s, the manifold X, has a presentation of the formX,.ufu ••- , /*;«)•

This follows from (6.1).

(7.3). LEMMA. / / He A((n, k, s), then there exists—a C°° non-degener­ate function f on H,f(dH) = s + (l/2), / has one critical point of index 0, value 0, k critical points of index s, value s and no other critical points.

This follows from (6.2). The proof of Theorem C then goes as follows. Take a nice function /

on M by (7.1), with X, denned as above. Note that X0 e Jin, q, 0) and ^(X,) = 0, hence by (7.2) and (1.2), Xx e J(n, k, 1). Suppose now that 7r,(Af) = 1 and n ^ 6. The following argument suggested by H. Samel-son simplifies and replaces a complicated one of the author. Let X'-, be the sum of X, and k copies Hlf • • •, Hk of DnlxS3. Then since rc^X,) = 0, (1.2) implies that X\ e H(n, r, 2). Now let /«: dD3 x Dni — 9tf< n 9*i for i = 1, • • •, fc be differentiate imbeddings such that the composition

ntfD3 x Dn~3) — icjfiHt x dX[) — icJ&Hi)

187

POINCARE CONJECTURE 405

is an isomorphism. Then by (3.3) and (3.4), j ; ^ ; , / , , •••,/*; 3) is diffeo­morphic to Xt. Since -3fs = x(X*> 9u '"» 9u 3) we have

X, = lt!LXuf» '' • »/*. Ox,-", 9i, 3) , and another application of (1.2) yields that Xte H(n, k + I — r, 3).

Iteration of the argument yields that X'm e JC(n, r, m). By applying (7.3), we can replace fir by a new nice function h with type numbers satis­fying Af0 = 1, Mi = 0, 0 < i < TO. Now apply the preceding arguments to -h to yield that hl[n - TO - (1/2),n] = XleJCn, ku TO). Now we modify h by (7.3) on X* to get a new nice function on M agreeing with h on Af — Xt and satisfying the conditions of Theorem C.

The proof of Theorem I goes as follows. Let V be a manifold with dV = V, - V„ n = 2m + 2. Take a nice function / on V by (7.1) with /(V t) = - (1/2) and f( V,) = n + (1/2).

Following the proof of Theorem C, replacing the use of (1.2) with (6.1), we obtain a new nice function g on V with gr( VJ = — (1/2), g( Vt) — n + (1/2) and no critical points except possibly of index m -j- 1. The fol­lowing lemma can be proved by the standard methods of Morse theory [12].

(7.4) LEMMA. Let V be as in (7.1) and f be a C°° non-degenerate func­tion on V with the same boundary conditions as in (7.1). Then

Z K = E ( - D ' A f , + * F i , where Xr, XF, are the respective Euler characteristics, and M„ denote the <7tb type number off.

This lemma implies that our function g has no critical points, and hence V, and V, are diffeomorphic.

8. We have yet to prove Theorems F and G. For Theorem F, observe by Theorem C, there is a nice function / on Af with vanishing type num­bers except in dimensions Ma, Af„, Mm+U Mn, and Af, = Af„ = 1. Also, by the Morse relation, observe that the Euler characteristic is the alternat­ing sum of the type numbers, Mm=Mm^. Then by (7.2), f~l [0, m-|-(l/2)]> / _ 1 [m + (1/2), 2m + 1] e JC(2m + 1, Af„, m) proving Theorem F.

All but the last statement of Theorem G has been proved. For this just note that Af - D"* is diffeomorphic t o / - ' [ 0 , m + (1/2)] which by (7.2) is in JC(2m, k, m).

UNIVSRSISTY or CALIFORNIA, BERKELEY

REFERENCES 1. R. BOTT, The iteMe homotopy of the clostieal group*, Ann. of Math., 70 (1959), 313-

337.

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406 STEPHEN SHALE

2. H. GLUCK, The weaifc Hauptvermutung for cells and sphere*, Bull. Amer. Math. Soc , 66 (1960), 282-284.

3. MORRIS W. HIRSCH, On combinatorial rubmanifolds of differentiable manifold*, to appear.

4. M. KERVAIRE, A manifold which doe* not admit any differentiate structure, Com­ment. Math. Helv., 34 (1960), 304-312.

5. and J. MlLNOR, Group* of homotopy spheres, to appear. €. K. A. KUROSH, Theory of Groups, v. 2, New York, 1956. 7. B. MAZUR, Oft embedding* of sphere*, Bull. Amer. Math. Soc., 65 (1959), 59-65. 8. J. MlLNOR, Differentiable structure* on spheres, Amer. J. Math., 81 (1959), 962-972. 9. , Somme* de varietes differentiable* et structure* differentiable* des sphere*,

Bull. Soc. Math. France, 87 (1959), 439-444. 10. , Differentiable manifolds which are homotopy spheres, (mimeographed), Prin­

ceton University, 1959. 11. E. E. MoiSE, Certain classical problems of euclidean topology, Lectures of Summer

Institute on Set Theoretic Topology, Madison, 1955. 12. M. MORSE, The calculus of variations in the large, Amer. Math. Soc. Colloq. Pub­

lications, v. 18, New York, 1934. 13. , The existence of polar non-degenerate functions on differentiable mani­

folds, Ann. of Math., 71 (1960), 352-383. 14. J. MUNKRES, Obstructions to the smoothing of piecewiee-differentiable homeomorphism*.

Bull. Amer. Math. Soc., 65 (1959), 332-334. 15. , Obttruction* to imposing differentiable structure*, Abstract, Notices of the

Amer. Math. Soc.. 7 (1960). 204. 16. J. NIELSON, Ober die Isomorphismen unendlicher Gruppen ohne Relationcn, Math.

Ann., 9 (1919), 269 272. 17. R. PALAIS, Extending diffeomorphitm*, Proc. Amer. Math. Soc., 11 (1960). 274-277. 18. C. D. PAPAKYRIAKOPOULOS, Some problem* on 3-dimensional manifold*, Bull. Amer.

Math. Soc., 64 (1958), 317-335. 19. H. SEIFERT and W. THRELLFALL, Lehrbuch der Topologie, Teubner, Leipzig, 1934. 20. S. SHALE, The generalized Poincare conjecture in higher dimension*, to appear. 21. , On gradient dynamical systems, Ann. of Math., 74 (1961), 199-206. 22. R. THOM, Let structures differentiable* des boules et des spheres, to appear. 23. , La classification des immersions, Seminaire Bourbaki, Paris, December, 1957. 24. , Des varietes triangutees aux variltes differentiates, Proceedings of the

International Congress of Mathematicians, 1958, Cambridge University Press, 1960.

25. , Les classes caracteristiques de Pontrjagin des vari&es triangulees, Topo-logia Algebraica, Mexico, 1958, pp. 54-67.

26. A. H. WALLACE, Modification* and eobounding manifold*, Canad. J. Math., 12 (1960), 503-528.

27. J. H. C. WHITBHEAD, On C'-complexe*. Ann. of Math., 41 (1940), 809-824. 28. , On the homotopy type of manifolds, Ann. of Math., 41 (1940), 825-832. 29. H. WHITNEY, Differentiable manifold*, Ann. of Math., 37 (1936), 645-680. 30. W. T. WO, 0» the isotopy of C> manifold* of dimension n in eudidtan (2» + 1)-

space, Science Record, Now Ser. v. II (1958), 271-275.

189

ANNALS OF MATHBMATICa Vol. 74, No. 3. November, 1961

Printtd in Japan

DIFFERENTIABLE AND COMBINATORIAL STRUCTURES ON MANIFOLDS

BY STEPHEN SMALE* (Received December 5, 1960)

1. By extensions of our methods of a previous paper [7], hereafter referred to as GPC, we prove the following theorems. They are all gener­alizations of results in GPC.

(1.1) THEOREM. Let C* be a contractible C°° compact manifold with simply connected boundary where n =t 3, 4, 5, 7. Then C is diffeomor-phic to the n-diak D".

(1.2) COROLLARY. Ifn^i, 5, 7, there is a unique C" structure up to diffeomorphism on the n-disk.

(1.3) COROLLARY. If n ^ 4, 7, and / ; Sn~i-*En is a differentiable imbedding of the sphere in euclidean space, then the closure C of the bounded component of EH — /(SB _ I) is diffeomorphic to D".

The second corollary is a strong version of the differentiable Schoenflies problem, n ^ 4,7. Mazur's theorem [3] had already implied that C was homeomorphic to D".

Two abelian groups, r n and A", studied by Milnor [4], Munkres [6], and Thom [8], have been found to be important in the theory of differentiable structures on manifolds. The group r n is the group of diffeomorphisms of Sn_1 modulo those which can be extended to Dn. It has been identified with the set of differentiable structures on S" compatible with the stand­ard triangulation of Sn (see GPC for more details and references for all these things discussed in the Introduction). The group A" is the group of those differentiable structures on S" which, minus a point, are diffeo­morphic to En. In GPC it was proved that A", n ^ 4 is the group of all differentiable structures on Sn .

On the other hand a group Sin of homotopy spheres of dimension n under "J-equivalence" has been studied by Kervaire and Milnor [2]. They have shown SCn is finite, n =£ 3, and have computed the order of Mn, 3 < n ^ 15.

(1.4) THEOREM. For n * 3, 4, 6, 7, St* = A" = r \ Also ST = A7, I* = A8.

(1.5) COROLLARY. There are a finite number of differentiable struc-* The author is an Alfred P. Sloan fellow.

498

190

STRUCTURES ON MANIFOLDS 499

tures on Sn, n =t 4, 6. We also obtain here (1.6) THEOREM. Combinatorial analogues of (1.1) and (1.2) are valid

with diffeomorphism replaced by combinatorial equivalence. (1.7) THEOREM. Combinatorial structures on spheres and cells are

unique if their dimension is not 4,5,7. That is, the Hauptvermutung is true for combinatorial n-manifolds which are homeomorphic to spheres or cells, n # 4, 5, 7.

We refer the reader to GPC, beginning, for the proof that (1.1)-(1.7) follow from the next two theorems.

(1.8) THEOREM. Let Mn be an (m — l)-connected, C~ closed manifold with n^2m-l, and (n, m) f. (4, 2), (3, 2), (5, 3), (7, 4). Then there exists a non-degenerate (nice) C°° function on M with type numbers M satisfying

M0 = M„ = 1 , Mi = 0 , Q<i<m,n — m<i<n. (1.9) THEOREM. Suppose M," and Af" are J-equivalent (m - ^-con­

nected closed C" manifolds with n = 2m or n = 2m + 1 and (n, m) 0 (4, 2), (3, 1), (6, 3). Then M? and M,n are diffeomorphic.

REMARK. From the methods used here, it will be clear that all the previous theorems will be true also for n = 6, 7 if the following likely hypothesis is true.

Hypothesis. Let / , g: S3 —♦ M" be differentiate imbeddings which are homotopic with

K0(M6) = ff,(Jlf) = KJM') = 0 .

Then / and sr are differentiably isotopic. 2. We emphasize that we are assuming the notation and terminology

of GPC. We will prove (2.1) EXTENDED HANDLEBODY THEOREM. Let n ^ 2s + 1, (n, s) 0

(4, 1), (3,1), (5, 2), (7, 3) and HeJ(n, k, s). Suppose

V=x(B',f» •••,/,;« + !) and 7Z,(V) = 0. Also if s = 1, let

ff1(tftf;/1,..-,/r-*;8 + l)) = l . Then VeJC(n, r - k, s + 1).

We first note that (1.8) follows from (2.1) and the arguments of GPC. Also, since the proof of (1.9) involves nothing beyond straight-forward

191

500 STEPHEN SMALE

application of methods of GPC and this paper, we omit it. Hence it re­mains to prove (2.1). We use the following special case of a very recent theorem of A. Haefliger [1].

(2.2) THEOREM (Haefliger). Suppose f, g: S* -»Af* are differentiable imbeddings which are homotopic, k > 3 and

K0(M) = 7r,(Af) = . . . = j r ^Af ) = 1 .

Then f and g are differentiably isotopic.

(2.8) THEOREM. Let y e 7c,,(Mu) where Mu is a C" simply-connected manifold and k > 2. Then y can be realized by a differentiable imbed­ding f: S* - M" .

As has been observed by Milnor [5], one can prove this by using the work of Whitney [9]. See also [1].

The following theorem extends Theorem (2.1) of GPC. The proof is the same but uses Theorems (2.2) and (2.3) above, instead of the weaker theo­rems of Whitney and Wu used there.

(2.4) THEOREM. Let n ^ 2s + 1, (n, s) * (4,1), (3,1), (6, 2), (7, 3), let a = (M, Q',f, •••,fr; 8 + l)be a presentation of a manifold V, and as­sume- K^Q) = 1 if n ^ 2s + 2, and 7r,(Q) = 0 i / « = 2« + l . Then for any automorphism a: Gr —»Gr, V realizes fjx. .

Now (2.1) has been proved in GPC except for the case » = 2s + 1, 8 > 3. Thus in proving (2.1), assume HeJ(2m + 1, k, m), m > 3. From §3 of GPC it follows that

He Sr x A"+1 + • • • + S? x D?+1 •

Let gu •••,£» be the corresponding generators of nm(H) and ht be the generators of nm(dH) corresponding to qt x dD?+\ g< e S". Thus ffu '",gk>Ki ''*i K) is an independent set of generators of the free abelian group nm(dH).

We can represent H in the form

where •$<: 0 D - x Dt

m+l - » dD***1

are imbeddings with the images of i/r< disjoint. Then the above h can be represented by ^ restricted to q( x dD?+l for some qt e dD?.

On the other hand

V=xH;f,--,fr) = x°)

192

STRUCTURES ON MANIFOLDS 501

where

U &A"+' x D?-~dH are imbeddings. Let

nJdH) = Gk + Hk

where Gt is generated by gu ••-,gk and Hk by hlt •••,hk, and let TT,: TtJdH) - Gk

be the projection. Since nm( V) = 0, izxfa is an epimorphism where

fy.Gr^7rJdH) is induced by a. Define an epimorphism g: Gr — Gk by fl-A = g„ i ^ A; and gDi = 0, i > k. Then by (4.1) of GPC there is an automorphism a of Gr such that nj„cc = g. Thus ^ / ^ ( A ) = & or

/ ^ A = 91 + 1 ^ 0 ^ . Then by (2.4), we can assume V = x(H; / „ • • • , fr) where the homotopy class of/, restricted to 9A*+1 x 0 in TtJdH) is p, + £3* a A for some set of a,.

Let r be the union of the subsets

A " x A " + \ •••, A " x A"+1

of i/ . Now it is clear that each h e icJdH) is the image of some class in Km(dH — (dH n r)) under the homomorphism induced by inclusion. The same is true also for gt. This implies that the class gt + J2* a< < n a s *^e

same property, or that there exists a map /,': dDm+l xO—dH-(dHnr)

which is homotopic in dH of the restriction of

/ , : 9Dr+1 x Am — dH to 9A"+l x 0. By (2.3) we can realize /,' by an imbedding. By (2.4) of GPC extended slightly by (2.2) of this paper we obtain a differentiable imbedding

/ / : 8A"+1 x A " — dH- (dHn O

such that x(H'< fl) a n d X(H; A) are diffeomorphic. If we generalize the definition x(Af; <plt ••-, <pk) of GPC to include the

case of handles of more than one dimension, we have x(H; ft) is diffeo­morphic to p , ; / , ' , fa, - -', -ft) where if, = x(D'm+1; yjrj. Then we have X(H; / ,) is diffeomorphic to %(#,; ^ „ • • •, fk) where H, = %,(#,; / , ' ) . By

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502 STEPHEN SMALE

the proof of (3.3) of GPC, H* is diffeomorphic to P m + 1 . Thus V is of form X(H3; / „ • • . , / , ) where H, = *(#,; +t, . . . , +„) is in M(2m + l,k-l,m). By induction on A; we get (2.1).

UNIVERSITY OF CALIFORNIA, BERKELEY

REFERENCES

1. A. HAEFUGER, DifferentiabU imbeddings, Bull. Amer. Math. Soc., 67 (1961), 109-112. 2. M. KERVAIRE and J. MILNOR, Groups of homotopy spheres, to appear. 3. B. MAZUR, On embeddings of spheres, Bull. Amer. Math. Soc, 65 (1959), 59-65. 4. J. MiLNOR, Sommes de varietes differentiables et structures differentiables des spheres,

Bull. Soc. Math. France, 87 (1959), 439-444. 5. , Differentiable manifolds which are homotopy spheres, mimeographed, Prin­

ceton University, 1959. 6. J. MUNKRES, Obstructions to the smoothing of pieeewise differentiable homeomorphisms,

Bull. Amer. Math. Soc., 65 (1959), 332-334. 7. S. SMALE, Generalized Poincare's Conjecture in dimensions greater than four, Ann.

of Math., 74 (1961) 391-406. 8. R. THOM, Des varietes trianguleis aux varietis differentiable. Proceedings of the Inter­

national Congress of Mathematicians, 1958, Cambridge University Press, 1960. 9. H. WHITNEY, The self-intersections of a smooth n-manifold in 2n-space, Ann. of Math.,

45 (1944), 220-246.

194

AjINAlJI OV AlATBBMATICa Vol. 75, No. 1. January. 1962

Printtd in Japan

ON THE STRUCTURE OF 5-MANIFOLDS BY STEPHEN SHALE*

(Received January 11, 1961)

The main goal of this paper is to give some theorems which relate to the problem of classifying C manifolds up to diffeomorphism. In the case of simply connected closed C 5-manifolds with vanishing 2nd Stiefel-Whitney class W1, we are able to apply our general theory to give a com­plete classification.

THEOREM. There is a 1-1 correspondence g> between simply connected C" closed 5-manifolds with vanishing 2nd Stiefel-Whitney class and finitely generated abelian groups. The correspondence is given by: <p(M) = F + ±T where Ht(M) = F + T is a direct sum decomposition of Ht(M) into a free and torsion part, and \T + JT is a direct sum decom­position of T.

There is an example of Wu [8] of a closed 5-manif old with W* =£ 0 which can be made simply connected by surgery [5] preserving the property W* =£ 0. Hence our classification does not include all simply connected 5-manifolds.

The classification is given as follows. For each non-negative integer q, we will define, in § 6, a certain closed 5-manif old Mf. For example Af0* = S* x S\ Mi = S' and f or q > 1 H,(Mt) = Z, + Zq.

THEOREM A. Every simply connected C" closed 5-manifold AP with vanishing 2nd Stief el- Whitney class is diffeomorphic to a sum of the M, sum in the sense of Milnor [4]. This sum can be written uniquely in the form

M' = Mi+ • • • + Mi + Mk\+ • • • + Mir

with ki | ki+l and ^ > 1 1 / Af' # S'. Furthermore, these manifolds are simply connected with W1 = 0.

The general theory here has numerous other applications. For example: THEOREM B. Every 2-connected closed C" 6-manifold is homeomorphic

toS*orS% x S ' + ••• + StxS\ The methods used in the proof of these theorems depend strongly on

our previous work [7], hereafter referred to as GPC. In connection with Theorem A, the negative result of Markov should

be noted; that is, in a certain sense the general classification problem for * The author is an Alfred P. Sloan Fellow.

38

195

5-MANIF0LDS 39

manifolds is not solvable [8]. In general we will use the notation of GPC. All manifolds are

C"; A*, •D". etc., denote m-disks. Imbeddings are differentiate imbed-dings.

1. In this section, we give some motivation for studying the set of handlebodies JC(2m, k, m) by showing how they arise while studying mani­folds. Roughly speaking, we reduce the theorems stated in the Intro­duction to the study of SC(2m, k, m). Recall that JC(2m, k, m) is the set of manifolds of the form H = xiD^, /,, • • • ,fk; m) where the /<: 0#" x Z><"— dD** are imbedding with disjoint images and if is obtained by identifying points under the / , and smoothing. Let «#(m) = U*-o<#(2m, k, m). We will always assume m > 1.

The following is Theorem 6 of GPC. (1.1) THEOREM. If M*" is an (m — l)-eonnected closed manifold, m # 2,

then M with a 2m disk removed is in SC(m), We note now a generalization of (1.1) that follows from GPC. (1.2) THEOREM. Let W* be an (m — l)-connected compact manifold,

m :£ 2, and dWbe non-empty and (m — 2)-connected. Then We Mm). Conversely every element of J((m) has these properties of W.

PROOF. By [6], take a nice function on W which on dW has value 2m + i. By GPC, /''[0, m + i] is in JC(m), thus one can modify / on / - I[0, m + i] to get a nice function g on JFwith type numbers satisfying Af, = 1, M( = 0, 0 < i < m using GPC (6.2).

To kill the type numbers Mit i> m, one proceeds as follows. It is suf­ficient to kill the type numbers M„ • • •, Jli -j of — g. Theorem 5.1 of GPC accomplishes this at least up to Af„_,. Thus suppose h is a nice function on W with vanishing type numbers except in dimensions m — 1, m, 2m, and is — i on dW. Then if Xt is the presentation of W defined by h (i.e., X, = h~\-i, i + i]), Xm.t = tf-X.-,, h~Km - t);/„ - . , / * ; m - 1) where Xm-t is diffeomorphic to dW x 7. Since dW is m — 2 -connected, each f is contractible in h~\m — f) and Xm~x is of the form X^-i + H, He JC(2m, k, m — 1) and the sum is along h~l(m — ).

Now Xm = xX—1» h~*(m ~ i)> ffi. • • •. Or) m) and x.-AXJ = 0. Thus the homomorphism defined in § 2 of GPC, Gr —»JT1._1(0XJ,_1) is an epimor-phism and, using the Theorems 2.1, 3.1 and 4.1 of GPC (as applied there), we can remove k of the gt'B and H. Finally by (6.1) of GPC, We JCm).

The converse is proved very easily by the method in the proof of (3.2), GPC.

(1.3) THEOREM. Let dJC(m) denote the set of manifolds of form dW,

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40 STEPHEN SHALE

We <#(m). Let Af * be a closed simply connected 5-manifold with vanish­ing 2"4 Stiefel-Whitney class W*eH'M,Zt). Then Af»ea#(3). Con­versely every element of dJ((Z) is a simply connected 5-manifold with TP = 0.

PROOF. First, by obstruction theory, it follows that M is a -manifold (i.e., the normal frame bundle of Af* c En has a cross-section) using both Tr Af) = 1 and W* = W* (Whitney duality) = 0. Now according to Milnor [5], this implies that M is the boundary of a -manifold W. By surgery [5] one can make W% 2-connected without changing dW%. Then (1.3) follows from (1.2).

To prove the converse it is sufficient to prove W1 of dW is zero for W e JC(2>) by virtue of (1.2). From obstruction theory, W is parallelizable, hence a 7-manifold and thus its boundary is a ff-manifold. Hence W1 of d W is zero.

Similarly one can prove: (1.4) THEOREM. The set of 4-connected closed W-manifolds 'coincides

with the set dJC(Q). 2. The goal of this section is to study JC(m) = Ur.*#(2m, k, m) (where

diffeomorphic manifolds have been identified). We recall from GPC that a presentation of a manifold V is a triple a — (Af*; /„ •••,/*; m) where V = %(<*), ft' 0-D" x 2?;-" — 9M are imbeddings with disjoint images and X(o) is formed by identifying points under / , and smoothing. Here Af will always be D** and we write a presentation in the form F=flt ••♦,/*) leaving Af = D** and m implicit. Thus one can think of a presentation as an imbedding of the disjoint union U*-i0A" * A" in 0D*". For such Ft X(F)e ^"O, and in fact JC(m) can be thought of as the diffeomorphism classes of x(F) where F ranges over all presentations of the above type fixing m and letting k vary.

On the other hand, we will say two presentations F = (/lf • • •, /*) and F' — (f,'", fi) are equivalent or F~ F' if there exists a "homotopy" of presentations F, = (/„, ••♦,/».), 0 t £ 1, F, = F, F, = F', F, for each t is a presentation and each/,, has a continuous differential. In other words F, is a differentiable isotopy of the disjoint union [Ji.fiD* x Dm

into 0D*". Let Jt(2m, k, m) denote the set of equivalence classes of presen­tations fixing m, k, and Jt(m) = \j7-«&(2,m, k, m). From the covering homotopy property for imbeddings (see ope, § 2) if F ~ F', then %(F) and Z(F') are diffeomorphic. Thus we have a natural projection iz\ Ji2m, k, m) — JC(2m, k, m) and jr: J(m) - J((m).

3. We study here algebraic analogues of Ji2m, k, m) and JC(2m, k, m).

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5-MAN1F0LDS 41

Let Q.(fc) be the set all k by k integer valued matrices, symmetric if m is even, anti-symmetric if m is odd. Elements Qm(k) can be considered as representatives of symmetric or anti-symmetric quadratic forms. Thus we say, a, 0 e Qm(k) are equivalent if a = ifty* where y is a k by k inte­ger valued matrix, y* its transpose. Let the set of equivalence classes of Qm(k) be denoted by Qm(k) and JT,: Q.(fc) -»Q.(fc) the projection.

Let F = ( /„ • • • , /») e «#(2m, k, m) and Af = x(.F). The /«define an inde­pendent base for Hm(M, D**). Let <p( be the inverse image of /< under the canonical isomorphism Hm(M) -* Hm(M, D*"). Then by their intersec­tion matrix, the 9>< define an element of Qm(k), &(F), defining a transfor­mation 3>: Jc(2m, k, m) —»<?„(&). One can also consider *(F) to be the matrix defined by cup product in Hm(M, 9M) using the basis /* where f* corresponds to <pt under Poincare duality.

The transformation 4> induces a transformation <p: JC(2m, k, m) —»QJJc) with the diagram commuting.

Jt(2m, k, m)-^-*Qmk)

I" b ^"(2TO, ft, m)-^-»Q.(ft) .

That <p is well-defined follows from the fact that diffeomorphic manifolds define equivalent cup product forms. Note ic and JT, are onto.

Later we will prove the following theorem. (3.1) THEOREM. Let m be odd, itta = jr,/9, a, /3 e Qm(k) and $F = a.

Then there is F' e Jt(2m, k, m), izF' = it F such that $F' = /9. (3.2) COROLLARY, If $: Jt(2m,k,m) — Q(k) is 1-1, m odd, then so is

PROOF. Let <pM = <pM'. Then let Feirl(M) and F'STITW), <pF ~ a, <pF' = /3. Since nxa = nfi there is F", by (2,1), with $F' = £ and KF" = itF. Since £ is 1-1 F" = F' and hence M = M'.

REMARK. Our proof of (3.1) is perfectly valid for m even, m =£ 2. We assume m odd to avoid discussion of the diagonal elements of a.

4. We define a link X to be an imbedding of Sk into S"""1 where Sk is the disjoint union of k copies Sr~l, •••, Sk

m~\ of the (m — l)-sphere. Call links X, X': Sk —»S**-1 equivalent if there is a differentiate isotopy X,: Sk —»S *"1 between X and X'. Fixing m and ft, let Am(k) be the set of equivalence classes of links.

If X: Sk —* S"*-1 is a link, let the restriction of X to S-"-1 be denoted by Xv. Let X<oX = 0, and for t =£ j , let XjoXy denote the linking number of

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42 STEPHEN SMALE

Xt and Xj, Let Qm(k)* be the subset of Qm(k) of matrices with zero diagonal ele­

ments. Then define a transformation L: Am(k) —* Qm(k)* by L(X) = (Xs°X,). Since X,oX, = x oX, for m even and X^x, = — x,oX< for m odd, L is well-defined.

(4.1) THEOREM. The transformation L: AJk) — <?„(*)* ** 1-1 and onto for all k and for all m > 2.

PROOF. If / : S* —»i?*"-1 is an imbedding, / induces a map 9>,:Sk xSk- S . - S — by <pt(x,y) = [/(*) -/(i/)]/ | | /(*)-/(y)| |where S* x SM = S» is the product with the diagonal removed. There are natural involutions T: Sk x S„ - S* — S„ x S„- S„ and T': S*»-* — S*"-1 denned by T(x, y) = (y, z) and 7" the antipodal map. The map <pf commutes with T and T or is equivariant. Then a theorem of Haefliger [2] (see also Wu [9]) says that if m > 2,f—><pf induces a 1-1 correspondence between differentiate isotopy classes of imbedding of Sk in RIm~l (or S**"1) and equivariant homotopy classes of Sk x S* — S» into S*"~*.

Consider now these equivariant homotopy classes. Since Sh is the dis­joint union of S,—\ • • •, S*""1. S» x S* - S* consists of S""1 x Sf-1 U S/-1 x S,—x, t * J, together with S?'1 x Sr_1 - Sr_1 each i.

There is a unique equivariant homotopy class of S*-1 x S'-1 — S~~l into S**"1. This can be seen as follows. These equivariant homotopy classes correspond to homotopy classes of maps of (S""1 x S*'1 — Sf'^/T = X into S*"",/ri = Y which map the 1-dimensional characteristic class of Y into that of X. Since F'(X, *«( Y)) = 0, t > 1, there is only one of these. Now fix i, j , i # j and consider the equivariant homotopy class of S?-1 x S'-1 U Sp-1 x S,"-t into S*"-1. It is clear there is a 1-1 corre­spondence between these equivariant classes and ordinary homotopy class­es of Si"-1 x Sr~l into S"'-1. The homotopy classification of S?~l x Sf-l

into S1"-' is given by the integers, the correspondence being the degree. Now it is well-known that the linking number of /(S""1) and /(S"_1) is characterized by the degree of g>, (defined above) restricted to S,"-1 x S,"-1. Putting these facts together we obtain (4.1).

Of course (4.1) is false if m = 2. Define^ map p: JC2m, k, m)—AJk) by p(F) = X where F =( /„•• • , /»),

/4 = \., f: dDr x 0 — aD*" is the restriction of ft and identifying S""1, S**-1 with dDr x 0 and 9-D*" respectively. Also define ft: QJk) — £.(&)* by setting equal to zero the diagonal elements of a matrix. Then the following lemma is a restatement of a well-known property of linking numbers.

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5-MANIFOLDS 43

(4.2) LEMMA. The following diagram commutes.

JC(2m,k,m)~^-*Qm(k) \p Pi

K(k) -^S.(fc)*. The following lemma is trivial. (4.3) LEMMA. The transformations p and p, are onto and Vi ** 1-1

if mis odd. (4.4)LEMMA. Ifm = Zorl,pis 1-1. It is sufficient for the proof of (4.4) to prove the following assertion. Assertion. Let / , / ' : 8D—1 x Dm-^Stm~1 be two imbeddings which

agree on dDm~l x 0, and suppose m = 3 or 7. Then / and / ' are differ-entiably isotopic.

Because the ideas of this proof have been frequently used, we only out­line it. Using a radial retraction of P", we can obtain g, g'\ 9Dm~1 x D"— S**_1, differentiably isotopic to / , / ' respectively, which have a common image T (a small tubular neighborhood of g[dDm-x x 0) = g'(2Dm-1 x 0)) and map the rayB R,,% - (p, rq) \ 0 £ r g 1, (p, q) e dDm x dDm into small geodesies issuing from g(p, 0) = ffip, 0). Then fixing a product structure on T, g and g* define elements g, Q' of w._,(0(m)). Since m — 3 or 7, this group vanishes, so 3 = 8' and the corresponding homotopy defines the desired isotopy between g and g*. This proves the assertion, and hence (4.4).

From (4.1), (4.2), (4.3), and (4.4) we obtain, (4.5) THEOREM. If m = 3 or 7, $ is 1-1 and onto. Then, by (3.2), we have, (4.6) COROLLARY. If m = Zor1, <p: 30,2m,k,m) — QJJc) is 1-1 and

onto. REMARK. The last corollary is a central result of our paper. It is true

also for m = 6 mod 8, if the elements of Qm(k) are permitted only even diagonal elements. This we will prove elsewhere using the periodicity theorem of Bott. For all other m, <p is not 1-1.

5. This section is devoted to proving (3.1). For this proof we will use the following lemma.

(5.1) LEMMA. Let F = Wu"',fh) e Jt2m, k, m) and xF = Me J((2m,k,m). For each i, i = l, •••,*, fix qedDt

m, ptedD? (with /<: 0D" x A" — OD*), and let h, g be the restriction offf to qtx dDf,

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44 STEPHEN SMALE

8 A" x Pi respectively. Let r c 3D1" be the union of the images of the ft and g: Sm~1-*dD*m — r any imbedding. Then gt = X *_,(i?0ff*) <» where gogi is the linking number, and gt, hit the homotopy classes of g, h, re­spectively in dM.

PROOF. Note first, in proving (5.1), it is sufficient by considering the inclusion 9D1" - r — dM to prove g* = £J.i(0°0<)V where g* and hf are the elements of iz^-JfiD** — r) defined by g and ht respectively.

Let Li be dD** minus the interior of the image of ft and a<: dD**—r—Lt be the inclusion. Then *._,(!<<) is infinite cyclic generated by a^h? = £,.

The following is a consequence of the equivalence of several ways of denning linking number.

(5.2) LEMMA. For each t, aa* = igogfa in ^^(A). Then (5.1) follows from (5.2) by considering the inclusion QD** — T—• L<. Now to prove (3.1), let M = itF where F = ( / „ • • • , /») e Jf(2m, k, m),

and let <pt be the element of HJJA) corresponding to ft as in § 3. The following automorphisms generate the automorphism group of HmM).

R: <Pi — -9>u <P< — <Pi, t > 1 Tt: 9>, — <pit 9>i —• <PU <Pi — <Pi, j * l , j * i ,

and S: 9i — <Px + 'Pi, 9i-+<Pi, t > 1 .

It is sufficient to prove (3.1) for the case that a is equivalent to /S by each of R, Tt and S.

We first consider the case for R. Let /,': 9 A" x A" — dD** be the im­bedding /,r' where / ^ r x e : 9 A" x A" — 9 A" x D-, where e: A" x A" is the identity, and f the restriction of an orientation reversing diffeo-morphism r: A* — A"- Then fof = -/,©/;. each i = 2, •«•, k where /,: 9A" x 0—flD** is the restriction of ff; and by taking F'=(//,/„ • • • ,/*), we prove (3.1) for this case.

The case that a and /9 are equivalent by T, is taken care of easily. Thus it remains to prove (3.1) for the case that a and £ are equivalent byS.

It is sufficient to find F' = (/,', /,_• •»_, fk) such that x(F) and *(.F') are diffeomorphic, and /,'<>/; = /,<>/< + /»°/i using (4.1).

Define If, to be the handlebody %(!>*; /» • • • ! /»)• Then JJ, is a subset of M in a natural way. We claim that /,' and /, are homotopic in 9-Hi. By (5.1), h = E*(/i°0<)fr,i a n d h = T,\(Jl°g<)K as elements of n.-tfHj, where &, A4, t = 2, • • , k are defined as in (5.1). Hence the homotopy class / i - /„ is 53*(/i°0<)fe«- 0 ° t h e other hand ?,, = £*(fft°0<)fci« by (5.1) where 0, is / , restricted to 9A* x p,. Since / , and 0, are differentiably

201

6-MANIPOLDS 45

isotopic in the complement of the images of fit i = 1,3, • • •, k, and gH = 0 in iz^JpHd, this implies that /,', - /„ = 0 as elements of x.-AdH,), thus proving our assertion.

Now apply (2.4) of GPC to obtain an extension /,' of f to ©A" * Dm

such that %(Hit /,') and %(J?«, /,) are diffeomorphic. This finishes the proof of (3.1).

6. For m = 3, 7, we define elements TP<*" e Jf(m), i = 0,1,2, • • as fol­lows. By (4.5), 0: Jt(2m,k,m)^Qm(k) is 1-1 and onto. Let AheQm(2) be the matrix (_ jj j \ fc = 1,2,3, and 4 , e Q„(l) be the matrix (0). Then define WT = 7r£-l(A<). Let dWf be denoted by Af?—l.

We can define a sum operation in JC(m). If H„ Ht e «#(m), let A*"-1. A*"-1 be sub-disks in dHu dHt respectively, and h: A*"-1 —A*""1 be an orientation reversing diffeomorphism. Then identify points which corre­spond under h and smooth to obtain Hx + Ht (see GPC and [7]). Then 8(fli + Ht) = 9Hj + 9Ht where the sum of closed manifolds is defined ex­plicitly in [4].

(6.1) LEMMA. The group H^Mf*-1) is isomorphic to Z\qZ + Z\qZ for q> land #„_,(#,?—') = Z. Furthermore, Hm.,(.M^' + M~-') = Hm^(MT-t) + Hm.,(Mr-%

PROOF. The first sentence follows from Poincare duality and the exact homology sequence of the pair (Wf*t AT*-1) following Milnor, [5, p. 14]. The second statement is a consequence of the Mayer-Vietoris sequence.

Our structure theorems for JC(m) and dJC(m) for m = 8 and 7 are the following.

(6.2) THEOREM. Letm = Sor7. If We J((m), then Wisdiffeomorphic to one of the following manifolds:

wr + ~- + wr + £;_, w?, KZ 1. K \ *<+l. q times

Furthermore no two manifolds of this type are diffeomorphic. This means, if m = 3 or 7, there is a 1-1 correspondence between

JC(m) and tuples q, ku • • •, K), q S 0, fcj 1, and kt divides fc^,. (6.3) THEOREM. Let m = 3 or 7. If Me3C(m) and is not S \ then M

is diffeomorphic to one and only one of the manifolds. Mr-' + 'y + Mr-1 + E,-^*?" 1 . d > 0, fc, 2, k( Ik(+l

q times These manifolds are in dJC(m).

Theorem A and A' of the Introduction follow from (6.3), (6.1), (1.3) and

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46 STEPHEN SMALE

the fundamental theorem of abelian groups. Theorem B follows from (6.2) as follows. By 1.1, if M is a 2-connected

closed 6-manif old, M minus a 6-disk is an element H of JCS). Since dH = S\ by (6.1) and (6.2), H must be diffeomorphic to J^^H!t, K = 1, all i, and Ht(H) is the direct sum of r copies of Z. But 23«_ S* x S' minus D* has the same property, so #and ]£,[St x S? — Z>' are diffeomorphic. This implies Theorem B.

The following theorem which can be found in [1], gives the structure of Qm(k) when m is odd. Together with (4.6) and (6.1), it yields (6.2).

(6.4) THEOREM. Let E be a free abelian group of rank n and 4> an alternating bilinear integer-valued form on E. Then there exists a base «u •••»«« of E,2rg.n such that

(1) *(e„ e,) = alt <&(«„ «,) = <*„•••, *(e,r-„ e,r) = 2r, where the a, are positive integers, a< | ai+1, i = 1, • •, r — 1.

(2) <I>(eif et) = Q,iSJ otherwise. (3) The a< are uniquely determined. It remains to prove (6.3). From (6.2), one obtains that M is diffeo­

morphic to a manifold of the form: MT-1 + ' • • + M^1 + l?tmlMp-1, *» 2s 1. fc* I ki+1.

Since J**—1 is diffeomorphic to S*—1, and S*—x + V—1 = V"-1 is true for any closed manifold V, we obtain Min the form of (1.3). The unique­ness follows from (6.1) and the fundamental theorem of abelian groups applied to Ht(M).

UNIVERSITY OF CALIFORNIA, BERKELEY.

REFERENCES

1. N. BOURBAKI, Elements de Mathematique, Livre II, Algebre, Chapitre 9, Formes Sea-quilineaires ct formes quadratiques, Hermann, Paris, 1959.

2. A. HAEFUOER, Differentiabl* imbedding; to appear in Bull. Amer. Math. Soc. 3. A. A. MARKOV, The problem of homeomorphy, Proceedings of the International Con-

grew of Mathematicians, 1958, Cambridge, 1960. 4. J. MlLNOR, Some* de variHkt differential)!** et structure* diffirentiables de spheres,

Bull. Soc. Math. France, 87 (1959), 439-444. 5. , Differentiate manifolds which are homotopy spheres, (mimeographed) Prin­

ceton University, 1959. 6. S. SHALE, On gradient dynamical systems, Ann. of Math., 74 (1961), 199-206. 7. , Generalized Poincari's conjecture in dimensions greater than four, Ann. of

Math., 74 (1961), 391-406. 8. W. T. WU, Clauee caracteristiques et i-earre* d'une variiti, C. R. Acad. Sci., Paris,

230 (1950), 508-509. 9. , On the isotopy of (^-manifolds of dimension n in euclidean (2* + l)-space,

Science Record, New Ser. II (1958), 271-275.

203

ON THE STRUCTURE OF MANIFOLDS.*

By S. SMALE.1

In this paper, we prove a number of theorems which give some insight into the structure of differentiable manifolds.

The methods, results and some notation of [13], hereafter referred to as GPC, and [12] will be used. These two papers and [14] can be considered as a starting point for this one. The main theorems in these papers are special cases of the theorems here.

Among the most important theorems in this paper are 1.1 and 6.1. Some conversations with A. Haefliger were helpful in the preparation

of parts of this paper. Everything will be considered from the differentable, equivalently C",

point of view; manifolds, imbeddings, and isotopes will be Cm.

Section 1. We give a necessary and sufficient condition for two closed simply connected manifolds of dimension greater than four to be diffeomorphic. The condition is h-cobordant, first defined by Thorn [16] for the combinatorial case, and developed by Milnor [9], and Kervaire and Milnor [7] for the differentiable case (sometimes previously A-cobordant has been called J-equivalent). I t involves a combination of homotopy theory and cobordism theory. More precisely, two closed connected oriented manifolds Mx

n, Man are

A-cobordant if there exists an oriented compact manifold W with dW (the boundary of IF) diffeomorphic to the disjoint union of Mi and —Mt, and each component of dW is a deformation retract of IF.

THEOREM 1.1. If n^. 5, and two closed oriented simply connected manifolds Mt

n and M2" are h-cobordant, then Mr and Ma are diffeomorphic by an orientation preserving diffeomorphism.

It has been asked by Milnor whether A-cobordant manifolds in general are diffeomorphic, problem 5, [9] . Subsequently, Milnor himself has given a counter-example of 7-dimensional manifolds with fundamental group Z„ A-cobordant but not diffeomorphic [10]. Thus the condition of simple-con­nectedness is necessary in Theorem 1.1.

• Received July 18, 1961. 1 The author is an Alfred P. Sloan Fellow.

387

204

388 8. SHALE.

Theorem 1.1 was proved in special eases in [13] and [14]. These special cases were applied to show that every sphere not of dimension four or six has a finite number of differentiable structures. The six dimensional case is taken care of by the following.

COROLLARY 1.2. Every homotopy 6-sphere is diffeomorphic to 8".

This follows from 1.1 and the result of Kervaire and Milnor [7] that every homotopy 6-sphere is ft-cobordant to S*.

COROLLARY 1.3. The semigroup of 2-connected closed 6-manifolds is generated by S* X S*.

This follows from 1.2 and [15]. Haefliger [2] has extended the notion of A-cobordant to the relative

case. Let Vu V2, Mt, M, be closed oriented, connected maaifoldfl with Vi C Mt,i — 1,2. According to Haefliger (Mx, V,), (M„ V2) are h-cobordant if there is a pair (if, V) (i.e., V C M) with dM — M1 — M2, dV = V, — V , and Mt-*M, Vt-*V homotopy equivalences. Then 1.1 can be extended to the relative case.

THEOREM 1.4. Suppose (Mf, TV) and (M2n,V2

k) are h-cobordant, jfc^5, »i(^i) '~ir1(Mi—V^ — 1 . Then there is an orientation preserving diffeomorphism of Mx onto M2 sending Vx onto V2.

By taking V« empty (the proof of 1.4 is valid for this case also), one can consider 1.1 as a special case of 1.4.

Actually we obtain much stronger theorems which will imply 1.4. The proof of 1.4 is completed in Section 3.

It would not be surprising if the hypothesis of simple connectedness in these theorems could be weakened using torsion invariants (see [10], for example).

Theorem 1.4 has application in the theory of knots except in codimension two.

Section 2. The main theorem we prove in this section is the following. Here we use the notation of GPC.

THEOREM 2.1. Let Mn be a compact manifold with a simply connected boundary component Q. Let V — x(Af, Q;f;m) where f: dD0

n X D0n'm-+Q is

an embedding, m > 2, n — m > 3. Suppose W — x(V, Qx;gu- • •,gri»» + 1) where Qt is the component of dV corresponding to Q and suppose that Hm(W, M) is zero. Then W is of the form X(M, Q; g\, , g'n-i \ m +1).

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8THCCTUBE OF MANIFOLDS. 389

Note that an example of Mazur [8] shows that the dimensional restriction is necessary here.

For the proof we use several lemmas.

LEMMA 2.2. Let M* be a compact manifold, Q a component of dM, n — m>l, V — xM,Q;f;m), W — x(V,Qx\gu- -,<7r;ro + l ) where Q, is the component of dV corresponding to Q and f: dD0

mX^on~m-*Q, gii dDt

mtl X A"-™"1 -*• Q, are imbeddings. Let F — q X -Do""* c v with q t dD0

m, dF C dV. Suppose dF does not intersect g^dDj**1 X 0), t — 1, • • •, r — 1 and grdDf**1 X 0) intersects dF transversally in a single point. Then W is of the form

x(M,Q;g'u- ,g,r-l;m +1).

Proof 2.2. In the proof of 2.2, we use without further mention, the fact that the diffeomorphism type of an n-manifold is not changed when an n-disk is adjoined by identifying an (n — 1) disk on the boundary of each under a diffeomorphiBm. See GPC, 3.4, J. Milnor, " Sommes de varietes differen-tiables et structures difterentiables des spheres," Bulletin de la Societe Mathe-matique de France, vol. 87 (1959), pp. 439-444 and R. Palais, " Extending diffeomorphism," Proceedings of the American Mathematical Society, vol. 11 (1960), pp. 274-277.

We may assume, using the uniqueness of tubular neighborhoods that dF does not intersect y , ( « D r " X ^ * ' ) , t — 1,- • - , r — 1 .

Since gridD™*1 X 0) is transversal to dF in dV, there exists a disk neigh­borhood L of o — gridlP^XO) DF, L-.AmXB*-m-\ where Am X 0 is a disk neighborhood of o in gridD™*1 X 0), 0 X 5""m_I a disk neighborhood of o in dF, with (o,o) corresponding to o.

Now there exists a disk neighborhood Dam of the point F f~l (D0

m X 0) in Do" X 0 so small that if N — Da

m X #o""m C V, then

(1) N n imagegt«= 0 , t — 1 , - • • ,r—1, and

(2) 2V n image 0 r C.L.

Since both Dam X o and Am X o (i. e. Am X 0) are transversal to dF in

dV, we may assume using a diffeomorphism of V, and restricting L, that Am X o, D9

m X o coincide, and that L coincides with image gr n 2V. The following statements are made under the assumptions that corners

are smoothed via "straightening the angle," Section 1 of GPC or better [9], Let K — N U D r -«- i C W.

206

390 S. SHALE.

We claim that KnCl(W—E) is diffeomorphic to an (n — l)-disk. First KnClW — E) is 8Da

m X (MV*1 X ZV1-*-1) —interior L or

dDam X D<rm U #i>m X D,"-""1

where Dt,m is 0IV 4 1 minus the interior of an m-disk. Furthermore KnCl(W—K) may be described as MV" X A.""1" with D r X ^ r " " " " 1

attached by an embedding h: dDbm X Dr"-""1 -»9fl ," X -Do""m with the

property that h(dDbm X 0) coincides with 0.DO

,B X c for some point c £ Do""1". In fact, h is the restriction of gr. This is the situation in the proof of 3.3 of GPC, where it was shown that the resulting manifold was a disk. Thus KDCl(W — K) is indeed an (n —l)-disk.

Since E is an n-disk, E D Cl(W — E) an (n— l)-disk, we have that W is diffeomorphic to Cl(W — E). On the other hand it is clear from the previous considerations that Cl(W— E) is of the form

x(M,Q;g\,- ,g'r-l,m + 1). This proves 2.2.

The next lemma follows from the method of Whitney [18] of removing isolated intersection points. The paper of A. Shapiro [11] makes this apparent (apply 6.7, 6.10, 7.1 of [11]).

LEMMA 2.3. Suppose JV»-m is a closed submanifold of the closed mani­fold Xn and f: Mm ->X is an imbedding of a closed manifold. Suppose also that M, N are connected, X is simply connected, n — m > 2, m > 2 and 6— f(M) oN is the intersection number of f(M) and N. Then there exists an imbedding f: M-*X isotopic to f such that f'(M) intersects N in b points, each with transversal intersection.

LEMMA 2.4. Let FaH~m~l be a submanifold of Q where Q is a com­

ponent of the boundary of a compact manifold Vn, n — m > 2. Let W — x(v>Q>9'm + 'i-) where 9• dDa

m+l X 2V~m~1 ->Q isan imbedding with b the intersection number gdDa

mtl X 0 ) °F0 . For an imbedding h: Sm-*Qn dW, there is an imbedding h': Sm-+Q (1 dW, isotopic to h in dW with h'(Sm)°Fo — h(Sm)°F0 + b, sign prescribed.

Proof. Let D be the closed upper hemisphere of S^, xo € ft/V""*"1 and H*, H~ be the closed upper, lower hemisphere respectively of g(dD0

mtl Xx»)* Then h is isotopic in dW n Q to an imbedding V: S^-^dW n Q, with K'(Sm) C\ g (dDo""1 X Xo) equal H* with the orientation determined by the ± 6 of 2.4. This follows essentially from R. Palais, Extending Diffeomor-phism, Proc. AMS, vol. 11 (1960), pp. 274-277, Theorem B, Corollary 1.

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STRCCTUBB OP MANIFOLD8. 391

Next let X be V followed by the reflection map H+-*B-, w that X, V: D-*dW are naturally lopologically isotopic. However W is an angle on dD. By the familiar process of " straightening the angle " we modify V: 5™ -* dW n Q to an embedding h': Sm-*dW C\ Q. Our construction makes it clear that ft' and h are isotopic in dW, and that h' has the desired property of 2.4.

We now prove 2.1. Let F be as in 2.2 and &< be the algebrais intersection number gi(dDm*1X0)°9F, t — 1,- • ,r. We first note that the 6( are relatively prime. This in fact follows from the homology hypothesis of the theorem.

r The proof proceeds by induction on 2 I &< I an<i is started by 2.3 and 2.2.

r «=1

Suppose 2.1 is true in case 2 I &t I is p — 1 > 0. We can say from the homotopy structure of W that Hm(W,M) is

Hm(V,M) wtih the added relations [Mty"*1]—0, t — l , - - , r , where [3Am t l] C f f m (7 , J f ) —Z and Fm(7,Af) is generated by (Da

m,dD<r). Since jffm(W,ilf) -= 0, [dD«m+1] are relatively prime. On the other hand,

since 2 ? 0m X 0 o ^ = l, we have that [dD^*1] — bt. So the bt, i = l,- • -,r

are relatively prime. Since the 6< are relatively prime, there exists, t0, ii, t o ^ t i with

I K I = I &«i I > 0. One now applies 2.4 to reduce \bk\ by | &<, | using the covering homotopy property as in Section 2 of GPC. The induction hypo­thesis applies and we have proved 2 .1 .

LEMMA 2.5. Lei n^2m + l, (n,m) ^ (4,1), (3.1) , (5 .2) , (7.3) , M» be a compact manifold with a simply connected boundary component Q andV—"x(M,Q;f;m) wheref: dDmXtDn-*n-+Q is a contractible imbedding. Let Qt be the component of dV corresponding to Q and W — x(^> Qu g\ *» + 1) where g: dDx

m*1 X IV~m-> Q\. Then if the homomorphism irm(V,M) ->Tm(W,M) induced by inclusion is zero, W is diffeomorphic to M.

We use the following for the proof of 2.5.

LEMMA 2.6. Let Y be a simply connected polyhedron and Z an (m — 1)-connected polyhedron. Then irm(Y V Z) — irm(Y) -\-ir„(Z).

This is a standard fact in homotopy theory. For example it follows from [6], V. 3.1 and the relative Hurewicz theorem.

Using 2.6 it follows easily that »*(&) — *m(Q) +irm(Sm). Then from the homotopy hypothesis it follows that the homotopy class y

of g restricted to 02V*+1 X 0 is of the form a + 0i where a€ wm(Q) and g%

generates irm(8m).

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392 8. SMALE.

Since P is contractible, V — M -\-H, where H is an (n— TO)-cell bundle over £"», and also Q1 = Q + dH. Then let / , : dD,n*z ->• Q be an imbedding representing a ond $7% : dJ)""1 -> 0ZF an imbedding intersecting 0F transversally in a single point where F is the same as n 2.2. Then by the sum construction we obtain g': dD™*1 X 0-> @i realizing y with the property that g^dD^'1 X 0) intersects dF transversally in a single point where F is the same as in 2.2. Application of 2.4 of GPC and 2.2 finishes the proof.

Section 3. Among other things, we apply the theory of Section 2 to obtain 1.4.

THEOREM 3.1. Let Wn be a manifold (not necessarily compact), n > 5, with dW the disjoint union of simply-connected manifolds M± and M, where the inclusion M(—*W are homotopy equivalences. Suppose j : V„—*Mi is the inclusion of a compact manifold Va into M which is a homotopy equiv­alence and there is an imbedding a: Cl(Mt — F0) X [1*2]—*W such that a) the complement of the image of a has compact closure and b)

aClMl — V,)X«) CMi » - M ,

a restricted to Cl(Ml— V0) X 1 ** j - Then a can be extended to a diffeo-morphism Af, X [1,2] -> W.

Proof of 3.1. Let h = [— i>n + i ] and replace [1,2] in the statement of 3.1 by I0, denoting the projection CJ(Jlf j — V0) C I0-*I0 by /0. We may sup­pose that points under a have been identified so that Cl(M1— 70) X ' « C W , Then by the results of [12] one can find a non-degenerate C" real function / on W such that a) / restricted to Cl(M1— V0)Xh is fa, b) at a critical point the value of / is the index and c) /(Afi) — — £, fM) — n-f- £

Let Z p — / _ 1 [ — i , p + i ] - ^ e wiN 8 n o w inductively that by suitable modifications of / which also satisfy a), b) and c), we can assume Afp is a product ilf 1 X I (or equivalently the modified / has no critical points of index ^Sp).

FirBt by 5.1 of GPC, note that we may assume that the function / has no critical points of index 0. Next by the method in Section 7 of GPC, using the fact that ?ri(Afi) =-wi(W) — 1, we can similarly assume that there are no critical points of / of index 1.

We are not quite yet in the dimension range where 2.1 applies, but we apply 2.5 to eliminate a critical point of / of index 2 if it occurs, as follows.

We have that Z , = X(X1, & ; / „ • •, fk;2), Xt — X(Z2 ,Q,;gu - , ? r ;3 ) where & — / " ' ( l i ) , Qf=f'1^i)- I t follows from the homotopy hypothesis

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STRUCTURE OP MANIFOLDS. 393

that each ft is contractible in Qr so that X2 is of the form Xx 4- E, H£9i(n, k, 2) (following notation of GPC).

The 0i's induce a homomorphism Or-*^tQi)- Let <f> be the composition

O r - > » i ( C i ) - » * i ( - y i ) - » » t ( ^ )

where the last homomorphism it obtained by identifying Xi to a point in X2.

Assertion. <f> is an epimorphism.

Suppose the assertion is false and ce£ir2(H) is not in the image of <f>. Then since ir2 (Z2) = T 2 ( ^ I ) + i r 2 ( H ) (by 2.6), the image of a under irt(H) - » J T » ( Z 2 ) -*in(2f,) is not in the image of ^ ( Z j ) ->ir2(Z2) -*ir2(Z»). But the last composition is an isomorphism since Z , — .if i X I, thus contra­dicting the existence of such an a. Hence the assertion is true.

Let yi,• • • ,yk be the generators of v2H) corresponding to / „ • • -,/». Then by 4.1 of GPC there is an automorphism /? of Or such that <fc/3(<?<) — y<, i=£ k, <j>0gi) — 0, t > k. By 2.1 of GPC it can be assumed that the gx are such that <f>(ffi) — yt, t ^ k, <f> ($r<) — 0, t < k.

Now apply 2.5 with W, V, M corresponding to

respectively. This eliminates the critical point of / corresponding to /* and by induction all the critical points of index two are eliminated.

Applying some of the previous considerations to n — / we eliminate the critical points of / of index n, n — 1.

Now more generally suppose / on Xp_t has no critical points where p^n — 3. Then since Hp(X^lfXp) — 0, 2.1 applies to eliminate the critical points of index p. Thus we obtain by induction a function / on W with critical points only of index n — 2, and which satisfies the conditions a)-c) above. By 7. 5 of GPC, / has no critical points at all. This proves 3.1.

COROLLARY 3.2. Suppose W* is compact, n > 5, dW the disjoint union of closed manifolds 1S\, Mt with each Jf<—*TF o homotopy equivalence. Suppose also V C W with

dV — V1 U Vz, VtCMt, V—VXI and ir^Af,— F,) - 1.

Then i: -*W can be extended to a diffeomnrphism of MxXl onto W.

Proof of 3.2. First t may be extended to T X / where T is a tubular neighborhood of Vx in Mr. Then apply 3.1 to W— 7 to get 3.2.

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394: 8. 8MALE.

We now can prove 1 - 4. First by 3.2 with 7 empty applied to V of 1.4 yields that V is diffeomorphic to V, X I- Now 3.2 applies to yield 1.4.

Section 4. The following is quit a general theorem and in fact con­tains 1.1 as a special case with fc = n — 1.

THEOREM 4.1. Suppose W* D Mk where W is a compact connected manifold and M is a closed manifold. Furthermore suppose

(a) ri(dW)-^(M)-1

(b) n > 5 (c) The inclusion of M into W is a homotopy equivalence. Then W

is diffeomorphic to a closed cell bundle over M, in particular to a tubular neighborhood of M in W.

We need a lemma.

LEMMA 4.2. Suppose B is a compact connected n-dimensional sub-manifold of a compact connected manifold V" with dBOdV —-0, wi(ftB) — TTI(37) —1 andH^(B) -*^E^(V) induced by inclusion is bijective. Then Q — Cl(V — B) has boundary consisting of dV, dB with the inclusions of dV, dB into Q homotopy equivalences.

For the proof of 4.2 we use the following version of the Poincare Duality Theorem, which follows from the Lefschetz Duality Theorem.

THEOREM 4.3. Suppose Wnis a compact manifold dW the disjoint union of manifolds M^ and Mi (possibly either or both empty). Then for all i, E^W^t) is isomorphic to H^(W,M2).

To prove 4.2 note Ht(Q, dB) — ff«(F, B) =- 0 and E<(Q, dB) — H*(V, B) — 0 for all i. By 4.3 then Ht(Q, dV) •= 0 for all i also. By the Whitehead theorem we get 4.2.

The proof of 4.1 then goes as follows. We can first suppose that M is disjoint from the boundary of W. Now let T be a tubular neighborhood of M which is alro disjoint from dW. Now apply 4.2 and 3.3 to Cl(W — T) with V of 3.2 empty. This yields tht Cl(W — T) is diffeomorphic to dT X I and hence W is diffeomorphic to T. We have proved 4 .1 .

THEOREM 4.4. Suppose 2n =i 3m -\- 3 and a compact manifold Wn has the homotopy type of a closed manifold Mm, n > 5, with »i(0W) — (Jf ) =-1. Then W is diffeomorphic to a cell-bundle over M.

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8TBUCTUHB OF HANIFOLDS. 395

Proof. Let / : M-+W be a homotopy equivalence. By Haefliger [1] , / iB homotopic to an embedding g: M-+W. Now 4.1 applies to yield 4.4.

Section 5. We continue with some consequences of 4 .1 . The next theorem is a strong form of the Generalized Poincare' Conjecture for n > 5 and was first proved in [14] except for n — 7 . This theorem follows from 4.1 by taking M to be a point.

THKOKEM 5.1. Suppose Cn is a compact contractible manifold with Tl (pC) — 1 and n > 5. Then C is diffeomorphic to the n-disk D".

For n — 5, if one knows in addition that dC is diffeomorphic to S*, then using the theorem of Milnor 0" — 0 and 1.1, one obtains that C is diffeo­morphic to 2?8.

The following is a weak unknotting theorem in the differentiable case. Haefliger [2] has given an imbedding (differentiable) of 3* in S* which does not bound an imbedded D*. On the other hand we have:

THBOBEM 5.2. Suppose S* C Sn with n — k > 2. Then the closure of the complement of a tubular neighborhood T of Sk in Sn is diffeomorphic to gn-k-i >< 2>»»i

The proof of 5.2 is as follows (the case n f§ 5 is essentially contained in Wu Wen Tsun [19]). I t is well-known and easy to prove that if X — Cl(S» — T), X has the homotopy type of S"-*-1. In fact T is diffeo­morphic to a cell bundle over Sk and the inclusion of the boundary of a fiber S0

n~k'1 into X induces the equivalence. Furthermore the normal bundle of S0

H'k~1 in 5" is trivial because tfo""*"1 bounds a disk in Sn. Now 4.1 applies to yield Theorem 5.2.

One can also prove some recent theorems of M. Hirsch [5], replacing his combinatorial arguments by application of the above theorems.

THEOREM 5.3 (Hirsch). / / / : Mxn —>M2

n is a homotopy equivalence of simply connected closed manifolds such that the tangent bundle of Mt is equivalent to the bundle over Mi induced from the tangent bundle of Mt by f, then Mt X D* and Mt X -D* are diffeomorphic for k > n.

One obtains 5.3 by imbedding Mr in M> X #* approximating the homo­topy equivalence and applying 4 .1 . The tangential property of f is used to conclude that a tubular neighborhood of Mi in Mt X&* is a product neighborhood.

THEOBBM 5.4 (Hirsch). If the homotopy sphere M* bounds a paralleliz-able manifold, then M*XD* is diffeomorphic to S* X -D*.

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396 8. SKALE.

One first proves that M" can be imbedded in S"*s with trivial normal bundle by following Hirch [4] or using "handlebody theory." Then apply the argument in 5.2 to obtain the complement of a tubular neighborhood of M* is diffeomorphic to 81 X Dntl. The closure of the complement of S* X D**1

in Sn*' is Sn X D\ thus proving 5.4.

Section 6. The main goal of this section is the following theorem.

THEOREM 6.1. Let M be a simply connected closed manifold of dimen­sion greater than five. Then on M there is a non-degenerate C" function with the minimal number of critical points consistent with the homology structure.

One actually obtains such a function with the additional property that at a critical point its value is the index.

6.2. We make more explicit the conclusion of 6.1. Suppose for each O ^ t ^ n , <T«I,• • •,<r»»(i),ni)' ',Ti«(i) 18 a set of generators for a corres­ponding direct sum decomposition of Hi(M), ay free, -ry of finite order. Then one can obtain the function of 6.1 with type numbers satisfying

Nt-p(i)+q(i)+q(i—l). By taking the q(i) minimal, the Mi becomes minimal.

In the case there is no torsion in the homology of M, 6.1 becomes.

THEOREM 6.3. Let M be a simply connected closed manifold of dimen­sion greater than five with no torsion in the homology of M. Then there is a non-degenerate function on M with type numbers equal the betti numbers of M.

We start the proof of 6.1 with the following lemma.

LEMMA 6.4. Let M* be a simply connected compact manifold, n > 5, n S^ 2m. Then there is an n-dimensional simply connected compact manifold Xm such that:

a) H,(Xm)— 0, j>m b) There is a "nice" function on Xm, minimal with respect to its

homology structure. In other words there is a C" non-degenerate function on Xm, value at a critical point equal the index, equal to m-\-\ on dXm>

regular in a neighborhood of dXm and the k-th type number Mn is minimal in the sense of 6.2.

c) There is an imbedding i: Xm—*M* such that

i(3Xm) n - i f - 0, if. B,(Xm) -> H,(Xm)

is bijective for j < m and surjective for j — m.

213

8TBUCTURE OF MANIFOLDS. 397

Proof. The proof goes by induction on m, starting by taking Xi to be an n-disk. Suppose, X^.,, to: X^-i ->M have been constructed satisfying a)-c) with 2k ^ n . For convenience we identify points under i0 so that Zfc_i C M. We now construct Xk, i: X*->M satisfying a)-c).

By the relative Hurewicz theorem the Hurewicz homomorphism

is bijective.

For the structure of Hk(M,Xk-\) consider the exact sequence

0 -*F*( t f ) -* * » ( * , * * - ! ) ->#»_»(*»_») >H+.tM) -*0.

Let Yu' " -,yp be a set of generators of Hk(M,Xk-i) corresponding to a minimal set of generators of Hk(M) together with a minimal set for Ker;'.

Represent the elements A"1(yi)< • " ^ ( y * ) by imbeddings

§t: (D*,dD><) - (Cl(M-X^dX^)

with §\(Dk) transversal to dX^ along gi(dDk), for example following Wall [17], proof of Theorem 1.

In the extreme case n — 2k, the images of <J« generically intersect each other in isolated points- These points can be removed by pushing them along arcs past the boundaries. Still following [17], the gt can be extended to tubular neighborhoods,

g(: (D>,dD*)XD"-*-+ (CUM—X^dX^).

Then we take X* to be x(2T f c .1 ; /„- • -,g't;h) where / « : d&XD*-* -> bD^x is the restriction of gt. I t is not difficult to check that Xk has the desired properties a)-c). This proves 6.4.

To prove 6.1, let Af« be as in 6.1 with n — 2m or 2m + 1. Let Xm C M as in 6.4, / the nice function on Xm and K~~Cl(M—Xm). Then Ht(M,X) — 0, i^m, so by duality HSK) — 0, j^n — m. By the Universal Coeffi­cient Theorem this implies that H_m_1(Z) is torsion free. Let >Vm-i C K be again given by 6.4 with g the nice function on Y^^.^. By 4.2 and 3.2 we can in fact assume that K and y«-m_i are the same, so M«— X„ U Y% m ,. Let /o be the function which is / on X„ and n — ^ on r»_«»-i. By smoothing /o along flX,, we obtain a C" function f. It is not difficult using the Uni­versal Coefficient Theorem and Poincare Duality to show that f may be taken as the desired function of 6.1.

The previous results of this section may be extended to manifolds with boundary.

214

398 S. 81CALE.

By the previous methods one may prove the following generalization of 6.1. We leave the details to the reader.

THEOREM 6. 5. Suppose Wn is a simply connected manifold with simply-connected boundary, n > 5. Then there is a nice function f on W (non-degenerate, value n -\- \ on dW, regular in a neighborhood of dW, value at a critical point is the index) with type numbers minimal with respect to the homology structure of (W, 0W).

Section 7. The goal of this section is to prove the following.

THEOREM: 7.1. Let f: WS-tWf be a homotopy equivalence between two manifolds such that the tangent bundle 2 \ of W\ is equivalent to f~lTt. Suppose also n>5, n c", 2m + 1, H^W^—O, i>m, ^(W^ =,rI(dW1) — ir^dWj)—1. Then W\ and Wt are diffeomorphic by a diffeomorphism homotopic to f.

Let g be a nice function on Wi with no critical points of index greater than m, whose existence is implied by 6.5. Thei we let X» — g'1[0, fc + $ ] , k — 0,1, • • •, m with Xm = W,. By 3.2 and 4.2 is is sufficient to imbed Xm

in W2 by a map homotopic to /. Suppose inductively we have defined a map /*_!: X* —* W2 homotopic to

/ with the property /*_i is an imbedding, k ^ m. Let Xk be written in the form x(Xk.1;gu- • -,gp;k) where gt: dDkXD"-k-+dXkl-1. Using the Whitney imbedding theory we can find / V i : Xk -> W2 homotopic to /*_!, which is an imbedding on Xk.1 and on the images g\Dk X 0) in X*. M we^-It remains to make / V i a n imbedding on a tubular neighborhood of each of the gi(Dk X 0), or equivalently on each of the g^D" X i?""*).

This can be done for a given i if and only if an element -y4 in T*-i(0(n — &)) defined by / V i in a neighborhood of gi(9Dk X 0) >8 zero-But the original tangential assumptions on / insure -y« —0 in this dimension range. The arguments in proving these statements are so close to the argu­ments in Hirsch [3] Section 5, that we omit them. This finishes the proof of 7.1.

Section 8. We note here the following theorem.

THEOREM 8.1. Let M2™*1 be a closed simply connected manifold, m > 2, with Hm(M) torsion free. Then there is a compact manifold W2"*1, uniquely determined by M and a diffeomorphism h: dW-*dW such that M is union of two copies of W with points identified under h.

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STRUCTOHB OF MANIFOLDS. 399

Proof. Let Wx2mtX C M be the manifold given by 6.4. Let W ^ 1

CM — W be ako given by 6.4. Then it is not difficult using homotopy theory to show that W„ Wt satisfy the hypotheses of 7.1. AIBO by previous arguments Wt is diffeomorphic to Cl(M—WJ. The uniqueness of Wx— W, is also given by 7.1. Putting these facts together we get 8.1.

Remark. I don't believe the condition on Hm(M) is really necessary here. Also in a different spirit, 8.1 is true for the cases m — 1, m — 2.

REFERENCES.

[1] A. Haeniger, " Plongement Differentiable des varietes des varietes," Commentari Mathematicii Helvetica, to appear.

[2] , "Knotted (41s — 1) sphere* in rtfc space," to appear. [3] M. W. Hirsch, " Immersions of manifolds," Transaction* of the American Mathe­

matical Society, vol. 93 (1959), pp. 242-276. [4] , " On manifolds which bound in Euclidean space," to appear. [6] , Diffeomorphisms of tubular neighborhood* (mimeographed), Berkeley,

1901. [6] E. T. Hu, Homotopy theory, New York, 1959. [7] M. Kervaire and J. Milnor, "Groups of homotopy spheres, I and II," to appear. [8] B. Mazur, "A note on some contractible 4-manifolds," Annal* of Mathematic*,

vol. 73 (1961), pp. 221-228. [9] J. Milnor, Differentiable manifold* which are homotopy spheres (mimeographed),

Princeton, 1959. [10] , Two complexes which are homeomorphio but combinatorial distinct

(mimeographed), Princeton University (1961). [ I I ] A. Shapiro, " Obstructions to the imbedding of a complex in a Euclidean space, I.

The first obstruction," Annal* of Mathematics, vol. 60 (1957), pp. 266-269. [12] S. Smale, "On gradient dynamical systems," Annals of Mathematics, vol. 74

(1961), pp. 199-206. [13] ."Generalized Poincare's conjecture in dimensions greater than four,"

Annals of Mathematics, to appear. [14] , "Differentiable and combiratorial structures on manifolds," Annals of

Mathematics, to appear. [15] , "On the structure of five manifolds," Annals of Mathematics, to appear. [16] R. Thorn, " Les classes caracteristiques de Pontrjagin des varieteB triangulees,"

Topologia Algebraico, Mexico, 1958, pp. 54-67. [17] C. T. C. Wall, "Classification of (n —1)-connected 2n-manifolds," to appear. [18] H. Whitney, "The Belf-intersections of a smooth n manifold in 2n-space," Annals

of Mathematics, vol. 45 (1944), pp. 220-240. [19] Wu Wen Tsun, "On the isotopy of C" manifolds of dimension n in Euclidean

( 2 n + 1)-space," Science Record, New Series, vol. II (1938), pp. 271-275.

216

A SURVEY OF SOME RECENT DEVELOPMENTS IN DIFFERENTIAL TOPOLOGY

S. SMALE

1. We consider differential topology to be the study of differenti-able manifolds and differentiable maps. Then, naturally, manifolds are considered equivalent if they are diffeomorphic, i.e., there exists a differentiable map from one to the other with a differentiable in­verse. For convenience, differentiable means C°; in the problems we consider, C would serve as well.

The notions of differentiable manifold and diffeomorphism go back to Poincare" at least. In his well-known paper, Analysis situs [27] (see pp. 196-198), topology or analysis situs for Poincard was the study of differentiable manifolds under the equivalence relation of diffeomorphism. Poincar6 used the word homeomorphism to mean what is called today a diffeomorphism (of class C). Thus differential topology is just topology as Poincarf originally understood it.

Of course, the subject has developed considerably since Poincare'; Whitney and Pontrjagin making some of the major contributions prior to the last decade.

Slightly after Poincar6's definition of differentiable manifold, the study of manifolds from the combinatorial point of view was also initiated by Poincare, and again this subject has been developing up to the present. Contributions here were made by Newman, Alexan­der, Lefschetz and J. H. C. Whitehead, among others.

What started these subjects? First, it is clear that differential geometry, analysis and physics prompted the early development of differential topology (it is this that explains our admitted bias toward differential topology, that it lies close to the main stream of mathe­matics). On the other hand, the combinatorial approach to manifolds was started because it was believed that these means would afford a useful attack on the differentiable case. For example, Lefschetz wrote [13, p. 3<5l], that Poincar6 tried to develop the subject on strictly "analytical" lines and after his Analysis situs, turned to com­binatorial methods because this approach failed for example in his duality theorem.

Naturally enough, mathematicians have been trying to relate these

An address delivered before the Stillwater meeting of the Society on August 31, 1961, by invitation of the Committee to Select Hour Speakers for Western Sectional Meetings; received by the editors November 28,1962.

131

217

132 S. SMALE (March

two viewpoints that have developed side by side. S. S. Cairns is an example of such.

In the last decade, the three domains, differential topology, com­binatorial study of manifolds, and the relations between the two, have all advanced enormously. Of course, these developments are not isolated from each other. However, we would like to make the follow­ing important point.

It has turned out that the main theorems in differential topology did not depend on developments in combinatorial topology. In fact, the contrary is the case; the main theorems in differential topology inspired corresponding ones in combinatorial topology, or else have no combinatorial counterpart as yet (but there are also combinatorial theorems whose differentiable analogues are false).

Certainly, the problems of combinatorial manifolds and the rela­tionships between combinatorial and differentiable manifolds are legitimate problems in their own right. An example is the question of existence and uniqueness of differentiable structures on a com­binatorial manifold. However, we don't believe such problems are the goal of differential topology itself. This view seems justified by the fact that today one can substantially develop differential topology most simply without any reference to the combinatorial manifolds.

We have not mentioned the large branch of topology called homo­topy theory until now. Homotopy theory originated as an attack on the homeomorphism or diffeomorphism problem, witness the "Poin-car£ Conjecture" that the homotopy groups characterize the homeo­morphism type of the 3-sphere, and the Hurewicz conjecture that the homeomorphism type of a closed manifold is determined by the homotopy type. One could attack the homotopy problem more easily than the homeomorphism one and, for many years, most of the progress in topology centered around the homotopy problem.

The Hurewicz conjecture turned out not to be true, but amazingly enough, as we shall see, the last few years have brought about a re­duction of a large part of differential topology to homotopy theory. These problems do not belong so much to the realm of pure homotopy theory as to a special kind of homotopy theory connected with vector space bundles and the like, as exemplified by work around the Bott periodicity theorems.

Of course, there are a number of important problems left in differ­ential topology that do not reduce in any sense to homotopy theory and topologists can never rest until these are settled. But, on the other hand, it seems that differential topology has reached such a satisfactory stage that, for it to continue its exciting pace, it must

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1963] DEVELOPMENTS IN DIFFERENTIAL TOPOLOGY 133

look toward the problems of analysis, the sources that led Poincar6 to its early development.

We here survey some developments of the last decade in differen­tial topology itself. Certainly, we make no claims for completeness. A notable omission is the work of Thorn, on cobordism, and the study of differentiable maps. The reader is referred to expositions of Milnor [2l] and H. Levine [14] for accounts of part of this work.

2. We now discuss what must be considered a fundamental prob­lem of differential topology, namely, the diffeomorphism classifica­tion of manifolds.

The classification of closed orientable 2-manifolds goes back to Riemann's time. The next progress on this problem was the develop­ment of numerical and algebraic invariants which were able to dis­tinguish many nondiffcomorphic manifolds. These invariants include, among others, the dimension, betti numbers, homology and homo-topy groups and characteristic classes.

For dimension greater than two, there was still (at the beginning of 1960) no known case where the existing numerical and algebraic invariants determined the diffeomorphism class of the manifold. The simplest case of this problem (or so it appeared) was that posed by Poincar£: Is a 3-manifold which is closed and simply connected, homeomorphic (equivalently diffeomorphic) to the 3-sphere? This has never been answered.

The surprising thing is, however, that without resolving this prob­lem, the author showed that in many cases, the known numerical and algebraic invariants were sufficient to characterize the diffeo­morphism class of a manifold. Generally speaking in fact, considera­ble information on the structure of manifolds was found. We will now give an account of this.

To see how manifolds can be constructed, one defines the notion of attaching a handle. Let M* be a compact manifold with boundary dM (we remind the reader that everything is considered from the C" point of view, manifolds, imbeddings, etc.) and let D' be the j-disk (i.e., the unit disk of Euclidean 5-space £•)• Suppose/: (3J?')XX?*-* —*M is an imbedding. Then X(M;f; s), "M with a handle attached by / ," is defined by identifying points under/and imposing a differ­entiable structure on M\J/D''X.Dn-' by a process called "straighten­ing the angle." Similarly if /<: (dD[) X£>T'—M, t = l , •• • , k, are imbeddings with disjoint images one can define X(M;fi, • • • , / * ; s). If M itself is a disk, then X(M;fu • • • , /»; 5) is called a handlebody.

(2.1) THEOREM. Let f be a C* function on a closed (i.e. compact with

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empty boundary) manifold with no critical points on f~l[ — «, «] except k nondegenerate ones on / - , (0) , all of index s. Then / - , [ — », t] is diffeomorphic to Xf~x[ — » , — <];/i, • • • ,/*; s) for suitable/,•).

(For a reference to the notion of nondegenerate critical point and its index, see e.g. [1 ].) This might well be regarded as the basic theo­rem of finite dimensional Morse theory. Morse [23] was concerned with the homology version of this theorem, Bott [ l ] , the homotopy version of 2.1. The proof of 2.1 itself is based on the ideas of the proofs of the weaker statements.

(2.2) THEOREM (MORSE-THOM). On every closed manifold W, there exists a C function with nondegenerate critical points.

For a proof see [41 ]. By combining 2.1 and 2.2 we see that every manifold can be ob­

tained by attaching handles successively to a disk (we have been restricting ourselves to the compact, empty boundary case only for simplicity).

The main idea of the following theory is to remove superfluous handles (or equivalently critical points) without changing the diffeo-morphism type of the manifold. For this one starts (after 2.2) with a "nice function," a function on M given by 2.2 with the additional property that the handles are attached in order according to their dimension (the s in D'XDn~').

(2.3) THEOREM. On every closed manifold there exists a function f with nondegenerate critical points such that at each critical point, the value off is the index.

This was proved by A. Wallace [44] and the author [38] inde­pendently by different methods. (For a general reference to this sec­tion see [34; 37].)

The actual removing of the extra handles is the main part and for this one needs extra assumptions. The next is the central theorem (or its generalization "2.4'" to include manifold with boundary).

(2.4) THEOREM. Let M be a simply connected closed manifold of dimension > 5. Then on M there is a nondegenerate and nice) function with the minimal number of critical points consistent with the homology structure of M.

We make more explicit the conclusion of 2.4. Let <r,i, • • • , <r,x,-), Tii, • • • . Tfc(o, O ^ i ^ n , be a set of generators for a direct sum de­composition of HiM), a a free, T<> of finite order. Then one can ob-

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tain the function of 2.4 with type numbers Mi (the number of critical points of index »') satisfying Mi=p(i)+g(i)+g(i—1).

The first special case of 2.4 is (2.5) THEOREM. Let M be a simply connected closed manifold of

dimension greater than S with no torsion in the homology of M. Then there exists a nondegenerate function on M vnth type numbers equal to the betti numbers of M.

We should emphasize that 2.4 and 2.5 should be interpreted from the point of view of 2.1. One may apply 2.5 to the case of a "homotopy sphere" (using 2.1 of course).

(2.6) THEOREM. Let M" be a simply connected closed manifold with the homology groups of a sphere, n > 5. Then M can be obtained by "glu­ing" two copies of the n-disk by a diffeomorphism from the boundary of one to the boundary of the other.

For n = 5, the theorem is true and can be proved by an additional argument.

Theorem 2.6 implies the weaker statement ("the generalized Poin-carl conjecture in higher dimensions") that a homotopy sphere of dimension ^ 5 is homeomorphic to 5", see [33]. Subsequently, Stall-ings [39] and Zeeman [50 ] found a proof of this last statement.

Theorem 2.4 was developed through the papers [34; 35] and ap­pears in the above form in [37]. Rather than try to give an idea of the proof of 2.4, we refer the reader to these papers. One may also refer to [2; 3] and [15].

The analogue of 2.4, say 2.4' is also proved for manifolds with boundary [37] and this analogue implies the A-cobordism theorem stated below. The simplest case of 2.4' is

(2.7) DISK THEOREM. Let Mm be a contractible compact manifold, n > 5, dM connected and simply connected. Then M* is diffeomorphic to the disk D\

We now discuss another aspect of the preceding theory, the rela­tionship between diffeomorphism and A-cobordism. Two oriented, closed manifolds Mu Mt are cobordant if there exists a compact oriented manifold W with 8W= Mi—Mi (taking into account orien­tations). Thorn in [40] reduced the cobordism classification of mani­folds to a problem in homotopy theory which has since been solved.

Two closed oriented manifolds Mi, Mt are A-cobordant (following Milnor [18]) if one can choose W as above so that the inclusions Mf-*W, i = l , 2 are homotopy equivalences. The idea of A-cobordism

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(homotopy-cobordism) is to replace diffeomorphism by a notion of equivalence, a priori weaker than diffeomorphism and amenable to study using homotopy and cobordism theory.

The following theorem was proved in [37], with special cases in [34; 35]. It is also a consequence of 2.4'.

(2.8) T H E A-COBORDISM THEOREM. Let Mf, AfJ be closed oriented simply connected manifolds, n > 4, which are h-cobordant. Then M% and Mt are diffeomorphic.

Milnor [20] has shown that 2.7 is false for nonsimply connected manifolds. On the other hand Mazur [16] has generalized 2.7 to a theorem which includes the nonsimply connected manifolds.

Theorem 2.8 reduces the diffeomorphism problem for a large class of manifolds to the A-cobordism problem. This A-cobordism problem has been put into quite good shape for homotopy spheres by Milnor [19], Kervaire and Milnor [12], and recently Novikov [24] has found a general theorem.

Kervaire and Milnor show that homotopy spheres of dimension «, with equivalence defined by A-cobordism form an abelian group 3C\ Their main theorem is the following.

(2.9) THEOREM. 3C" is finite, n-^3.

Kervaire and Milnor go on to find much information about the structure of this finite abelian group, in particular, to find its order for 4 ^ n ^ 18. The main technique in the proof of 2.9 is what is called spherical modification or surgery, see [44; 22].

Putting together 2.8 and 2.9 one obtains a classification of the simplest type of closed manifolds, the homotopy spheres, of dimension n for 5 ^ n ^ l 8 and for general n the finiteness theorem. This also gives theorems on differentiable structures on spheres. See [35] or [12].

Most recently, Novikov has found a very general theorem on the A-cobordism structure of manifolds (and hence by 2.8, the diffeo­morphism structure). We refer the reader to [24] for a brief account of this.

Lastly we mention some specific results on the manifold classifica­tion problem [36; 37].

(2.10) THEOREM, (a) There is a 1-1 correspondence between simply-connected closed 5-manifolds with vanishing 2nd Stiefel-Whilney class and finitely generated abelian groups, the correspondence given by M-*Free part Ht(M)+l Torsion part Ht(M).

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(b) Every 2-connected 6-manifold is dijfeomorphic to S* or a sum (in the sense of [18]) of r copies of S'X-S*, r a positive integer.

C. T. C. Wall has very general theorems which extend the above results to (n—1)-connected 2n-manifolds [43].

3. We give a substantial account of immersion theory because the main problem here has been completely reduced to homotopy theory.

An immersion of one manifold Mk in a second X* is a C" map / : M—*X with the property that for each />£A/, in some coordinate systems (and hence all) about p and f(p), the Jacobian matrix o f / has rank k. A regular homotopy is a homotopy /«: M—*X, 0 g ti£ 1 which for each t is an immersion and which has the additional prop­erty that the induced map Ft: Tu—*Tx (the derivative) on the tan­gent spaces is continuous (on Tu XI). One could obtain an equivalent theory by requiring in place of the last property that / ( be a difFer-entiable map of MX I into X.

The fundamental problem of immersion theory is: given manifolds M and X, find the equivalence classes of immersions of M in X, equivalent under regular homotopy. This includes in particular the problem of whether M can be immersed in X at all. This general problem is in good shape. The complete answer has been given re­cently in terms of homotopy theory as we shall see.

The first theorem of this type was based on "general position" argu­ments and proved by Whitney [46] in 1936.

(3.1) THEOREM. Given manifolds Mk, X", any two immersions f, g: M—*X which are homotopic are regularly homotopic if n ^ 2 i + 2 . Ifn^2k there exists an immersion of M in X.

Recent proofs of the second part of this theorem can be found In [17] and [28].

The first statement of 3.1 is equally true with 2Jfe+2 replaced by 2k+ 1. Most of the theorems in this survey on the existence of immer­sions and imbeddings can be strengthened with an approximation property of some sort. Although these are Important, for simplicity we omit them.

The first immersion theorem for which arguments transcending general position are needed was the Whitney-Graustein theorem [45] (proved in a paper by Whitney who gives much credit to Graustein). For an immersion / : S1—£*, Sl, E* oriented, the induced map on the tangent vectors yields a map of Sl into S l; the degree (an integer) of this map times 2x is called by Whitney the rotation number.

(3.2) THEOREM (WHITNEY-GRAUSTEIN). TWO immersions of Sl in

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E1 are regularly homolopic if and only if they have the same rotation number. There exists an immersion of Sl in E* with prescribed rotation number of the form an integer limes 2T.

The next step in the theory of immersions was again taken by Whitney [48] in 1944 with the following theorem.

(3.3) THEOREM. Every k-manifold can be immersed in Eu~lfor Jk > 1.

The proof of this is quite difficult and involved a careful analysis of the critical points of a differentiate mapping of M* into -E**-1. In this dimension, these critical points are isolated in a suitable approxi­mation of a given map, but have to be removed to obtain an immer­sion. The difficulties in the study of singularities of differentiable maps have limited this method, although, very recently, Haefliger [4] has used effectively Whitney's ideas in the above proof, both in studying imbeddings and immersions. We shall say more about this later.

Some of the recent progress in imbeddings and immersions can be measured by Whitney's statement in the above paper. "It is a highly difficult problem to see if the imbedding and immersion theorems of the preceding paper and the present one can be improved upon." He goes on to ask if every open or orientable M* may be imbedded in E1

and immersed in E*. The complex projective plane cannot be im­mersed in E*, but every open M* can be imbedded in E7. See Hirsch [8; 9]. Whitney finally asks if every M* can be imbedded in £*. Hirsch has proved this is so if AP is orientable [lO].

The next progress in the subject of immersions occurred in papers [30; 31] and [32] of the author in 1957-1959. The first generalized the Whitney-Graustein theorem for circles immersed in the plane to circles immersed in an arbitrary manifold, and here methods were introduced which soon led to the solution of the general problem mentioned previously.

Consider " based" immersions of S> = 0 £6 = 2T in a manifold X, that is those which map 0 = 0 into a fixed point xe of X and the posi­tive unit tangent at 6 = 0 into a fixed tangent vector of X at x0. To each based immersion of S1 in X, the differential of the immersion associates an element of the fundamental group of Tx, the unit tan­gent bundle of X.

(3.4) THEOREM. The above is a 1-1 correspondence between (based) regular homotopy classes of based immersions of Sl in X and ri(Tx).

In the next paper [3l] corresponding theorems are proved with S l

replaced by S*. A noteworthy special case of the theorem proved there

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is that any two immersions of S1 in £ ' are regularly homotopic. It is a good mental exercise to check this for a reflection through a plane of S* in £ ' and the standard 5* in £' . This check has been carried through independently by A. Shapiro and N. H. Kuiper, unpub­lished.

In [32], the classification, under regular homotopy, of immersions of 5* in £" is given (any k, n). This can be stated as follows.

If/, g: 5*—»£" are based immersions (i.e., at a fixed point xt of Sk, f(xo)=g(xt) is prescribed and the derivatives o f /and g at x« are pre­scribed and equal) one can define an invariant il(J, £)€***( C»,*) where T*(K„,*) is the 4th homotopy group of the Stiefel manifold of jfe-frames in £*.

(3.5) THEOREM. Based immersions / , g: 5*—£" are (based) regu­larly homotopic if and only if il(f, g) = 0. Furthermore, given a based im­mersion f:Sk—♦£" and flo£x*(K«.*), there is a based immersion g: S*-»£" such that Q(f, g) = fl0.

The content of Theorem 3.2 is that the homotopy group xt(V..*) classifies immersions of 5* in £*. Information on the groups can be found in [25]. An application of this theorem is that immersions of 5* in Eu are classified by the integers if k is even, the correspondence given by the intersection number.

R. Thorn in [42 ] has given a rough exposition of the proof of the previous theorem, which contributes to the theory of conceptualizing part of the proof.

M. Hirsch in his thesis [8 ] , using the results of [32 ] , has generalized 3.5 to the case of immersions of an arbitrary manifold in an arbitrary manifold. If Mh and X* are manifolds Tu, Tx their tangent bundles, a monomorphism <f>: Ty—*Tx is a fiber preserving map which is a vec­tor space monomorphism on each fiber. For each immersion/: M—*X the derivative is a monomorphism <f>f: Ty—*Tx-

(3.6) THEOREM. If n>k, the mapf-*<t>/induces a 1-1 correspondence between regular homotopy classes of immersions of M in X and (mono­morphism) homotopy classes of monomorphisms of Tu into Tx.

In this theorem one can replace homotopy classes of monomor­phisms of TM into Tx by equivariant homotopy classes (equivariant with respect to the action of GL(k)) of the associated fc-frame bundles of Tu and Tx respectively. Still another interpretation is that the regular homotopy classes of immersions of M in X are in a 1-1 cor­respondence with homotopy classes of cross-sections of the bundle associated to the bundle of i-frames of M whose fiber is the bundle of

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fe-frames of X. Recently Hirsch (unpublished) has established the theorem for the case n = k provided M is not closed.

Theorem 3.6 includes as special cases all the previous theorems mentioned here on immersions and has the following consequences as well.

(3.7) THEOREM. If n>k and Mk is immersible in £ n + r with a normal r-field, then it is immersible in £ \ Conversely, if Mk is immersible in £" then (trivially) it is immersible in £"+ r with a normal r-field.

(3.8) THEOREM. If Mk is parallelizable (admits k independent con­tinuous tangent vector fields), it can be immersed in £*+l . Every closed Z-manifold can be immersed in E*\ every closed 5-manifold can be im­mersed in E*.

Theorem 3.6 is the fundamental theorem of immersion theory. I t reduces all questions pertaining to the existence or classification of immersions to a homotopy problem. The homotopy problem, though far from being solved, has been studied enough to yield much in­formation on immersions through Theorem 3.6 as can be seen for example in Theorem 3.8. Most further work on the existence and classification of immersions would thus seem to lie outside of differ­ential topology proper and in the corresponding homotopy problems.

We note that Haefliger [4] has very recently given another very different proof of Theorem 3.6 under the additional assumption n > 3 0 + l ) / 2 . See also [7].

We return now to discuss very briefly some of the methods used to proved the theorems of the previous section. The first step is to intro­duce function spaces of immersions. If Mk, X" are manifolds, let Im'(M, X) be the space of all immersions of M in X endowed with the C' topology, 1 ^ r ^ °°. This means roughly that two immersions are close if they are point wise close and their first r derivatives are close. Of course Im'(M, X) might be empty! A point in Im'(M, X) is an immersion of M in X and an arc in Iml(M, X) is a regular homo­topy, so the main problem amounts to finding the arc-components of Iml(M, X) or ir9(Imx(M, X)). One now generalizes the problem to finding not only 7Ct(Imx(M, X)), but all the homotopy groups of Imx(M, X). The homotopy groups of ImT(M, X) do not depend on r and we sometimes omit it. To find these homotopy groups one uses the exact homotopy sequence of a fiber space and one of the main problems becomes, to show certain maps are fiber maps.

The following in fact is perhaps the most difficult part of [32].

(3.9) THEOREM. Define a map r: Im*(Dk, E*)-*Iml(dDk, £") by

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restricting an immersion of Dk to the boundary. If n>k + l, IT has the covering homotopy property.

Actually, one uses an extension of this theorem to the case where boundary conditions involving first order derivatives are incorporated into the range space of it. Since it was first proved, Theorem 3.9 has been generalized and strengthened. The general version, due to Hirsch and Palais [ l l ] is as follows.

(3.10) THEOREM. Let V be a submanifold of a manifold M, X another manifold and it: Im(M, X)—>Im(V, X) defined by restriction. Then it is a fiber map in the sense of Hurewicz (and hence has the cover­ing homotopy property).

A version of 3.10 is also proved with the boundary conditions men­tioned above.

An idea not present in the author's original proof of 3.9, but intro­duced by Thorn in [42], was to prove theorems of type 3.9 and 3.10 by first explicitly proving the corresponding theorem for spaces of imbeddings, this theorem being much easier and quite useful itself. The final version of this intermediate result is due to Palais [26].

(3.11) THEOREMS. Let M be a compact manifold, V a submanifold, and X any manfold. Let £(Af, X), S(V, X) be the respective spaces of imbeddings with the C topology, l£rg«>, and x: Z(M, X)—>Z(V, X) defined by restriction. Then it is a locally trivial fiber map.

Using Theorems 39 , 3.10 and an induction basically derived from the fact that the dimension of the boundary of a manifold is one less than the manifold itself, one obtains weak homotopy equivalence theorems. The most general one is due to Hirsch and Palais [ l l ] . Given manifolds M, X let K(M, X) be the space of monomorphisms of TM into Tx with the compact open topology. Then as described in §2, there is a map

a: Im(M, X) — K(M, X).

(3.12) THEOREM. The map a induces an isomorphism on all the homotopy groups is a weak homotopy equivalence) if dim ^ > d i m M.

Theorem 3.12 applied to the zeroth homotopy groups or arc-com­ponents of Im(M, X) and K(M, X) yields Theorem 3.6. Theorem 3.12 was first proved for Im(S", E") in [32].

4. An imbedding (or differentiable imbedding) is an immersion which is also a homeomorphism onto its image. A regular (or differ­entiable) isotopy is a regular homotopy which at each stage is an im-

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bedding. The fundamental problem of imbedding theory is; given manifolds Mk, X", classify the imbeddings of M in X under equiva­lence by regular isotopy. This includes the problem: does there exist an imbedding of M in X? Our discussion of imbedding theory is limited to work on this problem. The difficulty of the general problem is indicated by the special case of imbeddings of 5' in £ ' . This prob­lem of classifying "classical" knots is far from being settled (and of course we omit any discussion of this special case although it could well be considered within the scope of differential topology).

Again the first theorems are due to Whitney in 1936 and are proved by general position arguments [46].

(4.1) THEOREM. A manifold Mk can always be imbedded in £ " + I . A ny two homotopic imbeddings of M in Xu+i are regularly isotopic.

One can replace Xlk+1 by X""1"1 here. See [17] or [28] for recent proofs of the first statement of 6.1. In 1944, Whitney proved the much harder theorem [47],

(4.2) THEOREM. Every k-manifold can be imbedded in Eu.

The methods used in this paper have been important in subsequent developments in imbedding theory. A. Shapiro, in fact, has consider­ably developed Whitney's ideas in the framework of obstruction theory. Only the first stage of Shapiro's work is in print [29]. Besides being mostly unpublished, the theory has the further disadvantage from our point of view that it is a theory of imbedding for complexes and does not directly apply to give imbeddings (differentiable) of manifolds. On the other hand, Shapiro's work has in part inspired the important theorems of Haefliger that we will come to shortly.

Wu Wen Tsun in a number of papers, see e.g. [49], has a theory of imbedding and isotopy of complexes which overlaps with Shapiro's work. Shapiro's (unpublished) theorems on the existence of imbed­dings of complexes in Euclidean space seem much stronger than those of Wu Wen Tsun. On the other hand, Shapiro works with spaces de­rived from the two-fold product of a space, while Wu studies stronger invariants derived from the £-fold products. Also Wu Wen Tsun not only considers existence of imbeddings but isotopy problems as well, including the following one for the differentiable case [49]. The proof is based on Whitney's paper [47].

(4.3) THEOREM. Any two imbeddings of a connected manifold Mk in £**+i are regularly isotopic.

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Haefliger [4] has taken a big step forward in the theory of imbed-dings with the following theorems, proved by strong extensions of the work of Whitney, Shapiro and Wu Wen Tsun. Haefliger's main theo­rem can be expressed as follows.

An imbedding /:.!/—»•£" induces a map <f>t: MXM—M—*SK~i

(MXM—M is the product with the diagonal deleted) by #/(x, y) = (f(x)-f(y))/\\f(y) /(*)|l- Then clearly <t>f is equivariant with re­spect to the involution on MXM—M which interchanges factors and the antipodal map of S" -1.

(4.4) THEOREM. / / n>3(k + l)/2, the map f-*t>, induces a 1-1 cor­respondence between regular isotopy classes of M in E* and equivariant homotopy classes of MXM— M into S"~l.

The equivariant homotopy classes are in a 1-1 correspondence with homotopy classes of cross-sections of the following bundle E. Let M* be the quotient space of MXM— M under the above involution. The two involutions described above define an action of the cyclic group of order two on (MXM— M)XS*~*. The orbit space of this action is our bundle E with base M* and fiber Sn~l.

Haefliger actually proves 4.4 with £" replaced by an arbitrary manifold X".

Another of Haefliger's theorems is the following.

(4.5) THEOREM. / / Mk and X" are manifolds which are respectively (r — 1)-connected, r-connected and n*i2k — r-f-1 then

(a) if 2r<n, any continuous map of M in X is homolopic to an imbedding;

(b) if 2r<n + l, two homolopic imbeddings of M in X are regularly isotopic. Thus if n>3(k + l)/2, any two imbeddings of Sk in E" are regularly isotopic.

Hirsch has proved some theorems on the existence of imbeddings of manifolds in Euclidean space. Perhaps the most interesting is the following [lO].

(4.6) THEOREM. Every on en table 3-manifold can be imbedded in £ l .

We do not discuss here, in general, the highly unstable problem of imbeddings of Af* in X" where n^k+2 except to mention that 2.7 is relevant to the differentiable Schonflies problem.

Since this section was first written Haefliger has obtained several further important results on imbeddings; see [S].

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29. A. Shapiro, Obstructions to the imbedding of a complex in a euclidean space. I. The first obstruction, Ann. of Math. (2) 66 (19S7), 256-269.

30. S. Smale, Regular curves on Riemannian manijolds. Trans. Amer. Math. Soc. 81 (1958), 492-512.

31. , A classification oj immersions oj the two-sphere, Trans. Amer. Math. Soc. 90 (1959), 281-290.

32. , The classification oj immersions oj spheres in euclidean spaces, Ann. of Math. (2) 69 (1959), 327-344.

33. , The generalised Poincart conjecture in higher dimensions, Bull. Amer. Math. Soc. 66 (1960), 373-375.

34. , The generalized Poincart conjecture in dimensions greater than four, Ann. of Math. (2) 74 (1961), 391-406.

35. , Differentiable and combinatorial structures on manijolds, Ann. of Math. (2) 74 (1961), 498-502.

36. , On the structure oj S-maniJolds, Ann. of Math. (2) 75 (1962), 38-46. 37. , On the structure oj manijolds, Amer. J. Math, (to appear). 38. , On gradient dynamical systems, Ann. of Math. (2) 74 (1961), 199 206. 39. J. Stalling*, Polyhedral homotopy-spheres, Bull. Amer. Math. Soc. 66 (1960),

485-488. 40. R. Thorn, Quelques preprints gtobales des varittts difftrentiables, Comment.

Math. Helv. 29 (1954), 17-85. 41. , Les singularitts des applications difftrentiables, Ann. Inst. Fourier

(Grenoble) 6 (1956), 43-87. 42. , La classification des immersions d'apris Smale, Seminaire Bourbaki,

December 1957, ParU. 43. C T. C Wall Classification oj (n-1)-connected In-manifcids, Ann. of Math.

(2) 75 (1962), 163-189. 44. A. H. Wallace, Modifications and abounding manijolds, Canad. J. Math. 12

(1960), 503-528. 45. H. Whitney, On regular closed curves in the plane, Compositio Math. 4 (1937),

276-284. 46. , Differentiable manifolds, Ann. of Math. (2) 37 (1936), 645-680. 47. , The self-intersections of a smooth n-manifold in In space, Ann. of Math.

(2) 45 (1949), 220-246. 48. , The singularities of a smooth n-manifold in (,2n — \)-space, Ann. of

Math. (2) 45 (1949), 247-293. 49. Wu Wen Tsun, On the isotopy of O manifolds of dimension n in euclidean

(2n+i)-space, Science Records (N. S.) 2 (1958), 271-275. 50. E. C. Zeeman, The generalited Poincart conjecture. Bull. Amer. Math. Soc. 67

(1961), 270.

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The Story of the Higher Dimensional Poincare Conjecture (What Actually Happened on the Beaches of Rio)*

Steve Smale

This paper is dedicated to the memory of Allen Shields.

Although these pages tell mainly a personal story, let us start with a description of the "n-dimensional Poin-car* Conjecture." It asserts:

A compact n-dimensional manifold M* that has the homotopy type of the n-dimensional sphere

S" = xi R»" | Ml = 1

is homeomorphic to 5". A "compact n-dimensional manifold" could be

taken as a dosed and bounded n-dimensional surface (differentiable and non-singular) in some Euclidean space.

The homotopy condition could be alternatively de­fined by saying there is a continuous map f:M" -»S" inducing an isomorphism on the homotopy groups; or that every continuous map £.S* -» M", fc < n (or just k s n/2) can be deformed to a point. One could equiva­lency demand that M" be simply connected and have the homology groups of S*.

Henri Poincare Btudied this problem in his pio­neering papers in topology. In [13], 1900, he an­nounced a proof of the general n-dimensional case. A counter-example to his method is exhibited in a subse­quent paper (14] 1904, where he limits himself this time to 3 dimensions. In this paper he states his famous problem, but not as a conjecture. The tradi­tional description of the problem as "Poincare^ s Con­jecture" is inaccurate in this respect.

Many other mathematicians after Poincare have

* This article is an expanded version of a talk given at the 1909 an­nual joint meeting of the American Mathematical Society, and the Mathematical Association of America.

claimed proofs of the 3-dimensional case. See, for ex­ample, [18] for a popular account of some of these at­tempts. On the other hand, there has been a solidly developing body of theorems and techniques of to­pology since Poincare.

In 1960 I showed that the assertion is true for all n > 4, and this is an account of that discovery. The story here is complemented by two articles |15] and [16], but the overlap is minimal.

I first heard of the Poincare conjecture in 1955 in Ann Arbor at the time I was writing a thesis on a problem of topology. Just a short time later, I felt that I had found a proof (3 dimensions). Mans Samelson was

4 4 THE MATHOUTICAL INTELUCfNCU VOL 12. NO. 2 O 1990 SpiUgn-Vnlai New Yott

Steve Snule ^ ^ r f ^ k

Steve Stttale is a member of the National Academy of Sciences (USA) and of the American Academy of Arts and

| Sciences, He has been awarded the Veblen Prize for Ge­ometry (American Mathematical Society, \%5)f the Fields Medal (International Mathematical Union, 1966), and the Chauvenet Prize (Mathematical Association of America,

Steve Smalc, a correspondent for the Mathematical In­telligencer, will celebrate his 60th birthday this year. The

; Mathematical Intelligencer wishes him a happy birthday.

in his office, and very excitedly I sketched my ideas to him. First triangulate the 3-manifold and remove one 3-dimensional simplex. It is sufficient to show the re­maining manifold is homeomorphic to a 3-simplex. Then remove one 3-simplex at a time. This process doesn't change the homeomorphism type and finally one is left with a single 3-simplex. (JED. Samelson didn't say much. After leaving his office, I realized that my "proof" hadn't used any hypothesis on the 3-manifold.

Less than 5 years later in Rio de Janeiro, I found a counterexample to the 3-dimensional Poincare Conjec­ture. The "proof" used an invariant of the Leningrad mathematician Rohlin, and I wrote out the mathe­matics in detail. It would complement nicely the proof I had just found that for dimensions larger than 4, the result was true. Luckily, in reviewing the counterex­ample, I noticed a fatal mistake.

Let us go back in time. I was bom into the "Golden Age of Topology." Today it is easy to forget how much topology in that era dominated the frontiers of mathematics. It has been said that half of all the Sloan Postdoctoral Fellowships awarded in those years went to topologists. Today it would be hard to conceive of such a lopsided distribution. Topology of that time, in fact, had revolutionizing effects in algebra (K-theory, algebraic geometry) and analysis (dynamical systems, the global study of partial differential equations).

In 1954 Thorn's cobordism paper was published. That theory was used by Hirzebruch to prove his "sig­nature theorem" (as part of his development of Rie-mann-Roch). In turn, already by 1956, Milnor used the signature theorem to prove the existence of exotic 7-dimensional spheres. I followed these results closely. Also as a student I learned from Raoul Bott about Serre's use of spectral sequences and Morse theory to find information on the homotopy groups of spheres. A little later, Bott himself was proving his periodicity theorems, also with the use of Morse Theory.

I received my doctorate in Ann Arbor in 1956 with Bott. My first encounter with the mathematical world at large occurred that summer with the Mexico City meeting in Algebraic Topology. I had never been to any conference before. And this conference was a his­toric event in mathematics by any standard. I have not seen that concentration of creative mathematics matched. My wife Qara and I took a bus from Ann Arbor to Mexico City, originally knowing only Bott and Samelson. Before leaving Mexico, I had met most of the stars of topology. There 1 also met two graduate students from the University of Chicago, Moe Hirsch and Elon Lima, who were to become part of our story.

That fall I took up my first regular position as an instructor in the college at the University of Chicago, primarily teaching set theory to humanities students. I had good relations with the mathematics department and went to the lectures of visiting professor Rene

Steve Smalt with hi* ton Nat, Chicago, 19H-

Thom (whom I had met in Mexico Gty) on transvers-ality theory. I also pursued work in topology showing that one could "turn a sphere inside out."

Chicago was a leading mathematics center at that time, before Weil, Chem, and a number of other im­portant mathematicians left. An important part of the environment was created by the younger mathemati­cians, especially Moe Hirsch, Elon Lima, Dick Lashof, Dick Palais, and Shlomo Steinberg. I was lucky to be there during that period.

A two-year National Science Foundation (NSF) post­doctoral fellowship enabled me to go to the Institute for Advanced Study in the fall of 1958. Topology was very active in Princeton then. I shared an office with Moe Hirsch and we attended Milnor's crowded lec­tures on characteristic classes and Borel's seminar on transformation groups. I frequently encountered Deane Montgomery, Marston Morse, and Hassler Whitney, played go (with handicap) with Ralph Fox and met his students Lee Neuwirth and John S tailings.

Moreover, in the summer of 1958, Lima introduced me to Maurido Peixoto, who sparked my interest in structural stability. That interest led to an invitation to spend the last six months of my NSF in Rio de Janeiro at I.M.P.A (Instituto de Matematica, Pura e Apli-cada).

Thus, at the beginning of January 1960, with Clara and our children, Laura and Nat, I arrived in Rio to meet our Brazilian friends. We arrived in Brazil just after a coup had been attempted by an Air Force col­onel. He fled the country to take refuge in Argentina, and we were able to rent his apartment! It was an 11-room luxurious place, and we also hired his two maids. The U.S. dollar went a long way in those days.

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Most of our immediate neighbors were in the U.S. or Brazilian military. We could sit in our highly ele­vated gardened patio and look across to the hill of the favela (Babylonia) where Black Orpheus was filmed. In the hot, humid evenings preceding camatxU, we would watch hundreds of the favela dwellers winding down their path to dance the samba in the streets. Some­times I would join their wild dancing, which paraded for many miles.

Just one block in the opposite direction from the hill lay the famous beaches of Copacabana (the Leme end). I would spend the mornings on that wide, beau­tiful, sandy beach sometimes swimming, or, de­pending on the height of the waves, body surfing. Also I took a pen and pad of paper and would work on mathematics.

Afternoons I would spend at l.M.P.A. discussing differential equations with Peixoto and topology with Lima. At that time l.M.P.A. was located in a small old building on a busy street. The next time I was to visit

Rio, l.M.P.A. was in a much bigger building in a much busier street. l.M.P.A. has now moved to an enor­mous modern palace surrounded by jungle in the suburbs of Rio.

Returning to the story, my mathematical attention was at first directed towards dynamical systems and I constructed the "horseshoe" [15]. As I continued working in gradient dynamical systems, I noticed how the dynamics led to a new way of decomposing a manifold simply into cells. The possibilities of using this decomposition to attack the i'oincaie conjecture soon developed, and before bng all my work focused on that problem.

With apparent success in dimensions greater than 4, I reviewed my proof carefully; then I went through the details with Lima. Gaining confidence, I wrote Hirsch in Princeton and sent off a research announcement to Sammy Eilenberg. The box titled "Dynamics and Manifold Decomposition" contains a mathematical de­scription of what was happening.

4 6 TH>MATHSMATiOU.INriLUGINCEKV01. 12.NO 2.IW0

Dynamics and Manifold Decomposition Consider a manifold M", with some Riemannian boundary of a cell could be in higher dimensional metric, and some function f:M" -» R. Construct the cells. dynamical system defined by the differential equa- Next, from this dynamical system ! constructed tion dx/di - - g r a d / o n M . what i called handiebodies.

If p is a non-degenerate critical point of/, then the set Wp) of all points tending to p under the dy­namics as t -* =° is an imbedded cell; the same ap- y N f \ plies to the set W[p) of points tending to p as ( -* / \ / /-\ \ - «. In the 2-dimensional picture, both W[p) and J /—\ \ J L—] 1 W(p) are 1-celis or arcs). t ^ / t ^ ^ ^ ^ ^

w,, , This is a diagram of a 1-handlebody, which con-h~ ^ L ^ V*/~^— -~ \ s ' s ' s °^ Sickened l-cel!s attached to a 3-disk. Then / \ . y \ one can add all the 2-handles at one time, etc., until

f*\ the manifold is given a filtrated structure by these J ' Y j handles.

•C"i) V" L ~"~~J ™ s S'v e s a starting point for their elimination, a v ^ ^ ^-— ( / J _^-1 pair at a time, using finally the homotopy and di-

""""—-—: -" mension hypotheses onAl. Eventually, one obtains the following description

Suppose now that / has only non-degenerate crit- o f o u r o r i g i n a l m a n i f o ] d i w h i c h w a s t 0 ^ p r o v e d . ical points. We may assume that W'(p) H W(f) for all critical points p,q of/; or, one could say that "the **** stable and unstable manifolds of grad / meet trans- /^~~\ - _ ^ versally." Under this hypothesis, the V>F(p) give a L-——\ f \ nice decomposition of M. The bou ndary of a cell is a r ^ i _ ^ T l t _ _ _ J union of lower dimensional cells. Ren£ Thorn had f-—;—-j \^_^y earlier considered such a decomposition, but ^<J^S without our transversality hypothesis so that the

235

I was already planning to leave Rio for three weeks , , „ _ , ..,„ _* . _ j • i r n_ » ,rvcn-n. I also recall Zeeman having written accurately of in Europe during June of that year, 1960. There was L. . . . " . . . , ' .. , r . .'.. . ' , .. .. these events and giving a similar picture. the famous Arbeitstasung, an annual mathematics . , . r, t .-_. i IAF,II J ,_ TY- i >_-«_. . t_ < , After my announcement appeared, Wallace event organized by Hirzebruch. This was to be fol- . L . -a 1 Q ,~ „ u _ " ( „ . AM*M* „r , . . . , , . -,.. . . . t u t wrote me (Sept. 29, 1960) asking me for details ot lowedby a topology conference m Z-Jnch to which I f ^ ^ m e w f t e r e6

h e w a 8 b l o c k e d i n had been invited. The two meetings provided a good ^ ^ ^ , s e n (

6h i m m y p r e p r i n t s o f , h e p r o o f

which he acknowledged in October (I still have his letters).

My best-known work was done on the i „&& a mistake in the first draft of my proof, beaches of Rio de Janeiro. which I easily repaired; but that doesn't affect these

issues. Certainly Stagings and Zeeman, and Wallace

opportunity to present my results. For a change of have done fine work on this subject. Yet I do wish pace and with his consent, I have put two recent that mathematicians were more aware of the facts letters of Stallings to Zeeman in a box later in this ar- that I have just described. I am sending copies of tide. These letters describe well the events that took this to Stallings, Zeeman and Wallace, and to a few place in Europe. While my memory in general is con- other mathematicians, sistent with Stallings', 1 don't believe Hirsch helped Best regards, me as Stallings conjectured. I do recall spending some Steve Smale relaxed days in St. Moritz with Moe Hirsch and Raoul Bott, after a more dramatic and traumatic week in Milnor wrote back (as mildly modified and expanded Bonn. by him):

Sometimes I have become upset at what I feel are inaccuracies in historical accounts of the discovery of February 27, 1988 the higher dimensional Poincare conjecture. For ex- I ample, Andy Gleason wrote in 1964 |1 ]: ". . It was a \^At Steve, great surprise, therefore, when Stallings in 1960 (4) proved that the generalized Poincare conjecture is true I am very sorry that my attempt at sketching the for dimensions 7 and up. His result was extended by history of the Poincare conjecture was inaccurate. Zeeman (10) shortly thereafter to cover dimensions 5 This was partly a result of not doing my homework and 6." (I wasn't mentioned in his article.) properly, but more a matter of expressing myself

Paul Halmos in his autobiography (18), page 398) v e r v badly. What I should have said of course is writes of my anger with him. I am sorry that I was t n a t the Stallings proof, completed by Zeeman for angry with Paul, and I wish I could be more relaxed dimensions 5 and 6, was logically independent of about this subject in general. My recent correspon- y o u r proof; but it is certainly true that yours came dence with Jack Milnor illustrates the Issue. fo$t Similarly, I should have said that Wallace's

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ proof, for dimensions strictly greater than five, defi-Dear lack nitely came later, and was not really different from

your proof. I am writing about your article on the work of M. „ i s w o r t h n o t i l ) g t h a , t h e S m a [ e ( o r W a [ l a c e )

Freedman in connection with his winning the Fields f( a t , h e C M , o f t h e s t r o hypothesis of dif-Medal (Proc. of the Int. Congress 1986). There you say f e f en t iability, gives a much stronger conclusion. in discussing the "*-d intension a 1 Poincare Hy- N a m e , y i „ s h o w s ( h a t ^ s m o Q t h „ . m a n i f o ! d c o n . P o t n e s i s tains a smoothly embedded standard (n - 1)-

The cases n 1,2 were known in the nineteenth cen- sphere which cuts it up into two smoothly em-tury, while the cases n * 5 were proved by Smale, and bedded standard closed n-balls. The Stallings-iafeePmnwM-y6lby S 'a m n g S " ^ ^""^ ^ ^ " ^ Zeeman proof starts with the weaker hypothesis

that M is a combinatorial manifold, and obtains the The word independently seems inconsistent with weaker conclusion that M with a single point re-the history of that discovery. moved is combinatorial^ homeomorphic to the

Stallings in his preprint "The topology of high-di- standard Euclidean space. mensional piecewise-linear manifolds" writes Sincerely,

When I heard that Smale had proved the Generalized J o h n Milnor Poincare Conjecture for manifolds of dimension 5 or more, I began to look for a proof of this fact myself. I wrote him, thanking him, saying that I appreciated

his letter.

TUB MATHEMATICAL SVULUCINCBX VOl 12. NO 2. MO 4 7

After the Zurich conference, 1 returned to join my family in Rio and shortly thereafter took up my posi­tion in Berkeley.

During the next year I wrote several papers ex­tending the results, culminating with a paper proving the "Ji-cobordism theorem" in June 1961. A mathemat­ical survey of all these matters, with references, is given in [17].

Because of Serge Lang and an irresistible offer from Columbia, we sold our house and left Berkeley in the

4 8 THE MATHEMATICAL tOTELDG£NCHl V<X U . W H M 1

summer of 1961. After three years of studying various aspects of global analysis, we returned from New York to Berkeley because of the prospect of better working conditions.

That was the fall of 1964, and the Free Speech Move­ment (FSM) caught my attention. After the big sit-in, I helped obtain the release from jail of mathematics graduate students David Frank and Mike Shub.

The Vietnam war was drastically escalated early in 1965. I felt that the U.S. heavy bombing of Vietnam

236

March 10, 1988

John Stallings 1 Dear Chris,

I got your letter to Minor, which must be about something he said or wrote about Smale, you, and hands too much on matters in the foundations ol me in the 1960 era. I think I got some comment PL topology that hadn't been properly proved; and from Smale about this too, but I cleaned up my of- this complaint obsessed me for the next six or seven fice between then and now, and I have no idea years. Of Smale's talk, I remember thinking that he what Smale said; and I don't know what Mil nor hadn't really considered the possibility that there said. And probably I should just go back to sleep. could be awful presentations of the trivial group; in

Your memory of this is much more specific than other words, he was just going to cancel the 1-mine. I remember spring of 1960. I remember the handles with some 2-handles without thinking hard Smale rumor, which I disbelieved until Papakyria- about it; I thought this was a fatal error in Smale's kopoulos visited; I think Papa said something like, method, and this made me secretly very gleeful. Eilenberg had looked over Smale's proof of the However, the next week, at Zurich, Smale had fixed high-dimensional PC and thought it was OK. I that up; in fact, if I had memorized Whitehead's vaguely remember your giving a talk; it was about a "Simplicial Spaces, Nuclei and m-Groups," 1 would paper by Penrose, Whitehead, and Zeeman; and it have seen how to do this myself. My impression, had some sort of engulfing thing in it; and it mainly for which I have no evidence at all, is that Moe strikes me that you or somebody said that it was the Hirsch, who is probably a much better scholar than only joint paper jHC Whitehead had been involved Smale (in other words, Smale is the Mad Genius in, in which his name didn't come last in alphabet- and Hirsch is the Hard Worker), told Smale how to ical order. 1 remember sitting up in my little office in fix it up. The point was that you could add trivial the olden Maths, Institute at Oxford and staring at pairs of 2- and 3-cells, which on the 2-skeleton is the blackboard, trying to think about how to cancel equivalent to adding trivial relators; this is the extra, handles; for some reason this was how I thought delicate type of "Tietze transformation" which you Smale would go about it, but I don't think anybody need in order to manipulate the 2-cells algebraically told me that, I couldn't figure out how to get rid of to get them to cancel the 1-cells. If the dimension is the pesky 1-handles, because I know some awful high enough, this manipulation can be done geo-presentations of the trivial group. Then it occurred metrically. I guess it is in the delicate part of this to me that I could, so to speak, push all the trouble geometry that the question of whether Smale did it out to infinity, using the engulfing stuff plus what I in dimension 5 or not comes up, think of as an idea due to Barry Mazur in the After that, 1 remember enjoying your hospitality, Morton Brown formulation. changing at Bletchley, and doing a lot of nice geo-

Then I remember being driven to Bonn by loan metrical mathematics during the summer of 1960. James; there was some other American along too, I do not really feel the reason why people are so maybe Bob Hunter or Dick Swan, Somewhere in interested in what exactly happened then. After all, the middle of Holland, loan got out of the car to it is not a big nationalistic thing like Newton versus piss at a farm; we Americans thought this was very Leibniz. We have all done more mathematics since British and were too scared to do that ourselves. In then; and what I am really interested in is what I am Bonn, both Smale and I talked. I remember that A. doing and learning about now. Smale is a colleague Borel complained in my talk that I was waving my who really means a lot to Berkeley and to me.

237

was indefensible and threatened world peace. My in­volvement in the protest increased, and included orga­nizing teach-ins and militant troop train demonstra­tions. I became cochairman with Jerry Rubin of the VDC, or Vietnam Day Committee. (Our headquarters near campus was later destroyed by a bomb.)

Already during the fall of 1965,1 was becoming dis­illusioned with the VDC and returned to proving theorems. The subsequent part of this story is de­scribed in [16]. In Moscow, August 1966, I criticized

the U.S. in Vietnam (and Russia as well) to return home under a storm of criticism. The University of California stopped payment of my NSF summer salary.

Dan Greenberg [2] gives a full account of what hap­pened next.

Upon being informed of the agitation surrounding him, and the withholding of his check, Saul* sent to Connkk an account of his summer researches—an account which

Contmutd on pagt 51

THE MATHEMATICAL WTEUKSKB! VOL II. NO 1,1*0 4 9

the background of these high-dimensional topology KristOPACr ZCCIIlSn results (hat I have been discussing. The methods of

JHC Wbitehead, MHA Newman, V Guggenheim, etc., were taught in Fox's classes. Milnor had had his construction of exotic structures on S7 and was giving classes on differential topology. So the time

There is a funny thing about time. Not only does was ripe for something to happen. existence bifurcate infinitely towards the future, but Most of the graduate students seemed to be also towards the past. In other words, if I manage trying to show off to each other about their vast to cross the street and not get flattened by a truck, I knowledge and intellectual sharpness; sheaves and think, well, in an alternate universe I was flattened spectral sequences were big topics of conversation. by the truck; I'm just not on that path towards the Then came Mazur"s sphere-embedding theorem. future, but another me was there. Now, 1 had the Suddenly (in winter or spring of 57-58), Barry opportunity to receive a visit a few months ago Mazur reappeared, after a long stretch of god-from a mathematician I had last seen in 1977, knows-what in Paris, with his theorem. It was a namely Fassi. I remember his appearance in 1977 very easy and immediately understandable proof of very dearly. Now he looks 10 or 11 years older, but something that had seemed so complex that no one other than that very similar to how he used to look; dared conjecture it. The result was completed and except that nowadays he is wearing a different nose! polished up by others later on, but the Genius was So 1 think there were several Passis ten years ago; Mazur's; this Genius consisted in the audacity and my existence passed by one of them then, but simplicity of Callow Youth, mathematical mode. somehow a slightly different one came out of a dif- In mathematics there are plenty of people who ferent past to visit Berkeley recently.—And so I are quick and quite enough people who have assim-take these reminiscences of what happened 28 ilated detailed volumes of knowledge. I consider years ago with a little sense of amusement, and I the Mazur-like phenomenon to be far more attrac-presume that we are all coming together out of tive and important. After someone like Mazur slightly different pasts. makes his incursion into new territory, the mathe­

matical masses follow, licking up drops of the Yours, leader's sweat, computing lists and tables and ob-John R. Stallings sanctions and sheaves and spectral sequences. This

gives employment to many. I did my thesis on Grushko's Theorem, on a level

March 14 1988 that seemed at the time mundane in my old age, I „ „, . have learned to appreciate this thesis much more).

But what I admired and envied, and who I hoped to On March 10, I sent you a description of my imitate—Barry Mazur.

memories of PL days in 1960. Now, instead of Here is the story I make up about Smale's work trying to be purely factual, I want to add some on the high-dimensional PC. Smale had already things about my impressions and my philosophy. I been working in high-dimensional differential to-plan to send a copy of this to Smale and to Milnor. pology; he was just a few steps behind Milnor in

I was a graduate student at Princeton, 1956-59; the incursion into this subject. Then the PC idea not as well-prepared as some of the other students, came to him "on the beaches of Rio." and with an idea that doing math was much more I imagine that if I had been on the beaches of Rio, important than learning it. I learned something of the idea wouldn't have come to me so easily, be-

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5 0 THB MATHEMATKAL INTOJJGENCDt VOl 11 NO. 1 I 9 »

23

cause, as I pointed out before, I perceived a big unknotting theorems and h- and s-cobordism problem in getting rid of the 1-handles. 1 imagine theorems; most of these results have both engulfing Smale did not know enough to be worried by this, versions and handle versions. And they have, of and so he plowed into this big result. I do think course, later on given birth to new generations of there was a little bug in his proof; but this was a mathematical theories. non-fatal bug. The 1-handle problem had been, in a Now, both my theorem and Smale's are valid somewhat different context, dealt with by JHC from dimension 5 on up. They are easier to prove in Whitehead 20 years earlier. I really think that dimensions 7 and higher. The intuition that they Smale's Big Theorem was a matter of putting to- work in dimension 5 was there from the start. It gether a new audacity in high dimensions with was mainly a question of formulating the right techniques from olden times due to Whitehead and lemmas and making plausible arguments. With ref-Whitney, and with his own particular develop- erence to my theorem, your idea of "piping away ments. Perhaps the finishing-up of this theorem, the highest-dimensional singularities" was a good and the pushing onward to computations and ob- formulation. structions and tables) were done by others. In par- (It occurs to me that I am saying something here ticular, I think that Milnor had a great deal to do that would warm the cockles of the hearts of Thur-with creating the fundamental techniques of differ- ston and Gromov. The crimes of Thurston and ential topology and expounding them clearly; and Gromov, which consist of asserting things that are that the biggest fruit of this was to make the proof true if interpreted correctly, without giving really of Smale's Theorem quite solid. good proofs, thus claiming for themselves whole

Then there was the engulfing-theory proof of the regions of mathematics and all the theorems high-dimensional PC. I'll call this "my theorem." In therein, depriving the hard workers of well-earned fact, my theorem and Smale's theorem, although credit—something like these crimes was indulged they have a similar sound, are logically distinct, in by both Smale and me in 1960, because we knew Smale's theorem was that if a differentiable mani- some of these theorems were true, and said so, at fold is a homotopy-sphere, then its underlying PL- least in private, without really knowing how to structure is that of a sphere. My theorem was that if prove them, or perhaps without wanting to indulge a PL-manifold is a homotopy-sphere, then its un- in the labor this would involve.) derlying topological structure is that of a sphere. As I see it, Smale did his work on his own

There were several things which paved the way mostly, with perhaps some very minor collabora­tor my theorem. I was familiar with Mazur's agru- tion with Hirsch. You and I did a considerable ment (I think I was the first person he convinced; amount of collaboration in the summer of 1960, and I had come up with a funny fact, when I was an both in polishing the arguments down to dimen-undergraduate, that a group in which you could sion 5 and in using similar arguments for various define an infinite product was the trivial group, a unknotting results. Eventually, we published our trivial fact). And 1 was familiar with PL topology own papers separately, although there were little from Fox's courses. The mental block against pieces of joint work here and there. If we were proving something about high-dimensional ho- really concerned about who thought of what first, motopy-spheres had been reduced by hearing we should have had a secretary taking down all our about the fact that Smale's alleged proof was being conversations. Neither of us was interested in pro-favorably received. And then, as you pointed out, 1 ducing joint papers. But that collaboration was had the Penrose-Whitehead-Zeeman trick at my stimulating and helped us produce enough good fingertips. theorems for both of us.

Once I had my theorem, it seems to me that its Finally, one last word: After working on this sub-proof was much simpler and more obvious than ject for a few years, 1 lost interest in it. There were Smale's proof of his theorem. Although you and 1 these big, easy theorems. They were done. Full and others wrote up a lot about the foundations of stop. If someone else wants to lecture about them, PL topology, the truth is that there is nothing deep they are welcome to do so. If someone wants to use and interesting in this (comparable, say, to Sard's these theorems to compute and classify, they have Theorem in differential topology); all the stuff about my minor blessing. But it all sounds tiring and general position is somehow not too impressive or boring to me. hard to believe, in spite of the fact that the details are painful. What was interesting about the books Cheers, on PL topology were the end results, such as the John R. Stallings

239

Continued from page 49 will probably be a classic document in the literature of science and government. He quickly established that he had satisfied the requirement of 2 months of research for 2 months of salary.

Connick was Vice-Chancellor of Academic Affairs at Berkeley. Greenberg's "classic document" surprised me, but my letter did contain my most famous quote. As recounted by Greenberg, 1 wrote to Connick:

However, during the remainder of this rime I was also doing mathematics, e.g., in campgrounds, hotel rooms, or on a steamship. On the S.S. France, for example, I dis­cussed problems with top mathematicians and worked on mathematics in the lounge of the boat. (My best-known work was done on the beaches of Rio dr Janeiro, 1960!)

1 would like to repeat that I resent your stopping of my NSF support money for superficial technicalities. The reason goes back to my being issued a subpoena by the House Un-American Activities Committee and the subse­quent congressional and newspaper attacks on me.

Before long, I did receive the NSF funds and things quieted down. However, within a year, a much bigger explosion took place when the NSF returned to me my new proposal amidst Congressional pressures.

The articles [3 -7 ] in Science by Dan Greenberg chronicle these events. See also [11J and [12].

Eventually as the situation was settling down once more, my statement about the beaches of Rio sur­faced. This time it was the science adviser to President Johnson, Donald Horrug [9], who wrote in Science:

This blithe spirit leads mathematicians to seriously pro­pose that the common man who pays the taxes ought to feel that mathematical creation should be supported with public funds on the beaches of Rio de Janeiro or in the Aegean Islands.

(I also had visited the Greek islands in August 1966 but, of course, not with NSF money.)

I was very happy at the response of the Council and President C. Morrey of the American Mathematical Society. Money's letter [10] in Science started:

The Council of the American Mathematical Society at its meeting on 28 August asked me to forward the following comments to Science:

Many mathematicians were dismayed and shocked by the excerpts of the speech by Donald Homig, the Presi­dential Science Adviser (19 July, p. 248). His . . . com­ments about mathematics and mathematicians are . . . uncalled for. Implicit in Horrug's remarks about vacations on the beaches of Rio or the Aegean Islands was a thinly veiled attack on Stephen Smale. The allegations against Smale were adequately disproved by Daniel S. Greenberg in his articles in Science on the Smale-NSF controversy.

At the same time a letter to the Notices wound up:

. . . the policy of creating a major scandal involving the implied application of political criteria in grant administra­tion, apparently for the purpose of placating and warding off the demagogic attack of a single Congressman, is not a policy that will cause anyone (and least of all Congress) to have any great respect for the principles and integrity of

those who adopt it. The conspicuous public silence of the whole class of Federal science administrators with regard to the future effects of current draft policy and the Vietnam war upon American science has been positively deafening. In this context, Homig's apparent attempt to turn the discussion of the current crisis in Federal support for basic science into a hunt for scapegoats in the form of "mathematicians on the beaches" would be ludicrous if it were not so destructive.

Hyman Bass E. R. Kolclun F. E. Browder S. Lang William Browder M. Loeve S. S. Chem R. S. Palais Robert A. Herrmann M. H. Protter I. N. Herstein G. Washniuer

It was especially gratifying to see such support from my friends.

Let me end, as I did in Phoenix: "Thanks very much for listening to my story."

References

10. Morrey, C , Letter to the editor, Scienie 162 (Nov. 1, 1968) 514-515.

11. , The case of Stephen Smale, Notices of the Amer­ican Math. Soc. 14 (Oct. 1967) 778-782

12. , The case of Stephen Smale: Conclusion, Notices of the American Math. Soc. 15 (J»n. 1968) 49-52 and 16 (Feb. 1968) 297 (by Serge Lang).

13. Poincare, H., Oemres, VI, Gauthier-Villars. Paris 1953, Deuxieme Complement a L' Analysis Situs, 338-370.

14. , Cinquieme Compliment a L'Analysis Situs, 435-498.

15. Smale, S., On How I Cot Started in Dynamical Systems, The Mathematics of Time. New York: Springer-Verlag (1980).

16. , On the Steps of Moscow University, Math. Intel­ligencer 6 no. 2 (1984) 21-27.

17. , A survey of some recent developments in differ­ential topology. Bull Amer. Math. Soc 69 (1963) 131-145.

18. Taubes, G., What happens when Hubris meets Nem­esis, Discover, July 1987.

Department of Mathematics University of California Berkeley, CA 94720 USA

THE MATHEMATtCAl. INmUCSNCHt VCH. 12. NO 2. 1M0 5 1

1. Gleason, A., Evolution of an active mathematical theory, Science 145 (July 31, 1964) 451-457.

2. Greenberg, Dan, The Smale case: NSF and Berkeley pass through a case of jitters, Science 154 (Oct. 7, 1966) 130-133.

3. , Smale and NSF: A new dispute erupts. Science 157 (Sept. 15, 1967) 1285.

4. , Handler statements on Smale rase. Science 157 (Sept. 22, 1967) 1411.

5. , The Smale case: Tracing the path that led to NSF's decision. Science 157 (Sept. 29, 1967) 1536-1539.

6. , Smale: NSF shifts position. Science 158 (Oct. 6, 1967)98.

7. , Smale: NSF's records do not support the charges, Science 158 (Nov. 3, 1967) 618-619.

8. Halmos, P., f Want to be a Mathematician, an Automatho-gravhy. New York: Springer-Verlag (1985).

9. Hornig, D., A point of view, Scirna 161 (Jury 19, 1968) 248.

14 Stephen Smale and the Economic Theory of General Equilibrium* GERARD DEBREU

During a 9-year time interval, Stephen Smale's dominant research interest was the economic theory of general equilibrium. His bibliography for the period 1973-1981 contains 15 articles that addressed problems at the core of that theory or, in the case of two of them, that were directly motivated by those problems. The 11 other papers authored or co-authored by Smale during the same time interval dealt with topics ranging from a model of two cells to complexity theory. One of them was devoted to the game theoretical problem of the prisoner's dilemma.

Nearly half of Smale's 15 articles on the theory of general equilibrium appeared in the Journal of Mathematical Economics, and one was included in the May 1976 issue of the American Economic Review, after having been presented by invitation at the 1975 annual meeting of the American Eco­nomic Association. The latter recognition by the economics profession came at the same time as his appointment in 1976 as Professor of Economics at the University of California at Berkeley, concurrent with his appointment as Professor of Mathematics. Both accolades by economists were later followed by his election as a Fellow of the Econometric Society in 1983.

By 1981 Smale had left a mark on the economic theory of general equilib­rium that will be the subject of this lecture. After that, he published one more article on economics [1987] returning to that theory, a survey valuable to the reader of Smale's work on economics for the broad views, and the insights into his research that it gives.

The four closely interrelated areas of the economic theory of general equi­librium to which Smale contributed will now be briefly described. A more detailed discussion of each area and of Smale's contributions will then follow.

* I thank all those who commented on an earlier version of this paper, or on its presentation on August 5, 1990, and I am especially grateful to Donald Brown, Graciela Chichilnisky, John Geanakoplos, Andreu Mas-Collel, Carl Simon, Stephen Smale, Hal Varian, and Karl Vind for their remarks.

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132 G. Debreu

0. General Equilibrium The economic theory of general equilibrium describes the observed state of an economy as the result of the interaction through markets of two types of agents, consumers and producers. Each one of those agents decides on the basis of prices what quantity of each one of the various commodities to produce, exchange, and consume. The large number of commodities, the equally large number of their prices, and the large number of interacting agents make the use of a mathematical model imperative for that description. In 1874-77, Leon Walras provided the first explanation of general equilib­rium in the framework of such a model. Walrasian theory has been developed by several generations of economists; its present formulation (as in the Arrow-Debreu model [1954]) makes it possible to convey some of its main ideas concisely to mathematicians.

The number of commodities is assumed to be a given integer t. Once a measurement unit has been chosen for each commodity, and a sign conven­tion has been made to distinguish inputs from outputs, a commodity-vector za in the commodity-space Rf describes the quantity of each one of the t commodities that agent a decides to consume or to produce.

The ith consumer (i = 1,.... m) is characterized by a consumption-set Xh a non-empty subset of R\ and a total preference preorder <, on Xt. Those concepts have the following economic interpretation.

For the ith consumer, inputs are positive, outputs are negative, and his consumption is denoted by x( e R'. The /ith coordinate of x, is the quantity of the /ith commodity that he consumes if x* > 0, or the negative of the quantity that he produces if x* < 0. The consumption-set X, is the set of possible consumption-vectors. As an example, an x, having, over the next month, a small food input and a large (in absolute value) labor output may be impossible.

Given two consumptions x and x' both in A",: if the ith consumer restricted to consume x or x' chooses x\ we write x ^ x' and read **x' is at least as desired by the ith consumer as x." The preference relation ;<, is assumed to be a total preorder (i.e., reflexive and transitive).

The j'th producer (j = l, . . . ,n) is characterized by a production-set YJt a nonempty subset of Rf. The economic interpretation is now the following.

For the ;'th producer inputs are negative, outputs are positive, and his production is denoted by y e R'. The /ith coordinate of yt is the quantity of the /ith commodity that he produces if y* > 0, or the negative of the quantity that he consumes if y* < 0. The production-set Y, is the set of possible production-vectors. As an example, an input-output vector y may be impos­sible given the technological knowledge available to the yth producer.

The economy S0 is then described by the m preordered consumption-sets X,, by the n production-sets Yj, and by the total endowment-vector e e 9t' characterizing the given quantity of each commodity available to the econ­omy as a whole. Thus, / 0 = (X„ ^ , ) , . , m,(Yj)j-i „.e).

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14. Stephen Smale and the Economic Theory of General Equilibrium 133

In this description of the economy, the ownership of its resources is not specified. Note also that the particular case of a pure exchange economy corresponds to the situation in which every Y degenerates to 0. The only possible production-vector then is the origin.

An attainable state s0 of the economy S0 is a list of the decisions made by the m + n agents, compatible with the endowment-vector e. In mathematical terms, an attainable state of <f0 is a list s0 = ((x,)i=1 „, (yj)j=i J of m consumptions x, and n productions yt satisfying the conditions

(1) for every i, x, e X(; (2) for every j , >>,• e Yy (3)Zr- iX| -Z"- iJ i = «-

1. A natural preorder on the set of attainable states was introduced by Pareto who defined s0 as superior (or indifferent) to s0 if, for every consumer, his consumption in s0 is preferred (or indifferent) to his consumption in s0. In symbols:

s0 3 s0 if for every i, xt <, xj,

Except in trivial cases, the Pareto preorder is not total, and attempting to compare two attainable states s0 and s0, one may find that one consumer prefers s0, whereas another prefers s'0. This gives special importance to the concept of a Pareto optimum defined as an attainable state of g0 that is maximal for the Pareto preorder. If s0 is a Pareto optimum, it is then impos­sible to find an attainable state s'0 that is superior to s0 according to the unanimity principle formalized by the Pareto criterion. Except in trivial cases again, there are infinitely many Pareto optima which cannot be compared to each other according to the Pareto preorder.

The characterization of Pareto optima by means of prices is one of the fundamental insights provided by welfare economics. Let p be a price-vector associating the price p* with one unit of the /ith commodity. Thus, the value of a vector z in the commodity space Rf relative to a vector p in the (dual) price space R/ is the bilinear form p-z = Y,i-\ P*2*- A consumption x, in X, is in equilibrium relative to p for the ith consumer if x< is best according to <, in the set z€ X,\p-i ^ p x , . It is then impossible for the ith consumer to find in his consumption-set X, a commodity-vector preferred to x, unless he spends more than he does on xt. Similarly, ys in Ys is in equilibrium relative to p for the ;'th producer if [z e Yj] implies [p • z ^ p • y\ It is then impossible for the yth producer to find in his production-set Yj a commodity-vector yielding a greater profit than y;- does. An attainable state s0 of <f0 is in equilib­rium relative to p if every agent of the economy f0 is in equilibrium relative top.

One of the main goals of welfare economics is to give conditions on / 0 ensuring that with every Pareto optimum s0 is associated a price-vector p relative to which s0 is in equilibrium.

2. If the resources of the economy are privately owned, its description also

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134 G. Debreu

requires a specification of the endowment e, in R' of the ith consumer, as well as of the fraction Oy of the profit of the jxh producer allocated to the ith consumer. Thus, £ j l i e> = «; f°r every pair (i,j). 0y ^ 0; and for every ;, Zr-ietf = i.

The mathematical description of the private ownership economy & now is

f = ((*i» ^.C|)i.i m>(Yj)j-l «.(0</)<-l J -J-l «

In this new context, an attainable state s of & is a list s = ((*,),=, m,(yj)j=.i «.P) of m consumptions x,, n productions yp and a price-vector p satisfying the conditions:

(i) for every i, xt e Xt; (ii) for every j , yt e Yy,

(iii) £ j l i *< ~ Z"-i yj = Z" i 4; (iv) p belongs to the price-space.

Such an attainable state is an equilibrium of S if (a) for every j , p -ys is the maximum profit relative to p in Y, and (b) for every i, the consumption x, is best according to <, in the budget set

i eX" , |p -z^p-e ,+ t 0uP-yj\-J-i )

The inequality expresses that the value of the ith consumer's consumption is at most equal to the sum of the value of his initial endowment and his shares of the profits of the n producers.

A fundamental question arises here; namely, under what conditions on the private ownership economy & can one assert that there is an equilibrium state?

3. Beyond existence theorems the question of uniqueness of general equi­librium presents itself for investigation. The requirement of global unique­ness, however, turns out to be too demanding, and, instead, conditions on £ that guarantee generic local uniqueness of general equilibrium are searched for.

4. Define now a state a of the private ownership economy as a list a = (x,),.j m,(yj)j-i n>P) of m consumptions x„ n productions yJr and a price-vector p satisfying the conditions

(i) for every i, x , e ^ ; (ii) for every ;, y e Yy, (iv) p belongs to the price-space.

This definition differs from the definition of an attainable state of £ given earlier in that condition (iii) ^JL, x, — YJ=I yj ~ Z"-i ei *s n ° longer imposed.

From the foundation for a theory of dynamic stability being laid by the study of local uniqueness of general equilibrium, one can begin to examine in the space of states of / processes that converge to the set of equilibria of S.

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14. Stephen Smale and the Economic Theory of General Equilibrium 135

Alternatively, processes of that type can be investigated in the search for efficient algorithms for the computation of equilibria of S.

1. The Characterization of Pareto Optima

If, with every Pareto optimum s0 of the economy &0, one can associate a price-vector p such that s0 is an equilibrium relative to p, then prices acquire a new significance as a means of achieving Pareto optimality. This insight due to Pareto [1909] was studied mathematically by him and by several of his successors at first in a differential calculus framework. This phase ended in the 1940s with solutions, notably by Lange [1942] and Allais [19431, that gave conditions confirming (and qualifying) Pareto's insight.

In the early 1950s, Arrow [1951] and Debreu [1951] pointed out that an alternative approach via convex analysis, resting on Minkowski's supporting plane theorem, provided a proof that was more rigorous, more general, and simpler than the calculus proofs offered up to that time. Convexity properties of preference relations and of production sets played an essential role in that approach. When Aumann [1964] introduced into economics the measure theoretical concept of a continuum of economic agents, those convexity properties became consequences of Lyapunov's theorem on the convexity and compactness of the range of a finite-dimensional atomless vector mea­sure (Vind [1964]) in sectors of the economy all of whose agents are small. The case of an industry with large producers having nonconvex production-sets was not covered however. In the case of nonconvexities, another diffi­culty arises as well with the loss of an elementary property of convex sets for which a local maximizer of a linear function is also a global maximizes

The move of economic theory away from the differential calculus approach in the 1950s and 1960s was reversed when the methods of global analysis became basic to the study of generic local uniqueness of general equilibrium that will be the subject of Section 3. It is in this new context that Smale reconsidered the question of characterization of Pareto optima in a differen­tial setup, in a series of seven articles ([1973], [1974b], [1974d], [1975a], [1975b], [1976b], [1981]). Smale's reconsideration differs from the earlier characterization of Pareto optima by differential methods by its rigor, its generality, its new results, and its powerful techniques. Two main themes run through those seven publications. One is an abstract theory of optimization for several functions, the other is the application to economics of that ab­stract theory. Some of the main features of both themes will now be outlined.

Smale consistently uses for each consumer, say the i'th, a utility function u,: Xt -»R representing his preference relation ^ on Xt in the sense that ix ^< *'] is equivalent to [u,(x) ^ u,(x')]. The preference relation is observ­able, and, therefore, a better primitive concept than one of its representations. But the function u, can be used instead of the underlying relation <, if the conditions of one of the available representation theorems prevail.

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136 G. Debreu

The abstract theory of optimization for several functions is studied by Smale in the following terms. W is an open subset of R" and t>, vm are C2

functions from W to R. A point x' in W is said to be (Pareto) superior to a point x in W if for every i, t>,(x') ^ u,(x) and, for some j , Vj(x') > Vj(x). A (Pareto) optimum is a point x of W such that there is no point of W superior to x. A strict optimum is a point x e W such that [t>,(x') ^ v,(x) for every i] implies [x' = x]. A general theorem of central importance is given by Smale ([1975b] as the end product of a development started in [1973] and pursued in subsequent papers [1974b], [1974d], [1975a], and by Wan [1975].

In its statement, Dvt(x) denotes the derivative of r, at x, a real-valued linear function on R", and D2vt(x), the second derivative of v, at x, a quadratic form on/?*.

If xeW is a local optimum, then there exist Xt,..., Xm, non-negative, not-all zero, such that £ , X,Dv,(x) = 0.

Moreover, if £,/l,D2u,(x) is negative definite on the space zef?"|V« = l,...,m,XiDvlx)z = 0, then x is a local strict optimum.

In the case of a pure exchange economy S0 with commodities and m consumers, for every i the consumption set Xt of the ith consumer is taken to be P, the interior of Ri the closed positive orthant of R'. The utility function of that consumer is a C2 function ut: P-* R with no critical point, satisfying the conditions (i) differentiable monotonicity, (ii) differentiable convexity, and (iii) a boundary condition, for the statement of which the following notation and terminology are required.

The set of points of P indifferent to x for the ith consumer (the indifference surface of x) is u^(c), where c = u,(x). We denote by g,(x) the oriented unit normal vector to that surface at x,

gradu,(x) g,[X) ||gradu((x)||-

(i) The differentiable monotonicity condition then requires that all the coordinates of ^(x) be strictly positive.

(ii) The differentiable convexity condition says that for every xe P, the second derivative D2u,x) is negative definite when restricted to the tangent plane to the indifference surface through x at x.

(iii) The boundary condition says that every indifference surface is closed in/l'.

For the pure exchange economy S0, the set W of attainable states is the set of m-lists x = (x^-.-.x^) such that for every i, x, e P, and the sum of the consumptions x, equals the endowment-vector e of the economy <?0:

W=\xepAyXi = t\.

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14. Stephen Smale and the Economic Theory of General Equilibrium 137

For every i, the utility function u( on P induces the function vt on W ac­cording to the equality

Vjx) = Uj(Xj) for every xeW.

The main result of Smale [1981, Section 4] is obtained in part as an appli­cation of his abstract theory of optimization for several functions. Under the assumptions just specified on the pure exchange economy S0:

The following three conditions on a point x of W are equivalent

(a) x is a local Pareto optimum, (b) x is a strict Pareto optimum, (c) 0j(X;) is independent of i.

Let 6 be the set of xeW satisfying one of these conditions. Then 6 is a C1

submanifold of W of dimension m — 1.

The common value p of the vectors gf(x() is the price-vector relative to which x is an equilibrium. Of the corollaries of this result given by Smale, one will be singled out as an incisive version of the two fundamental theorems of welfare economics.

The pair (x,p)e W x S'"1 of an attainable state x of S0 and of a unit price-vector p is called a Pareto equilibrium of S0 if x is an equilibrium relative to p. The set of those Pareto equilibria is denoted by A. Then (Smale [1976b] and [1981]):

The map from A to W defined by (x, p) i-» X is a C1 diffeomorphism of A onto 6.

The main result and its corollary that have just been quoted, as well as the general theorem on optimization for several functions from which they are derived, go from well-known (and easy to prove) first-order conditions for a Pareto optimum to new delicate second-order conditions, and to the original characterization of the set 6 of Pareto optima as a submanifold of W diffeo-morphic to the set A of Pareto equilibria. They are representative of the approach to Pareto optimality taken by Smale. But they account for only a small part of the many, sometimes complex, contributions that he made to that subject in the seven articles that he wrote in that area. For instance, the results mentioned earlier call for generalizations several of which were pro­vided by Smale. In particular, closed consumption-sets Xt rather than open sets of the form P = Int Ri are covered in his work. So are economies with production. So are nonconvexities for consumers' preferences and for pro­ducers' production-sets. Smale's writings on Pareto optimality amount to an extensive mathematical study of a topic of central importance in eco­nomic theory.

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138 G. Debrcu

2. The Existence of a General Equilibrium The theory of general equilibrium must provide conditions on the private ownership economy S ensuring that the equilibrium state on which it is centered exists.

The existence problem can be approached along the following line. First, exclude by economic considerations the case of a zero price-vector, and note that two positively collinear price-vectors p, and p2 are equivalent in the sense that all the economic agents make the same decisions on the basis of p, as they do on the basis of p2. Having chosen a norm in the price-space, one can then restrict the price-vector to be in the set T of vectors with norm 1. Consider a price-vector p in T and the reactions to p of the agents of S. The ;'th producer (j = 1,..., n) tries to maximize his profit relative to p by choos­ing an element yt of Yj. The ith consumer (i = l,...,m) tries to satisfy his preferences ;<< by choosing an element xt of X, constrained by the inequality p-X( ^ p e , + £"=1 Oijf-yj. The set of p for which these operations can be carried out for every agent of £ is a subset II of f\ Given p in II, denote by yi a production chosen by the jth producer, and by xt a consumption chosen by the ith consumer, the corresponding excess demand is the commodity-vector

m n m

I *< - I yj - I ei-1=1 j - i i - i

That vector need not be unique, however, and one is, therefore, led to associate with p in II, the set £(p) of all the excess demand vectors in the commodity space Rf to which p can give rise. The vector p* is an equilibrium price-vector if and only if 0 e C(p*). Thus, the problem of existence of a gen­eral equilibrium can be formulated in the following manner. A correspon­dence C associating with every element p of II a nonempty subset C(p) of R' is given. Under what conditions on £ can one assert that there is a p* in n such that 0 e C(p*)? Several ways of dealing with this existence question have been proposed. Notably, starting in the 1950s (Arrow and Debreu [1954], McKenzie [1954], Gale [1955], Nikaido [1956], Debreu [1956]), many exis­tence proofs have been based on fixed point arguments.

Smale ([1974a], [1976c], [1981], [1987]) takes a different approach and studies first the simple case where for each p in the set

, » p e K ' + £ p » = l , the excess demand vector is unique. Consequently, an excess demand func­tion z from A, to Rr is defined, and one seeks an element p* in Aj for which z(p*) = 0.

The function z is assumed to be of class C2 for the application of Sard's theorem that will be the basis of the argument.

The function z is also assumed to satisfy Walras' Law: for every p e At one has p-z(p) = 0. This law is directly implied by the insatiability of consumers.

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14. Stephen Smale and the Economic Theory of General Equilibrium 139

Finally, a boundary condition is imposed: If p* = 0, then z*(p) ^ 0. If the Aith commodity is free, then its excess demand is non-negative.

Smale states [1976c], [1981], [1987]:

/ / the C2 function z satisfies Walras" Law and the boundary condition, then there is an equilibrium price-vector.

Smale's proof has several important new features that are now sketched in an extremely brief summary of the complex arguments he develops in the first section of [1981], and especially as he proves Theorems 1.2 and 1.3.

Define A0 = q e H' l^J- i l" = 0, and let ?(p) = z(p) - Q>*(p)]p . The map <p is a C2 function from A, to A0, and z(p) = 0 is equivalent to <p(p) = 0. The last equality, therefore, expresses that p is an element of £ = <p'l(0), the set of equilibrium price-vectors that must be proved to be nonempty. Next, define $(p) = <p(p)/||<p(p)||, a function from A, \£ to S ' - 2 , the unit sphere in A0.

Assume that there is a point p0 in the boundary of A, such that Dz(p0) is nondegenerate (a weak assumption that can be dispensed with). Sard's theo­rem implies that there is a regular value y of $ in S'~2 near <P(p0) and such that 0~l(y) is nonempty. According to the inverse function theorem, the set y = <p~l(y) is a 1-dimensional C2 submanifold, i.e., a C2 curve in A,. The curve y is closed in A, \£ . A component of y leaves 5A, at a point near p0 . It cannot leave A,. Because it has no endpoint, it must tend to £. Therefore, £ is not empty.

Smale's proof considers the existence problem that he studies from the viewpoint of global analysis of which he uses several basic results. It is con­structive in the following sense. The curve y is a solution of his Global Newton differential equation

dp D<p(p)jt = -Acp(p),

that will be discussed in Section 4, where A is +1 or — 1 and sign(A) is determined by sign (determinant of i)cp(p)). A discrete algorithm approxi­mating the solution of that differential equation, therefore, will yield price-vectors approximately satisfying the equilibrium conditions.

Smale then extends the preceding existence theorem and weakens the dif­ferentiability assumption to a continuity assumption, approximating a given continuous function by a sequence of C°° functions. He also shows how correspondences rather than functions can be included in his study by means of another approximation process. Thus, existence theorems having the full generality of results previously obtained directly by fixed-point methods are derived from the global analysis foundation that he laid in a solution that emphasizes constructive aspects.

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140 G. Debreu

3. The Generic Local Uniqueness of General Equilibria The theory of general equilibrium would be entirely determinate if the equi­librium price-vector were unique. But the conditions on the private own­ership economy S guaranteeing that uniqueness are severely restrictive; and weakening the requirement of global uniqueness to that of local uniqueness is not by itself sufficient to overcome the severity of those restrictions. Under differentiability assumptions, however, the property of local uniqueness holds for all economies outside a negligible set. The conjecture of a result of that type (Debreu [1970]) was the occasion, in the summer of 1968, for my first meeting with Smale who introduced me to global analysis and who extended that result in several directions in five articles ([1974al, [1974c], [1976b], [1979], [1981]).

The theorems that Smale states in [1981, Section 5] are indicative of sev­eral of the genericity results obtained in various contexts over the past two decades. His proofs are centered on a study of the equilibrium manifold which appears with slightly different definitions in the preceding five papers. In [1981], he adopts the following definition in the pure exchange case that he considers. As in Section 1, there are commodities and m consumers in the economy. The ith consumer (/ = l,...,m) has a fixed consumption set P = Int/?+, and a fixed utility function u,: P -»R which is assumed to be of class C2, to have no critical point, and to satisfy the differentiable monotonicity condition (i), the differentiate convexity condition (ii), and the boundary condition (iii). The ith consumer also has a variable endowment-vector e, in P. Thus, the parameter defining the economy is the m-list e = (e , , . . . , em) in Pm, whereas an allocation of the total resources £ , e, to the con­sumers is an m-list x = (x,,...,xm) in Pm such that £ , x , = £ ,e , . The equi­librium manifold then is the set Z of pairs (e, x) satisfying the equilibrium conditions:

Z = Ut,x)ePm x P"|Vi,0i(x,) = pandp-x i = p - e ( ; yx , = £ e i

For every e e Pm, denote also by £(e) the set x e P"|(e, x) e £ of equilib­rium allocations associated with e. The main results of [1981, Section 5] state that

(1)1. is a C1 submanifold of Pm x Pm of dimension (m. (2) There is a closed subset F of P" of measure zero such that iftfF, then the

set £(e) is finite. Moreover, £(e) varies continuously with e in J""\F.

The proof of (2) can be briefly outlined. Let n:Pm x Pm -* Pm be the projec­tion defined by n(e, x) = e; let n0 be the restriction of n to X; and let C c I be the closed set of critical points of n0. The set F = n0(Q of critical values of n0 has measure zero by Sard's theorem; it is closed because n0 is a closed map. The remaining assertions in (2) follow from the inverse function theorem.

Thus, every regular economy e outside the negligible critical set F has a finite number of equilibria which vary continuously in a neighborhood of e.

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14. Stephen Smalc and the Economic Theory of General Equilibrium 141

The argument that establishes the result is focused on properties of the equi­librium manifold E, and of the projection n0 (as is the approach indepen­dently taken by Balasko [1975]).

Smale generalizes the previous genericity theorems to economies with a fixed total endowment, with variable utility functions, with genera] consump­tion sets, and with production. He dispenses with convexity and monotoni-city assumptions by introducing the concept of an extended price-equilibrium in which the attainable state of the economy is only required to satisfy the first-order conditions for an equilibrium. The five articles in which Smale addressed the question of generic local uniqueness of general equilibrium are given a remarkable unity of method by their global analysis approach.

4. The Global Newton Method

Two major problems in the theory of general equilibrium have led to the investigation of processes converging to the zeros of the excess demand function.

The first is the description of the dynamic behavior of a private ownership economy out of equilibrium. Assume that by reacting to a price-vector p in Si'1 = R n S'"\ the non-negative part of the unit sphere in the price space, the agents of that economy generate an excess demand function z: S+"1 -♦ R'. The point p* is an equilibrium price-vector if and only if z(p*) = 0, and a simple example of a dynamic process of economic disequilibrium has been formalized as the differential equation

Processes of this type were introduced by Samuelson [1941], and exten­sively studied by Arrow and Hurwicz [1958], and Arrow, Block, and Hurwicz [1959]. Examples of Scarf [1960], however, showed that (1) may not be globally stable. This assertion was confirmed by the results of Sonnenschein [1972], [1973], Mantel [1974], and Debreu [1974] who proved, with in­creasing generality, that the excess demand function z generated by a private ownership economy is arbitrary (except for continuity and Walras' Law).

The second problem is the development of efficient algorithms for the computation of general equilibria. Scarf, who played a leading role in that area, proposed solutions based on combinatorial methods [1973].

In an article that stands out by its originality and by its importance among the contributions he made to the theory of general equilibrium, Smale [1976c] provides, in an abstract setting, a process that can be applied in the solution of the first problem and that yields a differential analogue to the combinatorial algorithms of Scarf, and Eaves and Scarf [1976]. An anteced­ent for that analogue appeared in an article by Hirsch [1963]. Simultane­ously, Kellogg, Li, and Yorke [1977] gave an algorithm converging to the set

2S3

142 G. Debreu

of fixed points of a function on a closed disk and starting from almost any point in the boundary.

Smale considers a function / defined on a closed set M, where M is a nonempty, bounded, connected, open subset of R", whose boundary dM is a smooth submanifold; the function / takes its values in R"; it is of class C2. The goal of the analysis is to give a process starting from a point in the boundary of M and converging to the set £ of zeros of / , and thereby to obtain a constructive solution of the equation f(x) = 0.

The process proposed by Smale is a solution of the "Global Newton" differential equation

Df(x)jt = -Mx)f(x), (2)

where Df(x) is the derivative of/ at x, and A is a real-valued function defined on M whose sign is determined by the sign of the determinant of Df(x) in the following manner.

Assume that for every x in dM, one has Det Df(x) ^ 0, and that Dfx)~lf(x) is not tangent to dM. For every x in dM, choose now sign(A(x)) so that the vector -A(x) Dfx) '/(x) points into M. It is assumed that (a) for every x in dM, sign(>i(x)) = sign(Det Df(x)) or (b) for every x in dM, sign(A(x)) = -sign(DetZ>/(x)).

At any x in M for which Det Df(x) ^ 0, the value A(x) is chosen so that in case (a), sign(A(x)) = sign(Det Df(x)); in case (b), sign(A(x)) = -sign(Det£>/(x)).

Smale's main result asserts that in these conditions, there is a subset £ of measure zero in dM, depending only on / , such that starting from x0 in dM but not in S, there exists a unique C1 solution q>: [t0,ti)-*M of (2) with \\d<p/dt\\ = 1 and rt maximal. Moreover, this solution converges to the set E of zeros o f / a s t tends to t,.

If, in addition, Df(x) is nonsingular for every x in £ (a property that almost all C1 functions f:M-*R" have), then £ is finite, the exceptional set X is closed, and the solution <p converges to a single zero x* of/.

The first of the two questions mentioned at the beginning of this section, the description of the dynamic behavior of a private ownership economy out of equilibrium, was considered by Smale as "the fundamental problem of equilibrium theory" [1977]. In the process formalized by differential equa­tion (1), no trading takes place in the economy as it seeks an equilibrium. Three articles ([1976d], [1977], [1978]) of Smale studied an alternative pro­cess in which pure exchange occurs at every instant at the current disequili­brium prices. This is a "non-tatonnement" process of the type surveyed by Arrow and Hahn [1971, Ch. 13] and by Hahn [1982, Ch. 16]. But Smale's new approach focuses on cone fields of derivatives, rather than on fields of derivatives. Thus, for every feasible allocation x = (x,, . . . ,xm) of the fixed total resources e in the interior of Ri among the m-consumers of the econ­omy, a cone Cx (of vectors tangent at x to the set W of feasible allocations) is

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14. Stephen Smale and the Economic Theory of General Equilibrium 143

given. A curve £ from the time-interval [a, b) to the set W is an exchange curve if and only if for every t, d£/dt e C(0. In [1976d], giving a central role to the concept of a cone field of derivatives, Smale studied conditions under which an exchange curve converges to an equilibrium allocation.

5. Smale and the Theory of General Equilibrium In his recent article on global analysis in economic theory [1987], Smale lists several of his motivations for the systematic use of differential methods in the treatment of general equilibrium.

(1) The proofs of existence of equilibrium are simpler. Kakutani's fixed point theorem is not used, the main tool being the calculus of several variables.

(2) Comparative statics is integrated into the model in a natural way, the first deriva­tives playing a fundamental role.

(3) The calculus approach is closer to the older traditions of the subject. (4) Insofar as possible the proofs of equilibrium are constructive. These proofs may be

implemented by a speedy algorithm, which is Newton's method modified to give global convergence. On the other hand, the existence proofs are sufficiently power­ful to yield the generality of the Arrow-Debreu theory.

This short survey has outlined some of the principal contributions that Smale made in response to those motivations. Several of them have become important, sometimes essential, parts of the reformulation of the theory of general equilibrium by differential methods that took place over the past 20 years (extensive references for which are the books by Mas-Colell [1985] andbyBalasko[1988]).

But Smale's economics period has also influenced the orientation of his mathematical research during the past decade. At the end of that period, his attention was attracted by the problem of the average speed of the simplex method of linear programming, to the solution of which he contributed two major articles ([1983a], [1983b]). Earlier, his interest in the Global Newton method had led to his collaboration with Hirsch on algorithms for finding the zeros of a function from R* to R" [1979b]. Both lines of work later merged in the approach to the theory of computation that has become the main subject of his research.

Bibliography [1943] Allais, M., A la Recherche d'une Discipline Economique, Imprimerie

Nationale, Paris. [1951] Arrow, K.J., "An Extension of the Basic Theorems of Classical Wel­

fare Economics," Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, J. Neyman, ed., University of California Press, Berkeley, pp. 507-532.

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[1954] Arrow, K. J. and G. Debreu, "Existence of an Equilibrium for a Com­petitive Economy," Econometrica, 22,265-290.

[1958] Arrow, KJ. and L. Hurwkz, "On the Stability of the Competitive Equilibrium, I," Econometrica, 26,522-552.

[1959] Arrow, K.J., H.D. Block, and L. Hurwicz, "On the Stability of the Competitive Equilibrium, II," Econometrica, 27, 82-109.

[1971] Arrow, K.J. and F.H. Hahn, General Competitive Analysis, Holden-Day, San Francisco.

[ 1964] Aumann, R J., "Market with a Continuum of Traders," Econometrica, 32,39-50.

[1975a] Balasko, Y., "Some Results on Uniqueness and on Stability of Equi­librium in General Equilibrium Theory," Journal of Mathematical Economics, 2,95-118.

[1975b] Balasko, Y., "The Graph of the Walras Correspondence" Econo­metrica, 43,907-912.

[1988] Balasko, Y., Foundations of the Theory of General Equilibrium, Aca­demic Press, New York.

[1951] Debreu, G., "The Coefficient of Resource Utilization," Econometrica, 19,273-292.

[1956] Debreu, G., "Market Equilibrium," Proceedings of the National Acad­emy of Sciences, 42, 876-878.

[1970] Debreu, G, "Economies with a Finite Set of Equilibria," Econo­metrica, 38, 387-392.

[1974] Debreu, G., "Excess Demand Functions," Journal of Mathematical Economics, 1,15-21.

[1974] Dierker, E. Topological Methods in Walrasian Economics, Springer-Verlag. New York.

[1982] Dierker, E., "Regular Economies," Handbook of Mathematical Eco­nomics II, KJ. Arrow and M.D. Intriligator. eds., North-Holland, Amsterdam, pp. 795-830.

[1976] Eaves, B.C. and H. Scarf, "The Solution of Systems of Piecewise Lin­ear Equations," Mathematics of Operations Research, 1,1- 27.

[1955] Gale, D., "The Law of Supply and Demand," Mathematica Scandina-vica, 3,155-169.

[1982] Hahn, F.H., "Stability," Handbook of Mathematical Economics. II, K.J. Arrow and M.D. Intriligator, eds., North-Holland, Amsterdam, 745-793.

[1963] Hirsch, M., "A Proof of the Non-retractibility of a Cell onto its Boun­dary," Proceedings of the American Mathematical Society, 14, 364-365.

[1979] Hirsch, M., "On Algorithms for Solving f(x) = 0" (with S. Smale), Communications on Pure and Applied Mathematics, 32, 281 312.

[1977] Kellogg, R.B., T.Y. Li, and J. Yorke, "A Method of Continuation for Calculating a Brouwer Fixed Point," Fixed Points. Algorithms and Applications, S. Karamardian. ed., Academic Press, New York.

[1942] Lange, O., "The Foundations of Welfare Economics," Econometrica, 10,215-228.

[1974] Mantel, R., "On the Characterization of Aggregate Excess Demand," Journal of Economic Theory, 7, 348-353.

14. Stephen Smale and the Economic Theory of General Equilibrium 145

[ 1985] Mas-Colell, A., The Theory of General Economic Equilibrium. A Differ­ential Approach, Cambridge University Press, Cambridge.

[1954] McKenzie, L.W., "On Equilibrium in Graham's Model of World Trade and Other Competitive Systems," Econometrica, 22,147 161.

[1956] Nikaido, H., "On the Classical Multilateral Exchange Problem," Met-roeconomica, 8, 135-145.

[ 1909] Pareto, V„ Manuel d'Economie Politique, Giard, Paris. [1941] Samuelson, P.A., "The Stability of Eauilibrium: Comparative Statics

and Dynamics," Econometrica, 9,97-120. [ 1960] Scarf, H., "Some Examples of Global Instability of Competitive Equi­

librium," International Economic Review, I, 157-172 [1973] Scarf, H., (with the collaboration of T. HansenX The Computation of

Economic Equilibria, Yale University Press, New Haven, CT. [1973] Smale, S., "Global Analysis and Economics I, Pareto Optimum and a

Generalization of Morse Theory," Dynamical Systems, M.M. Peixoto, cd., Academic Press New York.

[1974a] Smale, S., "Global Analysis and Economics HA, Extension of a Theo­rem of Debreu," Journal of Mathematical Economics, 1,1-14.

[1974b] Smale, S., "Global Analysis and Economics III, Pareto Optima and Price Equilibria," Journal of Mathematical Economics, 1,107-117.

[1974c] Smale, S., "Global Analysis and Economics IV, Finiteness and Stabil­ity of Equilibria with General Consumption Sets and Production," Journal of Mathematical Economics, 1, 119-127.

[1974d] Smale, S., "Global Economics and Economics V, Pareto Theory with Constraints," Journal of Mathematical Economics, 1, 213-221.

[1975a] Smale, S., "Optimizing Several Functions," Manifolds Tokyo 1973, University of Tokyo Press, Tokyo, pp. 69-75.

[1975b] Smale, S., "Sufficient Conditions for an Optimum," Warwick Dynami­cal Systems 1974, Springer-Verlag, Berlin, pp. 287-292.

[1976a] Smale, S., "Dynamics in General Equilibrium Theory," American Eco­nomic Review, 66, 288-294.

[1976b] Smale, S., "Global Analysis and Economics VI, Geometric Analysis of Pareto Optima and Price Equilibria under Classical Hypotheses," Journal of Mathematical Economics, 3, 1-14.

[1976c] Smale, S., "A Convergent Process of Price Adjustment and Global Newton Methods," Journal of Mathematical Economics, 3,107-120.

[I976d] Smale, S„ "Exchange Processes with Price Adjustment," Journal of Mathematical Economics, 3,211-226.

[1977] Smale, S., "Some Dynamical Questions in Mathematical Economics," Colloques lnternationaux du Centre National de la Recherche Scienti-fique. No. 259: Systemes Dynamiques et Modiles Economiaues. Centre National dc la Recherche Scientifique, Paris.

[1978] Smale, S., "An Approach to the Analysis of Dynamic Processes in Economic Systems," Equilibrium and Disequilibrium in Economic The­ory, G. Schwodiauer, ed., D. Reidel Publishing Co., Boston, pp. 363-367.

[1979] Smale, S., "On Comparative Statics and Bifurcation in Economic Equilibrium Theory," Annals of the New York Academy of Sciences, 316, 545-548.

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[1980a] Smale, S., "The Prisoner's Dilemma and Dynamical Systems Asso­ciated to Non-cooperative Games," Econometrica, 48,1617 1634.

[1980b] Smale, S., The Mathematics of Time: Essays on Dynamical Systems, Economic Processes, and Related Topics, Springer-Verlag, Berlin.

[ 1981 ] Smale, S., "Global Analysis and Economics," Handbook of Mathemat­ical Economics I, KJ. Arrow and M.D. Intriligator, North-Holland, Amsterdam, pp. 331-370.

[1983a] Smale, S„ T h e Problem of the Average Speed of the Simplex Meth­od," Proceedings of the Xlth International Symposium on Mathemati­cal Programming, A. Bachem, M. Grotschel, and B. Korte, eds., Springer-Verlag, New York, pp. 530-539.

[1983b] Smale, S., "On the Average Number of Steps of the Simplex Method of Linear Programming," Mathematical Programming, 27, 241-262.

[1987] Smale, S., "Global Analysis in Economic Theory," The New Palgrave, A Dictionary of Economics, John Eatwell, Murray Milgate, and Peter Newman, eds., Macmillan, New York, Vol. 2, pp. 532-534.

[1972] Sonnenschein, H., "Market Excess Demand Functions," Econometrica, 40,549 563.

[1973] Sonnenschein, H., "Do Walras' Identity and Continuity Characterize the Class of Community Excess Demand Functions?," Journal of Eco­nomic Theory, 6, 345-354.

[1964] Vind, K., "Edgeworth Allocations in an Exchange Economy with Many Traders," International Economic Review, 5, 165-177.

[1874-77] Walras, L., Elements d'Economie Politique Pure, L. Corbaz and Com­pany, Lausanne.

[1975] Wan, H.Y., "On Local Pareto Optima," Journal of Mathematical Eco­nomics, 2, 35-42.

Reprinted from: DYNAMICAL SYSTEMS

© i m ACADEMIC MESS. INC., NEW Y O K AND IONDOM

Global Analysis and Economics I: Pareto Optimum and a Generalization of Morse Theory*

STEVE SMALE Department of Mathematics University of California at Berkeley Berkeley, California

One has considered for centuries the problem of maximizing a function via differential calculus. Morse theory could be regarded as a globalization of this problem. Relatively recently, economists have considered in a spe­cial case the problem of "optimizing" several functions at once, obtaining in this way what is called the Pareto optimum. Our goal here is to place this problem in the setting of global analysis, or several differentiable real functions on a manifold. We extend the notion of Pareto optimum to a larger set which we call the critical Pareto set 0. This set 0 is the analogue and generalization of the set of critical points of a single differentiable function, while the old Pareto optimum is the analogue and generalization of a maximum of a single function. This expansion of the economists' setting allows for the systematic introduction of calculus and global differentiable methods to the process of optimizing several functions on a manifold. For example, we obtain a natural notion of dynamics in this setting to generalize that of a gradient flow. Our approach contrasts with the more usual equilibrium or static approach mathematical economists take toward the study of a pure exchange economy. This development proceeds very naturally; arbitrary choices seem unnecessary.

531

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532 STEVE SMALE

Precisely the problem we consider is the following: One is given real differentiate functions ut: W -* R defined on a manifold W, say for i = 1, . . . , m. What is the nature of curves <p: R -* W with the derivative (dldt)(Ui o (p)(t) positive for all i, t ? For what x e W does there exist such a <p with <p(0) = x ? The critical Pareto set 0 is defined as the set of * e W, for which there is no such <p. The main problem is the study of 0. Another way of looking at this is how and when can one gradually improve the values of several functions simultaneously? One could consider this subject as part of game theory.

These questions lead to attractive mathematical problems, but especially one obtains a new way of studying utility, Pareto sets in economics, where traditional assumptions of convexity and monotonicity need not play such a key role. Also, I believe that the questions of optimizing several functions at once transcend economics; in other social problems, optimization of several functions rather than one permits one to go beyond a one-dimensional point of view. The question is one of many values in partial conflict versus maximization of a single value.

Our intention is to follow this article by one on price systems in eco­nomics treated from a related point of view.

I would like to warn the reader of the slightly tentative nature of some of the later parts of this paper. For example, while Section 5 seems simple and straightforward enough, the axioms of the theory of stratified sets for these cases haven not been carefully verified. Also "Theorem" 2 of Section 6 will need hard work before it can be regarded as a solid theorem (if true).

Let me end this introduction by thanking Gerard Debreu for getting me into this subject and for many helpful discussions.

1

Here we give a review of the notion of pure exchange economy and the classical Pareto optimum. The mathematician reader can skip this sec­tion if he is not interested in economics or concrete applications and the econometrist reader will know these things. For more details, one can see Debreu [3].

Commodity space will be an open set in cartesian space R1. There will be / different commodities in this pure exchange economy, each measured in quantity by a real number (fixing a unit of measurement) which can be considered a coordinate on R1. We are concerned with only positive

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GLOBAL ANALYSIS AND ECONOMICS I 533

amounts of each commodity and thus define commodity space as the positive orthant P of Rl. So P — x e Rl\ x has each coordinate positive. A point of P represents a bundle of commodities, which might be pos­sessed, for example, by a certain consumer in the economy.

It is assumed that there are a finite number of consumers, say m, with the possessions of the ith consumer denoted by X; e P. An unrestricted state of the economy is a point * = ( * , , . . . , xm) in the cartesian product P1" (a manifold of dimension ml). We suppose, however, that the total resources in this economic model are fixed, say a point to e P. Thus the attainable states form a subset W of P™ defined by W = x e Pm\Y.x\ = to. W is an open subset of an afrtne subspace with compact closure in (R')m. W is the basic state space we consider in this paper.

Each consumer is supposed to have his preferences represented by a function u : P -* R, his utility function which we suppose as differentiable as necessary. Thus consumer i prefers * / to xt if and only if u^Xi) > «j(Xj)- Consumer i is indifferent to commodity bundles in the same level surface of Wj. Hence the u?(e) are called indifference surfaces. In fact, it is these surfaces that are given primarily in economics, and it is these surfaces on which our analysis ultimately rests. However the u, are convenient for purposes of communication.

Let 7ti: Pm -* P be the projection n^x) = xt; then we have induced functions on W, still denoted by uit defined as the composition

Inclusion nt u< W /»» — P — R.

One considers exchanges in W which will increase the utility of each consumer or increase each u on W. A state x e W is called Pareto optimal if it has the property that there is no x' e W with Uj(x') > "i(x). all 1, and Uj(x') > ttj(x), some j . The idea is that if x e W is not Pareto, then it is not economically stable; there will be some trade which will take place and tend to make it Pareto.

In the succeeding sections, we consider this situation more generally; we study a manifold W with m real-valued smooth functions «, , . . . , um

defined on W and look at this situation from an (extended) Pareto point of view.

We end this section with a proposition communicated to me by Truman Bewley.

PROPOSITION Using the notation of this section, let utility functions uit i = 1, . . . , m, be defined and continuous on P and suppose:

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534 STEVE SMALE

(a) "convexity": uj"1!^ °°) *s strictly convex for each i, c; (b) "monotonicity": Define for x', x e P, x' > x if x' — x e P and similarly for y'.yeK*. Then x' > x implies «(*') > u(x), where « = ( « i " » ) •

Under these conditions the set of Pareto optima is homeomorphic to a closed (m — l)-simplex.

Furthermore, condition (a) (using either P or P) is sufficient to insure that the critical Pareto set 0 (of Section 2) coincides with the set of (clas­sical) Pareto optima.

2

Here we consider W a smooth (C°°) manifold with smooth functions UJ: W -* R, i = 1, . . . , m, where m < dim W. A prime example is the manifold W of attainable states of a pure exchange economy of the previous section where the u, um are utility functions of m consum­ers. The main goal of this section is to introduce a differentiable extension and generalization of the Pareto optimum which we call the critical Pareto set, 0.

Let u: W-»• P™ be defined by u = (ut, . . . , um) and Pos c Rm be the set of (y,, . . . , ym) 6 /?"» such that >\ > 0, each i.

Now let H(x) = DuW-^Pos), where Du(x)': TXW) — #» is the de­rivative of u at x, considered as a linear map from the tangent space of W at x to Rm. Thus i/(x) is an open cone in TXW). Then the critical Pareto set 6 is defined by 6 = x e W\ H(x) is empty. Clearly 0 is a closed subset of W.

We have the following alternative descriptions for H(x). Let

Hi(x)=veTx(W)\Dui(x)(v)>0.

Then i/(x) = n< /AC*) and, of course, also

tf(*) = i> 6 TXW) | Z)u(*) («) e Pos.

Thus one sees that q>: R —- W has increasing (infinitesimally) utility for each i if and only if q>'(t) e H(q>(t)), all r. We say that <p is admissible in this case. It is clear that an admissible curve does not admit any type of recurrence; e.g., if <p: [a, b] -* W is admissible then <p(a) ^ <p(b). An admissible curve can be thought of as a sequence of small trades in the example of Section 1.

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GLOBAL ANALYSIS AND ECONOMICS I 535

Thus x e 8 is the condition that there is no curve through x increasing infinitesimally all the Wj's. If m = 1, then 8 is precisely the set of critical points of the function u. The field of cones x -* H(x) on W is continuous in the following sense.

PROPOSITION (trivial) Let X be a continuous vector field on W with X(x0) e H(x0), some x0 e W. Then X(x) e H(x) for all x in some neigh­borhood of x0.

Note that if x is Pareto in the classical sense then x e 8. So we have not lost anything. But our critical Pareto set 8 is bigger in general than the old; we consider now certain natural subsets of 6 which are significant economically.

Suppose q>: [a, b) -* W is an admissible path and X\vs\t^,y(t) - x. Then we say that <p ends at x. In this case we also say that <p starts at <p(a) = w. If w £ 8, let 8(tv) be the subset of 8 of x for which there exist admissible <p starting at w and ending at x. Thus 8xv) is the set of Pareto critical points accessible from the initial state w by a sequence of (in­finitesimally) small trades.

Next we can define naturally in our context the notion of stability for x e 6. Say x e 6 is stable or x e 8S if given a neighborhood U(x) of x in 6, there exists a neighborhood V(x) in W such that if q> is any admissible path in W starting in V(x) and ends in 0, then <p ends in U(x). Clearly Bs will be the most significant part of 0 from the economic point of view. Note that 6S is an open subset of 6. Note also that for m = 1, just as 6 is the critical point set, 6S is the set of local maximums of «, at least in the nondegenerate case.

Let 6s(io) = 6S r\ 6(to). We will later show that frequently one can assert that 03(v>) 0 .

It seems to be the case that if the conditions of the finale proposition of Section 1 are satisfied, then 8 = 6S.

3

The goal of this section is to develop the idea of a "Hessian" in our context. Using this wc are able to obtain a criterion for a point x e W to be a stable Pareto point. For the case of one function u: W —* R this amounts to saying that x 6 W is a local maximum, and stably so, if the first derivative Du(x) is zero and the second derivative is negative definite.

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536 STEVE SMALE

The context we are in now is that of a smooth map u: W -* Rm from a manifold to euclidean space. It is convenient to make an assumption, the rank assumption, as follows: DEFINITION Say that * e 6 c W satisfies the rank assumption if rank Du(x) > m — 1 and that u satisfies the rank assumption if x does for all xeB.

Some remarks are in order. First it is clear that if x e 0, then rank Du(x) < m — 1. Thus one could write equality in the rank assumption instead of inequality.

We make a little excursion into the way we use the expressions "almost all" and "generic property." One can put a natural O-topology on the space of maps u: W -* R1*, 1 < r < oo, which makes this space a Baire space [1]. A Baire space has the property that a countable intersection of open dense sets is dense and is called a Baire set. "Almost all" refers to being an element of some Baire set. A generic property for u is one that is true for all elements of some Baire set.

For almost all u in this Baire sense, almost all points of 0 will satisfy the rank assumption. However, one cannot say that it is a generic property for u to satisfy the rank assumption.

On the other hand if (dim W + 4)/2 > m, then almost all u will satisfy the rank assumption. The above facts follow from Whitney or Thorn (cf. Calabi [2] or Levine [4]). In the case of a pure exchange econ­omy, this dimension condition will always be met (at least if there is more than one commodity) as a simple counting argument shows. Thus the rank assumption docs not seem too serious a restriction.

We now define the Hessian of u: W -* Rm at x e 6. Suppose x e 0 c W satisfies the rank assumption. Then Du(x): Tx( W) -> /?" has rank m — 1. The second derivative can easily be shown to define an invariantly (in­dependent of chart) defined symmetric bilinear form on Ker Du(x) <z TX(W) with values in the one-dimensional vector space Rmjlm Du(x). This form denote by Hx. The one-dimensional vector space Rmjlm Du(x) has a canonically defined positive ray (or orthant, etc.) from the fact that image Du(x) does not intersect Pos c: Rm. This last is key for our whole development and allows us to define negative definite, index, nullity, etc., for x. These ideas do not extend to the theory of singularities of maps because in this general case there is no natural definition of a positive part of Rm/Im Du(x).

We can now state our theorem. Suppose here that * g dO, that is, Im Du(x) n Pos = 0 .

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GLOBAL ANALYSIS AND ECONOMICS I 537

THEOREM Suppose u: W -* R* is C", r large enough and * is in the critical Pareto set 9. Suppose also that x satisfies the rank condition and the generalized Hessian Hx is negative definite. Then x is in the stable Parcto set, i.e., x e Bs.

For example, the proof could go via the normal forms in Levine [5]. In fact no doubt a direct simple argument using Taylor's theorem would work with r = 2.

Note from similar considerations that if x is as in the theorem and if an admissible curve q>: [0, 1) -» W has x in the closure of its image then actually <p(t) —* x as t -» 1.

Note finally that if x e 6 satisfies the rank condition, we have defined for x, the index, nullity, nondegeneracy, as the index, nullity, nondegen-eracy, respectively, of the bilinear form Hx.

4

We devote this section to four examples:

EXAMPLE 1 Let W be E*, the euclidean plane with norm || || given by an inner product. Suppose m — 2, with ut: E* -* R given by

«i(*)= - I I * II*. «•(*)= - I l * - * b l l ' , where x0 e E* is some fixed point in E1, x0 ^ 0. The 0, x0 are "satiation points" for numbers 1 and 2, respectively, i.e., points of maximum happiness. Then one can check through the definitions to see that 0 — 6S

= the closed segment between 0 and x„. Any admissible path which is complete in the obvious sense will end at 6. This example is illustrated in Figure 1, where we also mark off 6(a), and show a typical admissible path.

• <W)

Figure 1

365

538 STEVE SMALB

(o,o) «;

Figure 2

EXAMPLE 2. Edgeworth box This is a standard example from eco­nomics and fits into the case of Section 1, where we suppose there are two commodities, two consumers.

Suppose the total resources are denoted by to e P, with to = (to', to"), to' the amount of the first commodity, to" of the second. Then a state x e P* is of form * = (*,, * , ) , *, e P, xt e P, and we can write *4

= (*/, xi'), where * / is the amount of the first commodity owned by the first consumer, etc. Then W is the set of (*,', Xi, xt', xj') which satisfy 0 < * / , 0 < xi', and * / + xt' = to', x[' + *i' = to". Thus we could describe Walso with coordinates *,', *i', with 0 < *,' < to', 0 < xi' < to" as in Figure 2, where the level curves of ux, « , , respectively, are drawn where we suppose the standard convexity conditions of traditional eco­nomics are met. Here 6 — 6a = the arc of common tangents of these level curves. In this case 6 is called the Edgeworth contract curve. The situation is locally as in Example 1, away from the end points.

EXAMPLE 3 Here we relax the convexity assumption on at in the previ­ous example. Suppose in fact thai u, is as in Example 2 and uL has the

Fifnra 3

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GLOBAL ANALYSIS AND ECONOMICS I 539

Figure 4

qualitative features in Figure 3. Then arguing via common tangent one constructs 6, 6S as in Figure 4 (0 the circle and segment without any arrows).

Some admissible curves are drawn with arrows. These indicate how the circle part of 6 gets divided into a stable and unstable part. For some w e W, 0(w) may have 2 components. Also part of ds may not be classical Pareto optimal, and one sees how global problems enter into trading, versus sequences of small trades. Some acquaintance with Whitney [8] is helpful in understanding this example.

EXAMPLE 4 Here W = S2, the unit sphere in Ra, « , , ut are two coor­dinate axes so that u: W -* R* is projection into the appropriate coor­dinate plane n. Then 6S is the intersection of the first quadrant of n with S* and the only other part of 0 is the intersection 0' of the third quadrant of n with 5* (see Figure 5). Admissible curves will tend to go from 0' to 6S. This suggests the possibility of a Morse theory for this problem; we pursue this later. Note that a neighborhood of ds is the same as Example 1.

Figure 5

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540 STEVE SMALE

5

The goal of this section is to give some picture of the structure of 6, 0S, and then to construct a generalization of a gradient vector field.

For this first part it is useful to assume that u: W -* K* satisfies the jet transversality condition of Thom (cf. Levine [4]). This is a generic property and means that the derivatives of u are transversal to the mani­folds of singular jets. If in addition, the rank assumption is also true for u, as we shall assume, then the set of points x of W where Du(x) is not surjective, forms a submanifold Sx of W of dimension m — 1. Further­more, this Si contains a sequence of submanifolds Sf, q = 1, 2, 3, . . . , m, with dimension St

9 — m — q. See Levine [5] for details of this case of the theory of singularities of maps. Since 6 c S , , these results have a strong bearing on the nature of 0.

We need a further generic property on u as follows:

DEFINITION Say that u: W -* # " satisfies transversality condition Ax if it satisfies jet transversality and if the restriction w/5,: 5, -* /P" has its first derivative transversal to all the coordinate subspaces of R".

The point of transversality condition Ax is that it ensures reasonable behavior as Im Du(x) passes into Pos c R" or gives us a reasonable structure to dd. In fact the following proposition would seem to be valid.

PROPOSITION Suppose u: W -* /P" satisfies the rank assumption, jet transversality, and transversality condition Ax. Then (if not empty!) the critical Pareto set 6 is an (w — 1 )-dimensional manifold with corners in the sense of Cerf (thesis) or stratified set in the sense of Thom [7]. Thus 0 has the structure of an (m — 1 )-dimensional manifold 6t together with its boundary dd = 8 — 0t. The boundary is a union of submanifolds of dimension <m 1.

Compare this with the examples of Section 4. The reader will be able to construct more interesting examples with m = 3.

To obtain similar information for 0S, we introduce a generalization of transversality condition Alt another generic property:

DEFINITION Say that u: W -* /P" satisfies transversality condition A if it satisfies jet transversality and if the restriction ujS^: <S,» —- IP" has its first derivative transversal to all the coordinate subspaces of R", for q = 1, 2, . . . . m.

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GLOBAL ANALYSIS AND ECONOMICS I 541

Actually for our immediate purpose q = 1 and 2 would be good enough. Apparently the following proposition is valid.

PROPOSITION Suppose u: W -* Rm satisfies the rank assumption, jet transversality, and transversality condition A. Then 6S, the stable Pareto set, if not empty, has the structure of an (m — 1 )-dimensional stratified set. Furthermore, 6S can be characterized as the (open) set of x e 6 with index and nullity zero.

The simplest cases are in Section 4. Given u: W -* Rm, a gradient vector field for u is a smooth tangent

vector field X defined over W with the property that X(x) e H(x), for x $ 6, and X(x) = 0, for x e 0. It is an easy argument to show that one can construct a gradient vector field for any u.

6

The goal of this section is to try to gain a global theory of the critical Pareto set and stable Pareto points.

THEOREM 1 Suppose u: W -* R* satisfies the rank assumption, jet transversality, and transversality condition A with W compact. Then for any to & W with to fc 6, ds(to) ^ 0 .

We give an outline of how a theorem generalizing Morse theory to this case might so. This Morse theory perspective requires the introduction of the notion of cycle. S* always denotes the n-sphere.

DEFINITION Suppose u-.W-*- Rm. Then a cycle is a continuous injective map / : Sl -* W such that Sl can be written as a finite disjoint union of intervals / . and on each /«, either «< of is nondecreasing each i, or the image of I, is in 0.

The thoughtful reader will be able to construct an example of u with a cycle, even a map u: S* -» R* satisfying the rank and transversality conditions.

Define 6k to be the set of x e 6 with index A and nullity zero, and 6n

the set of x e 6 with nullity positive. Next let Jii ik be the coordinate subspace defined by setting

yx, . . . , yk all equal to zero in R*. Let

2 ^ t = x e Si | Z)«(a;)(7,i,(51)) is not transversal to n^ it.

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542 STEVE SMALE

By our transversality assumptions all the ^i1,...,it are manifolds and meet transversally with the Sf in St.

Let Ak = U(<1 <4> £^ ik and let A = Am_Y — Am_t. Then A will be the main part of the boundary of 0 (i.e., the union of the (m — 2)-dimensional strata of 36).

We want to assign a -\- or — to each point of A. This goes as follows: Let * e id. Then say that xe A+if the outward normal to u(A) in «(0)

at u(x) is in the closure of Pos c #". Then let 0' = the closure of (0„U/1+) and 0/^0' n » , .

"THEOREM" 2 Suppose u:W-+R* has no cycles, satisfies the rank assumption, jet transversality, transversality condition A with W compact. Let M< = 2JA dim Hi-X(Qx, 0/) with coefficients an arbitrary field. Then the Mj satisfy the Morse relations. That is if Bt denotes the 'th Betti number of W,

M0>Ba, M1-M0>Bl -B0,

Note that a Morse function u: W -* R clearly satisfies the hypotheses of "Theorem" 2. Thus "Theorem" 2 contains the usual Morse theory.

We check now how this specializes to m = 2. First observe that Mo = dimJ7o(0o,0o'),

Ml = dim #,(00, 0O') + #„(«,, 0,'), Mi = dim /fx(0<_,, 0_t) + dim Wo(0<, 0/) for 0 < * < n = dim W,

and Mm=dim/71(n_,,0;_I). This follows from the fact that 0 is one-dimensional and the index

is strictly less than «. Now what are the possible cases for the topology of (0*, 0/) ? Let us

call points of 5,* and A+ both generalized cusps by a slight abuse of language. Then we have for the components 6x

k of 0*: Case 1. dx

k is a circle with d empty. Let ak denote the number of these. Case 2. 0A* is an interval no endpoint a generalized cusp. Let at denote the number of these. Case 3. 0 / is an interval with one endpoint a generalized cusp. Let Pi denote the number of these.

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Journal of Mathematical Economics 1 (1974) 1-14. O North-Holland Publishing Company

GLOBAL ANALYSIS AND ECONOMICS IIA Extension of * theorem of Debren

S. SMALE University of California, Berkeley, Calif. 94720, U.S.A.

Received 29 June 1973

Introduction This note is a drastic revision and extension of the author's paper (1972). We

recall the following theorem of Debreu (1970) for a pure exchange economy: One is given C1 demand functions,/, fm, of m consumers, satisfying a

certain boundary condition. Then for almost all initial allocations of commodities to the m consumers, there are only a finite number of price equilibria.

Our goal is to remove the hypothesis of C1 demand functions, and in fact to remove the need for any well-defined demand functions. In their place, we study utility functions directly; our result is again finiteness and stability for the price equilibria. This is proved only for almost all initial allocations and almost all choices of utility functions. Our class of utility functions include those which define C1 demand functions.

We use a different boundary condition than Debreu. No convexity or monotonicity assumptions are made on the utility functions.

However, throughout, we do assume that they are of class C2 (have continuous second derivatives) and also that they have no critical points. Also we work with commodity bundles which may have some coordinates zero.

In carrying out the above program we introduce an extension of the notion of price equilibria, which we hope to develop in future papers in the series. This note while written in the spirit of Smale (1972 and 1973) does not depend on these papers.

Conversations with Debreu have been very helpful in writing this paper.

Sect. 1. We start by describing the space S? of states of an economy. The first ingredient is commodity space P = x = (x1 , . . . , xl) e R'\x' ^ 0, each /. Note that we allow the coordinates of points in P to be zero. Since xeP represents a bundle of goods and it is natural for an economic agent to have

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2 S. Smalt, Global analysis and economics

zero amount of some good, P seems preferable to P, the interior of P. Our use of P is in contrast to Debreu (1970) and Smale (1972), and a number of other recent papers in this area.

In any case, for us, a state of an economy associates to each of its m agents a commodity bundle or a point xt of P, I » 1 , . . . , m. Thus a state has a com­ponent a point x of the Cartesian product (Pf, x = (xt xm), x, e P. Note our use of subscripts and superscripts.

The other component of a state is a price system. We consider a price system to be represented by a non-trivial linear function p on R' = P which associates to a commodity bundle xeP,it& value p(x) e R, (R the real numbers). Thus one can write p = (p1,..., p*) with p(x) = J P'JC'. Two such linear functions p.qait considered to define the same price system provided Xp = q for some real number X > 0. We let 5 denote the space of normalized vectors; thus

S=peR'\M = l,

where M2 = £ (pO2-One can think of S as just a sphere of dimension /— 1. In this note it is supposed

that eachp1 £ 0; 5+ = pe S\p - (p1 p'^p1 J> 0 will denote our space of price systems. However, our methods and results apply to the case where S+ is replaced by S.

Finally we write our space of states S? as the product space y = (Pf x S+. So one interpretation of a state of an economy is the set of goods of each agent together with a price system.

A utility function for us will be a C2 function u:P-+ R and the space of all such functions will be denoted by C\P, R). It should be said what it means for a function u:P -* R to be C1 or C2 at a point of the boundary dP of P. Here dP <= xe P\x = (xl, .., x'), some xl - 0 or dP - P-P. Then u is said to be C on P if there is an open set G in Rl containing P and an extension of u to «': 0 -* R such that «' is C. Furthermore the first and second derivatives of u sXxeP, Du(x) and D2u(x) will be defined to be those oft/. It is easy to see that such derivatives are independent of the choice of «'. The first derivative Du(x) at x e P is a linear map from R' to R, which by using coordinates of R' can be considered as a vector in Rl, the 'gradient* of u,. The second derivative D2u(x) of u at x e P is a symmetric bilinear form on R'.

On the linear space C2(P, R) of utility functions we put the C2 Whitney topology [see Peixoto (1967)] which we recall at the end of this section. Utility functions considered here are supposed to never have a zero derivative. Thus we work with the open set U of C2(P, R)

U= ue C2(P, R)\Du(x) # 0 , all xeP. An economy associates to each of m agents, his or her initial allocation rteP

and his or her utility function w, e U, I = 1, . . . , m. Thus the ith agent prefers the

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S. Smalt, Global analysis and economics 3

commodity bundle xePto yePif and only if u,(x) > u&y). The indifference surfaces for that agent are uf*(c) c P, as c varies over the reals. In fact all of the analysis here ultimately rests on these surfaces ufl(c) and uses the functions u, as a convenient description of the indifference surfaces. In summary an economy is a point (r, u) of (Py x (U)m where (r,«) = (rx , . . . , rm,«,,..., u») with rt, ut respectively the initial resources and utility function of the /th agent

Given an economy (r, u) e (/*)" x (/)", certain states in Sf are singled out as (classical) /v/ce equilibria. The set of these states will be denoted by EJf, u) <= ST and is defined as follows.

Given an economy (r, u), the attainable states are those (x, p)eS? satisfying Ex, = £r (the total resources of the economy are fixed). The budget set of the /th agent at price system p is the set (xeP\p(x) ^ /Kr,) (sometimes the set x e P\p(x) = piryt). In words this budget set is the set of commodity bundles available to the /th agent with his resources, when a certain price system is given. Then for an economy (r, u), a state (x, p)e&is called a price equilibrium if it is attainable and if for each i the commodity bundle x, is a maximum of the utility function «, on the budget set of the (th. agent.

Given an economy (r, u), we say that an attainable state (x,p) is an extended price equilibrium or (x, p) e E„(r, u) if for each / — 1, . . . , m, x, e Bi>r — y e P\p(y) =p(rD) *nd x, is a critical point of the restriction u\Bt, (i.e., Du,\BKr) (x4) = 0).

Remarks. (1) Although some of the xt may be ondP, the definition still makes sense. (2) Example 4 of sect. 4 has an extended price equilibrium which is not a classical price equilibrium. (3) It is easy to check that EJj, u) => Ea(r, u) n [(F)" xS+ ] . That is, a classical price equilibrium, (x,p% where the x, are not on the boundary dP, is also an extended price equilibrium. Thus in case the number of extended price equilibria is finite, the same will be true for the classical ones defined on P. This permits a favorable comparison of our results with those ofDebreu(1970).

A regular economy (r, u) e (P)m x (ITf is defined by an analytic condition in sect. 3. In particular a regular economy (Proposition 4) has the property that it has only a finite number of extended price equilibria and they are stable (in a global sense) in the parameters (r, u) of the economy.

Theorem 1. The subset of regular economies is dense and open in the space (Py x (C/)" of all economies.

A related theorem is:

Theorem 2. There exists an open dense subset Y of the space (IT)" of m-tuples of utility functions such that for each ue Y, (r, u)isa regular economy for almost all initial allocations r. Furthermore Y contains the set of all m-tuples of utility functions which define C1 demand functions.

Here almost all can be interpreted as all except for a closed set of measure 0.

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The next section discusses what it means for a utility function *to define a C1

demand function'. By adding this last statement in Theorem 2 we are able to include the original Debreu theorem except for minor variations.

The Whitney topology is discussed more thoroughly in the last section of Peixoto (1967). Here we give the definition of this topology on C2(P, R)- It is sufficient to define neighborhoods of zero since C2(P, R) is a linear space. Such a neighborhood Nk is denned by each strictly positive continuous function h:P-*R. Then/6 Nh provided

|A*)|<*W allxeJ, (1) iDf(x)\\ < h(x) allxeJ*, (2) \\D2f(x)\\ < h(x) silxeP. (3)

For each x e P, Df(x), D2/(x) can be interpreted as elements of certain finite dimensional vector spaces. The norms || || are any fixed norms chosen on those vector spaces.

Sect. 2. Toward proving Theorems 1 and 2, we define a map $m: SP -* (S)m+i

for each HE (CO1" by:

Then if A is the diagonal in (S)M+1,

A = (vi ym+i)e(S)m+1\yi = y2 = • • • = j> .+ i ,

\j/~ '(A) is the set of states where the price vector and all the derivatives of utilities point in the same direction. Note that \j/u is C1 and in particular is differentiate even at points s e ST where Sf is not a manifold (that is those s — (xl3.. .,xm,p) with one or more xt, p in dP).

We recall that \j/„: ST -> (S)m+1issBddtobetransversaltoA c (S)™+1 provided that for each s with il/m(s) - y e A, the image of D^ijis) and r,(A) as subspaces of T,((S)"+1) span r,((S)"+1). See e.g. Abraham and Robbin (1967) for a development of these ideas of transversality. Note that although £f is not exactly a manifold, the notion of transversality still makes sense in our context.

It follows from the implicit function theorem that if ip¥ is transversal to A, then V'i"1(A) is a submanifold of £f with codimension V'J'HA) = codimension A [again see Abraham and Robbin (1967)]. Here codimension A = dimension (S)m+1—dim A, etc. We note also that the notion 'submanifold' of y is a slight extension of the usual notion. A precise definition would be as follows: Take an open set € of Sf in RXT x S. Then a 'submanifold' of ST is a subset of & of the form V rs& where Kis a submanifold of 6.

Proposition 1. Let Y be the set ofue (U)m such that ^.: Sf -» (5)*+1 is

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S. Smalt, Global analysis and economics 5

transversal to A. Then Y is open and dense in (C/)™ and Y contains the set of m-tuples of utilities which define a C1 demand function.

We remark that the Y of Theorem 2 will be taken to be the Y of this proposition.

Proof of Proposition 1. Openness follows from elementary properties of transversality and the topologies chosen (essentially from the fact that surjective linear maps are an open set of all linear maps).

Next the density of Y in (£/)* is a consequence of the Thom transversality theorem [Abraham and Robbin (1967)] (which in turn is based on Sard's theorem).

Finally we check the statement on 'demand defining utilities' being in Y. Let us see what this means. Given a utility function u:P -* R, then a corre­sponding demand function f: S+xR+ -* P (R+ denotes the positive reals), if it exists, is defined by f(p, w) = the maximum x of u on the set y e P\p(y) = w. In particular the map <f>: P -* S+xR+, tf>(x) = (p, w), defined by

6( 1 = ( DUX) Du(x)ix)\ 9W \\\Du(x)f ||2Mx)|i; must be locally invertible if u defines a Cl demand function; the derivative Dtf>(x) of <f> at each point must be a linear isomorphism.

Then, in fact, under these circumstances the composition

R> _ i * f L ^ rp>w(5+ x R+) J ^ i ° = > Tp(S+)

is surjective. If this is true for each uit it follows that \jfu is a regular map (i.e. surjective

derivative everywhere). This implies that \(iu is transversal to A. Summarizing: If each «! defines a C1 demand function, then ^„: S? -* (S)m+1 is transversal

to A. This finishes the proof of Proposition 1. Remark. Suppose that Du(x) ePforsdlxeP. Then a necessary and sufficient

condition on u in order that it locally define a Ci demand function at x e P is that the second derivative D2u(x) restricted to the kernel of Du(x) is negative definite (as a bilinear symmetric form). The following hypothesis on each «j, i = \,...,m implies that (ult..., um) e Y.

Hyp. (on u:P-*R) For each xeP, D2u(x) restricted to kernel Du(x) [i.e. the set of v e Rl such that Du(x)(v) = 0] is a non-degenerate bilinear form.

Compare this to Debreu (1972) where a related full discussion can be found. We remark also that for C2 u: P -* R to define a global demand function f:S+xR+ -* P, as Debreu (1972) assumes, the indifference hypersurfaces do not intersect dP. Thus in terms of utility functions, the boundary conditions in Debreu's (1970) are far removed from ours.

We wind up this section with another proposition. Counting dimensions using Proposition 1 and the implicit function theorem, we have:

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6 S. SmaU, Global analysis and economics

Proposition 2. If the utilities (ut,..., u j « « e Y, then the set of states (x,p) of Sf such that for all i, Dufocb and p point in the same direction, is a submanifold ofS? of dimension m+l—l.

Sect 3. We finish the proof of the main theorems. Toward this end, for an initial allocation r e Py, r «* (r t , . . . , r„), let

2, = (*,/>)eST\Z*, = £r t ,pixd - prd, / - 1 m-l). Thus 1, is the set of attainable states relative to r with a budget condition. For an economy (r, u), 1, contains the classical price equilibria (except for certain abnormal situations) and the extended price equilibria. Note that the condition /K-O "= /KO is omitted because it follows from the others.

Let T be the subset of (Py x ST: r-(r,s)e(PyxSr\selt).

Proposition 3. (a) T is a submanifold of(Py x Sf of dimension 2ml-m and (b) the projection ny: (PyxS? - S? restricted to T is a regular map (i.e. at every point ofT the derivative is surjective).

Proof. Consider the map : (Py x S? -* R' x R" ~1 defined by 4>(r,x,p) = C^Xi-^r^pix^-piri),.. .,pixm-i)-pirm^)).

We claim that is a regular map. Its derivative at (r, x, p) is given by: D4>(r,x,p)(r,x,p) = ( E * , - £ f „ #*,-/-,)+;>(*,-f,),

/ = 1 m— 1). This has the following meaning: (f,x,p) is a tangent vector of (PyxSf at (r, x,p). Thus f = (r t , . . . , r„) can be interpreted to be an element of (R*y, ft e Rl, similarly x is an element of (R'y and p is a vector in Rl perpendicular top.

To show that D<f>(r, x, p) is surjective, it is sufficient to solve the following system of equations for given a e El and real numbers bt,..., bm _ t:

£*/-£'« = <*. P(Xi-rd+p(x'i-r'd = *<» i «= 1, . . . , Am-i.

For this simply set/ and f, = 0, let * , , . . . , Jc„_1 satisfyp(xd = b( (this can be done since p # 0) and set xm = a— XT-V xt.

Thus <f> is regular and T = <ft~ 1(0) is a submanifold of the required dimension. (A similar argument shows that Er is a submanifold of Sf of dimension

m/—m.) To prove Proposition 3(b) first note that the tangent space T(rtX>r)(T) <=

TirwXtP)((Py x ST) may be described as the set,

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S. SmaU, Global analysts and economic* 7

/ = l , . . . , m - l , where 5 , = peR',plp.

To see that Jt,|r: r -»5^ is regular, it is sufficient to show that given (r, x, p), x, p, the following equations can be solved for f,

E*i = £?j»P(*r-',i)+/>(*i-',i) = 0. ' = 1 « - ! • This is done by choosing ft,i= 1 , . . . , m— 1, such that/^f,) = piX^—pXx,—^ (this can be done since/> T4 0); then let fm = Y?-i xt—YJ-\ fi- Thc proposition is proved.

Corollary. Let u e Y Proposition 1) and Z, = (r, x,/>) e r|^,(x, p) e A. 7%«i Zaua submanifold of T of dimension ml.

Proof of Corollary. ijiu is transversal to A by Proposition 1 and the com­position T -♦ (P)mxS? -» Sf is surjective by Proposition 3. It follows that the composition T -* (/>)* x y -» & -» (S)"+1 is transversal to A. The corollary follows.

Debreu has pointed out to me that in the case of demand functions, this corollary is contained in: F. Delbaen, 1971, Lower and upper-hemi-continuity of the Walras correspondence, mimco., Free University of Brussels, Brussels, Belgium, p. 56. Also Debreu adds that under similar hypotheses, Balasko proves that Z, is contractible in: Y. Balasko, 1973, Equilibrium space of a family of economies, mimeo., Electricity de France, 2 Rue Louis Marat, Paris, France.

Definition. An economy (r, «) will be called regular provided (a) u e Y and (b) nQ restricted to Z, has r as a regular value where nQ: (P)m x & -* (Pf is the projection [think of Q as (£)"].

Proposition 4. For a regular economy (r, «), the set of extended price equilibria E„(r, u) is a finite set which mooes continuously in (r, u).

In other words, if (r0, u0) is a regular economy, then the extended price equilibria may be labeled xt(r0, u0)t i = 1, . . . , k. Furthermore there is a neigh­borhood N of (r0,«0) in (P)mx(U)m, and continuous functions a:N-*S?, i = 1 k such that ot,(r, u) label precisely the elements of Em(r, u). Also the elements of N are regular economies.

Proof of Proposition 4. The manifold Z„ has dimension ml (previous corollary) the same as dimension (P)m. Since r is a regular value of UQ\Zm the inverse image of this map by the implicit function theorem is a submanifold of dimension zero or is a discrete set. By the definitions this set is Ea(r, u). Since this set is bounded it is also finite. Also by Sard's theorem, the set of critical values of nQ|Z. is closed, nowhere dense and of measure zero. Finally the implicit function theorem gives the stability property.

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8 S. Smale, Global analysis and economies

This proposition and its proof also yield Theorems 1 and 2, taking into account Proposition 1 of sect. 2.

Remark. If (r, u) is an economy and (x, p) e £„(/, u), then by definition x, is a critical point of u, on the plane y e P\p(y) = p\rd). If (r, u) is a regular economy then that x, is a non-degenerate critical point of u, on this set

The converse is false as example 3 of the next section shows. Here one has a clearly non-regular economy and every critical point of u, on y\py) = prfi is non-degenerate.

Sect 4. Given here are 4 examples of Edgeworth boxes, displayed geo­metrically rather than analytically. We do not claim originality for these examples.

^ U2 " contunt \ ~* ui » constant uj » contunt

Fig . l

U) -conjtint

EB number 1. This is the standard classic situation. Consider the case of an economy with two goods and two agents with a fixed initial allocation (rt, r2). Furthermore suppose that the utility functions have the classical convexity and monotonidty properties. Then the Edgeworth contract curve or set of Pareto optima [see e.g. Smale (1973)] is the curve of common tangents in the two-dimen­sional space W = x e (P)2\x1+x2 •» rl+r2. See fig. 1.

Now in this context a price equilibrium (x, p) can be represented by a line in the figure through r, and x1 which is tangent to the curve ut at xt as in fig. 2. This line is the set of y in P such that p(y) =» p(r).

One can check that the defining conditions for a price equilibrium are verified. EB number 2. One can see in the same terms why one cannot expect unique­

ness of price equilibrium, even in the most classical of situations, fig. 3.

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S. Smale, Global analysis and economics 9

As usual dotted line represents curves u2 = constant, and curved solid lines Mj = constant. The lines through r, are straight.

EB number 3. Next we see how the price equilibria can be infinite in number, and in fact constitute a one-dimensional set, fig. 4. One extrapolates so that while r, is fixed, xt fills out the points on the curve AB. Note that in this example, that for each price equilibrium (xi,x2,p), ut has xt as a non-degenerate maxi-

Fig.2

Fig. 3

mum on the set yePlutiy) = ut(r,). Of course by the main theorem, the example as a whole is degenerate.

Fig. 4 is drawn as follows. One first chooses rt, the arc AB and the three straight lines drawn through r t . After that the utility curves are drawn appro­priately. Of course one must imagine indifference curves filling up the figure, but always tangent at q along AB to the line joining rt and q.

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10 S. Smote, Global analysis and economics

EB number 4. We now relax the convexity hypothesis on the utility for u2 to see that one may lose existence of a classical price equilibrium. In this case one still has an extended equilibrium, fig. 5. But u2'» utility is minimized (I) on

Fig. 4

his budget set at equilibrium. Note that the corresponding Pareto optimum is 'classical' and stable in the sense of Smale (1973).

One can obtain the other ut = constant and u2 = constant curves by transla­tion in the direction of the vector/? (or perpendicular to the line rxx^).

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S. Smote, Global analysis and economics 11

References

Abraham, R. and J. Robbin, 1967, Transversal mappings and flows (W.A. Benjamin, New York).

Debreu, G., 1970, Economics with a finite set of equilibria. Econometrics 38,387-392. Debreu, G., 1972, Smooth preferences, Econometrica 40,603-616. Peixoto, M„ 1967, On an approximation theorem of Kupka and Smale, Journal of Dif­

ferential Equations 3,214-227. Smale, S., 1972, Global analysis and economics II, Extension of a theorem of Debreu, mimeo.

(University of California, Berkeley). Smale, S., 1973, Global analysis and economics I, Pareto optimum and a generalization of

Morse theory, Salvador symposium on dynamical systems (Academic Press, New York).

Appendix

This appendix gives an existence theorem for price equilibria in the context of our paper. We prove that under a certain boundary condition on the utility function, and a monotonidty assumption, there will always exist an extended price equilibrium. This is without using any convexity of the utility functions, u,. If furthermore one assumes enough convexity so that the u define C1 demand functions, then the extended price equilibria obtained are classical ones. Com­pare example 4 of sect 4.

Perhaps a more extended analysis which studies price equilibria on dP could substantially relax the boundary condition.

The boundary condition is a condition on a pair (r,u)ePxU (notations as in the body of paper).

Say that (r, w) satisfies (BC) if: (BC) For each x e dP, Du(x)(r-x) > 0. Here Du(x)(r—x) can be interpreted as the 'dot product' of the gradient,

gradw(.x)andr—x. The boundary condition has the effect of making the indifference surfaces

become parallel to the boundary as they near the boundary near oo, (the proof of the proposition, part (c) below illustrates the boundary condition in an example).

Next say that u e U satisfies the Monotonidty Hypothesis if: Monotonicity Hypothesis: Du(x)e i* for all x e i*. In what follows, this could be weakened to: (MB) There is an open half space H in Rl and Du(x) e H for all x e P. Let & be the set of all economics (r, K) in (Py x (U)m with the property that

for each 4 u, satisfies the Monotonidty Hypothesis and (rt, u,) satisfies (BC) [here r = (r , , . . . , rJ and u = (K, , . . . , uj]. Let &D be the subset of & of economics (r, u) with the property, for each /, u, has the convexity property: For each x e P, the restriction of the second derivative, &uAx)\YjnDuAx) is negative definite (see sects. 2,3; this amounts to saying that u, defines a C1

demand function locally).

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12 S. Smau, Global analysis and economics

Theorem, (a) An economy (r, u) in & has an extended price equilibrium. (b) An economy (r,u) in FD has a classical price equilibrium and in fact each extended price equilibrium is classical and conversely), (c) If in addition (r, u)e^ is regular, then there is an odd number of extended price equilibria x,p).

The extended price equilibria of the theorem have the property all of the xt are in the interior PofP where (xt,..., xj) = x.

An article related to the above theorem, in the context of demand functions, is: E. Dierker, 1973, Topological methods in Walrasian economics, CORE Discussion Paper no. 7301, Heverlee, Belgium.

Toward proving the theorem we define a different topology on C2(P, R), the C2 compact open topology. It is sufficient to define a system of neighborhoods of 0. Let K, = xe^llxl <q), q = 1 ,2, . . . then a neighborhood #..« of 0 is defined by some 8 > 0 and some q = 1,2, . . . as follows : p-e Nttt if

Sup|c(x)| <e , .

Sup iIZM*)!! <«,

Sup ||I>2K*)I < *•

Compare sect. 1. This topology is used on all function spaces in the following.

Proposition, (a) !F is arcwise connected, (b) If(r,u)eP and (x,p) is an extended price equilibrium for the economy (r, u), x = (x , , . . . , xj then no xtedP. (c) There exists a regular economy (r0,u0)e& with precisely one extended price equilibrium.

Proof of proposition. Let (r,«), (/•', u') e 9. We construct an arc in & con­necting (r,«) to (r,«') and another arc connecting (r,«') to (r', u"). Let r = (rx , . . . , O , « = («! , . . . , O . ete- Define for each t, 0 ^ t ^ I, and for each i = 1, . . . , m, uu = tut+(l -t)u't. Then u„ surely is in C\P, R) for each t. It is also continuous in t. This is true in the compact open C2 topology, but is not true in the Whitney topology.

Note that for each xeP,

Du£x) = tDuM+Q-ODuXx);

thus Du„(x) ,* 0 using the Monotonicity Hypothesis. So u, e U. At the same time we see that u„ itself satisfies the Monotonioity Hypothesis. Next it is checked directly that (BC) is true for (r,, uu). We thus have constructed an arc (r, u,) in f> from (r, u) to (r, uT).

Define r„ = trt+(l — Or/. Again (5C) is satisfied for each (ru, u',) and we have found an arc between (r, u) and (r',«'). Thus ^ is arcwise connected.

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5. SmaU, Global analysis and economies 13

Consider (b) of the proposition. If (x, p) e Enr, u) then for each i = l m, p(xi)=p(rl), />(/•,-*,) = 0, and hence Dufad(r,-xt) = 0. Therefore by (BC), x, e dF. This proves (b).

For (c) define u': P -* R by

«'(*) = II (*'+1) = (^ + 1)(*2 + 1) • • • (*'+l),

and r' e P, by r' = (/,..., /). Then we claim fa,«') satisfies (BC) and M' satisfies the monotonicity hypothesis. Differentiating,

Clearly Z)K'(.X) e P for each x e P (in fact DK'(->0e P) and so that monotonicity hypothesis is satisfied. To check (BC), observe

Du'(x)(r'-x) ^ l-x' l-xl

V(x) j c , + i + * * ' + * ' + r thus it is sufficient to show

' / ' x'

provided xedPor at least one of the x' is zero. Suppose x* = 0. Then 1 I ' xl

To finish, choose fa u) e (f)" x ((/)" with r = fa,..., r j = fa,..., /•') and « = («! , . . . , wm) = («',. -i "')• Then it is easily seen that the economy fa u) has a unique price equilibrium (x, p) with x = r and p = (1/VA • • •. l/VO* and i s

regular. This proves the proposition. We now prove the theorem, part (a). This goes by a degree argument. See for

example: O. Milnor, 1965, Topology from the differentiable viewpoint, University of Virginia Press, Charlottesville, Virginia, for a reference on degree_ theory. The idea is that any economy in & is connected by an arc to (r0, u0) and along this arc the price equilibrium moves in a somewhat continuous fashion, but cannot slide off the boundary of £f because of the boundary condition. The following makes this more precise.

For each fa u) e P, define a(r>1(): ST -+ (S)m+i x Rl x /T" 1 by

<*i,,.)(x,p) - (<A, , (* . / ' )>X>I-2>I . / ' (*I )

-Pird, • • -,/'(Jf«-i)-^(''*-i)).

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14 S. SmaU, Global analysis and economics

Then (x, p) e E9x(r, u) if and only if «(,,,)(*, p) n A x 0 x 0. Furthermore it can be seen that if (r, w) is regular then a(r>>) is transversal to A x 0 x 0.

From the fact that (r, u )e&, *(r'ty(d&) n A x 0 x 0 = ^, and the degree of a(r>M) is defined. If (r, u) is regular, this degree is the number of points in a(r>>)

-1(A xO xO), each counted with proper sign. By (c) of the proposition, the degree of a(,0>ao) is one. Some treatments of degree consider the analogous case where A x 0 x 0 is replaced by a single point which is a regular value; but degree theory carries over to our case as well. Finally the degree is a homotopy invariant, so that the degree of each a(r>a) is one, if (r, u) e ?. Thus (a) of the theorem follows pf £„(/•,«) is empty, then (r, u) is a regular economy!]. Part (c) follows similarly. Also part (b) is a consequence of the fact that under the hypothesis (r, u) 6 &D, every extended price equilibrium is classical (w, must be a maximum at xt on the budget set by the second derivative condition). This proves the theorem.

284

GLOBAL ANALYSIS AND ECONOMICS HI Pareto Optima and price equilibria

S. SMALE University of California, Berkeley, Calif. 94720, U.S.A.

Received September 1973

Among other things a global version of the fundamental theorem of welfare economics is proved. One starts with a pure exchange economy with fixed total resources where hypotheses of differentiability, convexity, and monotonicity are made on the utility functions. Let A be the set of price equilibria where the initial allocation coincides with the final one. Then the map which assigns to such a price equilibrium, the corresponding allocation is a diffeomorphism (a complete correspondence) between A and the set of Pareto Optima.

1. Introduction Here we examine in a calculus context the relation between Pareto Optima

and price equilibria for a pure exchange economy. In doing so we develop from the abstract Pareto theory of Smale (1973a, b) sufficient first- and second-order conditions for a state of a pure exchange economy to be a stable Pareto point (section 3). It is also shown under quite general conditions in a pure exchange economy that the set of infinitesimal Pareto points is an (m— l)-dimensional manifold, where m is the number of consumers (section 4).

No convexity assumptions on the utility function are needed. A basic result of equilibrium theory is that if (x, p) is a price equilibrium,

where x is an allocation and/> a price vector, then x is Pareto Optimal; conversely, given a Pareto Optimum x, there is a p with (x, p) a price equilibrium. See Debreu (1959) for a general theory of Arrow and Debreu to this effect. In sections 5 and 6 we give versions of this theorem in terms of corresponding concepts denned by first and second derivatives. Putting this kind of theory into a calculus context brings it closer to classical methods of maximization and calculation. Elsewhere we hope to continue to develop the problem of putting equilibrium theory in economics into a systematic calculus framework. In this paper, for example, no production is considered and essentially there is no analysis of points on the boundary of commodity space (where some commodity is zero). In future accounts we hope to remedy these omissions.

The last three sections of this paper depend rather heavily on parts of Smale (1974).

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108 S. Smalt, Global analysis and economics III

D. Barden, C. Simon, and C. Titus are preparing a paper which relates to some of this one.

2. We recall the abstract setting of Pareto theory [see e.g., Smale (1973a, b)]-

One is given smooth (i.e., C, r £ 2) real-valued functions «t um defined on a manifold W (W could be taken as an open set of some Euclidean space in all that follows). Let u: W -» R" be the map u = (M, , . . . , u j . where Rm is real Cartesian m space.

An admissible curve <p:(a,b) -* W is then a C1 curve with the property that the derivative of «, along q> is positive for each /, i.e., D«,(qj(f)X<if>'(0) > 0. A point x of W is called a Pareto point or an infinitesimal Pareto point or x e 6 provided there is no admissible curve through x.

Define Pos by Pos «= y e R"\y - (yt ym), yt > 0. Then x e 6, if and only if Du(x)(v) * Pos for all v e TX(W). Here Du(x): TX(W) -> RT is the first derivative of u at x and TX(W) is its domain, the set of tangent vectors of W at x [if W is an open set of Euclidean space E, then one can take TXW) = E).

Then 9 is an extension of the set of classical Pareto Optima; it has a characteri­zation given by the following proposition:

Proposition A. First-order proposition, abstract. Smale (1973b). A necessary and sufficient condition for x to belong to 6 is that one of the following equivalent conditions hold. (a) The Du,(x) do not all lie in the same open half-space [in the dual linear

space T?(fV)l (b) There exist Xt ^ 0, / = 1 m, not all zero, such that £ A,DH,(;C) = 0.

The prime example of the abstract system is the pure exchange economy. In this case one lets

P - (x l , . . . , xl) e Rl, x' > 0 be 'commodity space', and

W = xePT\x^(x1 x j , x,eP, 2 > , = r,

where r is a fixed element of P. Then W is the space of states of a pure exchange economy with total resources r. In this paper we omit consideration of both production and (for the most part), behavior on the boundary dP of commodity space.

If x e W, then x = zY,.. ., xn) and xt has an interpretation as the commodity vector which gives the resources of the rth agent, i = 1 m.

The preferences of the ith agent are supposed expressed by a 'utility function',

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S. Smalt, Global analysts and economics HI 109

a C2 function u,: P -* R. Thus the ith agent prefers x in P to x' in P if and only if ut(x) > M,(X') (with u, we follow a notational suggestion of Carl Simon and others). The following is assumed throughout the paper:

Hypothesis. The u, have no critical points; otherwise said, Du((xt) / 0 for all /, x, e P.

Let u,: W -* R be the function defined by u,(x) = u^x,), where x = (*i,. . ., xm). This puts the pure exchange economy into the abstract setting and Proposition A applies.

Proposition B. First-order proposition for the pure exchange case. Let u:W-*Rmbeas above, a pure exchange economy. Then xe9 if and only if there exist Xt > 0, / = I,. . .,m with XtDu,(x,) = a constant, independent ofi. In other words, all the Du;(*j) point in the same direction in Rl.

For the proof of Proposition B consider the first derivative Du(x): W -* .ft1", where

W'mXeRlr\x-'(xl,...,SJ, xteR\ £ * , = (>

is the space of tangent vectors to W. One can write Dux)(x) = (DwjOtjX*!) D i i J O W ) for x e W, x e W. Proposition A gives the condition for x e 9 as

£ XtDQ&xXX) = 0, all xeW.

Let in fact x = (0,. . ., 0, xk, 0 , . . . , 0, -xk). Then the condition becomes

XkDuk(xk)(xk) = XJXiJxJiXj, all xk e R', or

XkDuk(.xk) = A^Dfi^J.

Proposition B follows. Remark. The condition in Proposition B is well known to mathematical

economists. For example, it is explicit in Samuelson's Foundations (1971); also, see Henderson and Quandt (1958), Intriligator (1971) and Malinvaud (1972) for this and related material. Actually, in Pareto (1896/7), one finds first-order conditions for a Pareto Optimum, although perhaps not in a very clear or rigorous way.

Related to the above Proposition B is the following proposition of linear algebra which allows one to consider the possibility of critical points of utility functions in the theory.

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Rank Proposition. Suppose one is given linear maps a,: Rl -* R, i = 1 , . . . , m. Let W « (t>, vj e (R')m\ £ v, « 0, and a: W -* Rn the restriction of (<Xi aJ. Then: (a) If all the a( are not zero, then corank a = 0 or 1. Furthermore, corank o = 1

if and only if there are real numbers kk& 0,k = 1 m with A|0t( indepen­dent ofi.

(b) Suppose at least one of the at is zero. Then corank a = the number of ct,'s which are zero.

To obtain an alternate proof of Proposition B, one may apply the rank proposition to a = Du(x) restricted to W.

For the proof of the rank proposition, let Y « yx,..., ym.j) e (R'y*'1 and imbed r - * H"by

O-i, • • . , * . - . ) - ( * , • • . . * . - i . ~ " | > . ) e (*')".

Using this F basis for W we have

<^CFI ym-i) - ( ^ i ) am-iC^m-i). - £ B».0O) •

Now for part (b) we may suppose by changing indices that am = 0. Then (b) follows from this expression for a.

We next consider part (a); thus suppose that all the a( are not zero. Project Rm -+ R"~l by setting the last coordinate equal to zero to see that corank a ^ l . If corank a = 1, then by the expression for a there exist A, not all 0 with

m—1 m - 1 I lt«i(yt) = *m Z amCVi). all^ = (ylt.. .,ym.J. l I

The rest follows. We remark that in this section and in fact this whole paper, the analysis rests

only on the preference relation defined by the utility functions and not on the utility functions.

3.

The goal here is to give second-order (sufficient) conditions for Pareto Optima in a pure exchange economy. First, we recall the results of the abstract theory from Smale (1973b).

As in section 2, one is given a C2 map, u: W -* Rm which defines 9. Say that a point x of 9 is stable or that x e 9S if every admissible curve which

starts near x stays near x. In other words, x e 9S provided, given a neighborhood U of x, there is a neighborhood V of x such that every admissible curve starting in V stays in U. An example of a classical Pareto Optimum which is not in 0S is given at the end of this section.

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S. Smale, Global analysts and economic) III 111

If x e 6, we say that xcdQ provided the image of Du(x) intersects the closure of Pos in Rm only at the origin (since xefl, this image does not intersect Pos). For simplicity now, assume the rank of the linear map Du(x) is m— 1 (corank 1). Then the generalized Hessian Hx at x is a bilinear symmetric form on the kernel of Du(x), real-valued, and defined up to a positive scalar. This generalized Hessian Hx is the second derivative of u at x interpreted appropriately [see Smale (1973b)].

Proposition A. Second-order, abstract. Smale (1973b). Let u: W -* Rm, xed, x$dO and assume corank Du(x) = 1. Then if Hx is negative definite, xe6s. Furthermore, one has the expression for Hx,

Hx = ZXiD2ui(x), where the A( come from the first-order condition (Proposition A of section 2).

Remark. It must emphasized that the domain of Hx is the kernel of Du(x).

Proposition B. Second-order, pure exchange. Let u: W -* RT be given from a pure exchange economy (section 2), where the utility functions ut have no critical points. Suppose x e 8, but x $ dd. Then x e 0S ifHx is negative definite and one has the expression for Hx,

# , = EA,D2w,(*), where the A( come from Proposition B of section 2.

Remarks. (1) Proposition B follows easily from the previously stated propositions of section 2 (including the Rank Proposition) and Proposition A.

(2) One must be especially careful of the domains of Hx and D2u,(xt). The domain of Hx is the set of n e (R')m with £"-1 >7< = 0, rjt e R', n = (t\t,..., nm), and t\ e Ker Du(x). The condition that n e Ker Du(x) amounts to tj, e Ker Du,(x,), each J =• 1 , . . ., m. Summarizing:

Corollary 1. In Proposition B we may rewrite Hx by

Hx(t],n)= t A.D^XjXr,,,/!,),

where n = (tjt,. .., rjj, fi = (/z,,. . . , nm\ tj,, n, are vectors in R' such that D«,(x,)(r\i) = 0, DufadiPi) = 0, each i, and £ tj, = £ n, = 0.

(3) Compare the result to the case of classically denned utility functions. A utility function u:P-*R satisfies a differentiable form of convexity if the

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112 5. Smalt, Global analysts and economics III

second derivative D2u(x), restricted to the kernel of the first derivative Du(x\ is negative definite. This is precisely the condition that u define locally a C1

demand function [see Smale (1974)] and implies that u~l[b, oo) is strictly con­vex for each b. Thus from Corollary 1 and Proposition B we have:

Corollary 2. Suppose that in a pure exchange economy, the utilities satisfy the above 'differentiable' convexity condition. Then 9 coincides with 9S; every 'first-order Pareto point' is stable.

(4) We give an Edgeworth Box example in pictures to show why ordinary convexity is insufficient to guarantee that 9 is the same as 9S. The example also shows that even a classical Pareto Optimum is not necessarily stable.

Fig. 1. Dotted lines are u2 ~ constant, solid lines are Q\ = constant.

The classical Pareto set as well as 9 consists of the segments cdand ab. In fact, ab is part of a straight line on which both uY and u2 are constant. Except for ab the indifference curves for u2 are strictly curved. It can be seen that ab (at least except for ab n cd) is not part of 9S. This example gives a good motivation for the introduction and study of 8$. Note also that a slight perturbation of the utility functions destroys ab as part of the Pareto Optima. One may smooth the corners to make w, and u2 C2.

(5) Another illustrative Edgeworth Box example is given by taking both utility functions defined by

ut(x,xi) = xxi, J - 1 , 2 .

This yields a very classical Edgeworth Box picture. Corollary 1 (and Proposition B) apply to yield that every point in 9 is also in 9S. On the other hand, the economist's trick of maximizing a weighted sum of utility functions A1C1+A2«2

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S. Smale, Global analysts and economics III 113

fails to exhibit the Pareto Optima for any Xt, k2. That is to say for no ^ , X2 > 0 does /IjW, + A2M2 have its maximum at a Pareto Optimal point.

(6) While many economists have discussed second-order conditions in con­nection with Pareto Optima, I have found nothing in the literature comparable to Proposition B and its corollaries. For example, Pareto (1896/7) writes down many second derivatives. Samuelson (1971, pp. 236-240) has a good discussion where he states' . . . the first-order conditions are really of secondary importance as compared to the full inequalities implied by an optimum position'. But Samuelson doesn't state explicitly the second-order conditions for an optimum.

4.

The goal here is to discuss the structure of 8 in the case of a pure exchange economy where the utility functions have no critical points. The main result is that this set is 'generically' a submanifold dimension m— 1, where m is the number of agents. It is always such a submanifold in the case of the previously mentioned differentiable convexity condition on the utility functions (and hence in this case 0S is also a submanifold).

Main Proposition. Suppose that u: W -* R* denotes a pure exchange economy, where u is an element of the space ofm-tuples of utility functions (U)m (see below). For a dense open set 0 in (U)m, the Pareto set 6 is a submanifold of dimension (m — 1). Furthermore, 0 contains the set of all utility m-tuples which have sufficient convexity to define locally a C1 demand function. In this last case (as we have seen) 6 coincides with the stable Pareto setOs.

For detailed background on the spaces involved in this proposition see Smale (1974). We recall briefly that P is the closure of P in Rl so that P = x e R'\x = (x1 xl), x' £ 0 and that 8P - P-P. Differentiation of functions on dP follows Smale (1974), and U is the set of all real-valued C1 (utility) functions u:P-+R with no critical points. The Whitney topology is imposed on U and (U)m is the Cartesian product of U with itself m times.

Toward the proof, define a set A of linear maps by

A = a: (Rl)m -* Rm, where a is of the form a = ( a , , . . . , am),

a , : /? ' -* R,oci # 0.

Define W=ye (Rl)m\y - (tt ym\yt e *' , I>< = 0 and B = a 6 A\ corank o| W = 1. Thus 3 consists of those linear maps in A whose restriction to W have rank (m—1).

Proposition. B is a closed submanifold of A of dimension l+m—l.

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114 S. Smalt, Global analysis and economics III

For the proof, one simply uses the rank proposition of section 2 to give an alternate description of 3 as

B = a e A\* = (a , , . . . , a„), there exist A, > 0 and ce R' with A,a, = c.

Let B0 be the subset of B where the X, are all positive. Then B0 is a component of B.

Now if u: W -* R" comes from a pure exchange economy and xeW, the first derivative Du(x) is an element of A. This follows from the fact that u is the restriction of a map («!,.. . ,«„): (Pf -» Rm, where the ut e I/. Since B is a submanifold of A, the following hypothesis as a condition on u makes sense. A reference on transversality is Abraham and Robbin (1967).

Transversality Hypothesis. w.W-^R" coming from a pure exchange economy has the property that Du is transversal to B.

An example where this fails is the Edgeworth Box example of Remark (4) of section 3, or any box where u, = w2 « a constant along some segment.

Proposition. The Transversality Hypothesis is true for an open dense set ofue(U)m. '

This follows from Abraham and Robbin (1967) - compare Proposition 1 of section 3 of Smale (1974).

Since 6 = x e W\Du(x) e B0) we have as a corollary (implicit-function theorem):

Corollary. For a pure exchange economy, d is a submanifold of dim (m— 1) provided ueO-an open dense set of(Uy*.

We remark finally that if each individual utility ut:P-*R satisfies the con­dition D2u^x) is negative definite on the kernel of Du,(*) for each xeP [our differentiable convexity condition discussed in section 3 and Smale (1974)], then u: W -» if" satisfies the Transversality Hypothesis and so 0 is a submanifold.

There is an Edgeworth Box example in Smale (1973a), where u: W -* R1

satisfies the transversality condition but not any convexity conditions. In that case 6 consists of a circle and an arc, 8S two arcs.

5. Recall some of our perspectives from Smale (1974). An economy (r, u) is given

by m-tuples of initial allocation r « (r l t . . . , r j , r, e P and utility functions « = («i , . . . , i?J. The states of the economy are points (x, / ) ) e y » (Pf x S+

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S. Smale, Global analysis and economics HI 115

with x = (xj, . . ., xm), (x( £ P) an allocation and a price system p = (p1,.. .,/>'), p' ^ 0, £Q>')2 = 1- An extended price equilibrium x,p)e En(r, u) is a state with the property that £ xi = £ r, and «, on the budget set x, e Plp(xt) = />(r,) has x, as a critical point. At least if each x, e P, a classical price equilibria is also an extended price equilibria. We hope in the future to make an analysis of classical price equilibria, where some of the x( e dP.

If (x, p) is a state as above, then it is consistent with earlier sections to call x a Pareto point (infinitesimal) or x e 6 if x e 0 for the map u: W -* Rm, where W = xe (P)m\ £ x, = £ ri)>and " i s t h e restriction of (u, «J: (P)m ->• Rm. Our first result is that if (x, p) is an extended price equilibria, then no admissible curve passes through x or:

Proposition A. Suppose that in the pure exchange economy (with price systems) the individual utilities «,: P -* R satisfy Dw,(x,) e P for each i and each x,eP (a'monotonicity'hypothesis). If(x,p) eEtx(r, u), thenxed.

One may substantially relax the monotonicity hypothesis. For the proof let (x, p) e Etx(r, u). Then, as in Smale (1974), there exist

Xlt...,\m>0 with A,DM,(X() = p, each i. Thus by section 2, Proposition B, since yl,DM,(x() is independent of i, x 6 8.

There is a converse to Proposition A.

Proposition B. Suppose again that the w, satisfy the monotonicity condition of the preceding proposition and we P. Let xeO relative to the pure exchange economy u: W -» JT, W = x e (P)m\ £ x, = w, u the restriction ofl&i «J : (P)m -* R". Then there exists a price system peS* such that (x,p)eEcx(r, u), where r = x.

Proof. If x e 9, by Proposition B of section 2, there are X, > 0 with AjD«,(x,) independent of/. Lttp0 = Xflu^Xt), some /, and/» = p0/\\Po\\ • T n c rest follows.

6. In this 'second-order' version of the preceding section, we show that a 'stable'

price equilibrium determines a stable Pareto point. Suppose that (x, p) is an extended price equilibrium for an economy (r, «).

Then we say that (x, p) is stable or (x, p) e Es(r, u) provided that u, has x, as a non-degenerate local maximum on the set B = x e P\p(x) = pr,). Here that x, is a non-degenerate local maximum means the second derivative of «J, on B at x, is a negative definite symmetric form.

Proposition A. Suppose the monotonicity hypothesis of Proposition A of the previous section is satisfied for an economy (r, u). If(x,p) e Es(r, u), then x is a stable Pareto point ofx e 6 s .

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116 S. Smalt, Global analysis and economics III

Proof. This is a consequence of Proposition B of section 3 and its Corollary 1. The converse in any sense is false by example EB#4 of Smale (1974). In this

non-convex Edgeworth Box, x e 0S> while the utility of the second trader is minimized on his budget set.

If the differentiable convexity condition on the utility functions (see section 3) is satisfied, then it is easily shown that Enr, u) coincides with £s(r,«) and 8 = 9S (section 3). Thus in this case the situation reverts to the first-order theory of the previous section.

Throughout, the monotonicity hypotheses may be relaxed as one wishes. Finally, we remark that the example in section 3 can be extended to show that

there is an economy (r, «) with (x, p) a classical price equilibrium for (r, u) but x is not in0 s .

7. We give a global setting for the previous two sections. Suppose that a utility m-tuple u e (U)m satisfies the genericity hypothesis of

Proposition 1 of section 3 of Smale (1974). Then I = x, p) e ST | A,D«,(x,) = p, X, > 0 is a submanifold of Sf of dimension m + l— 1 (Proposition 2, same reference). Following the spirit of Smale (1974), consider the map

f.Z-+ R' defined by f(x, p) = ][>,.

Let re Rl be a regular value of/ (If sufficient differentiability of the u is assumed, almost all r in P will be regular values.) It follows that

A = (x,p)eZ\Ydxt = r

is a submanifold of I of dimension m— 1. One can think of A as the set of 'non-tatonnement price equilibria'.

Let W = x e (^)"| £ x, = r, and 6 be the Pareto set of the pure exchange economy associated to u: W-* Rm. Define g: A -* Why g(x,p) = x.

Proposition. Under the above conditions 6 is an m— l-dimensional submanifold of W compare section 4) and g is a diffeomorphism of A onto 9. Furthermore, defining As as the set of stable elements of A, As is an open submanifold of A, andg(A^) c 0S. However, gA^ is not necessarily all of '95.

References Abraham, R. and J. Robbin, 1967, Transversal mappings and flows (Benjamin, New York). Debreu, G., 1959, Theory of value (Wiley, New York). Henderson, J. and R. Quandt, 19S8, Microeconomic theory; A mathematical approach

(McGraw-Hill, New York). Intriligator, M., 1971, Mathematical optimization and economic theory (Prentice-Hall,

Englewood Cliffs, N. J.).

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S. Smale, Global analysis and economies III 117

Maliavaud, E., 1972, Lectures on microeconomic theory (North-Holland, Amsterdam). Pareto, V., 1896/7, Manuel d'economie politique (Rouge, Lausanne). Samuelsoo, P., 1971, Foundations of economic analysis (Atheneum, New York). Smale, S., 1973a, Global analysis and economics I; Pareto optimum and a generalization of

morse theory in 'dynamical systems' (Academic Press, New York). Smale, S., 1973b, Optimizing several functions, Proceedings of the Tokyo Manifolds Con­

ference (to appear). Smale, S„ 1974, Global analysis and economics IIA; Extension of a theorem of Debreu,

Journal of Mathematical Economics 1,1-14.

B

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Journal of Mathematical Economics 1 (1974) 119-127. © North-Holland Publishing Company

GLOBAL ANALYSIS AND ECONOMICS IV Finiteness and stability of equilibria

with general consumption sets and production

S. SMALE University of California, Berkeley, Calif. 94720, U.S.A.

Received March 1974

In the spirit of Debreu's work on economies with a finite number of price equilibria, it is proved that under generic conditions, economies contain only a finite number of price equili­bria when consumption sets are permitted to contain zero amounts of certain commodities. Also similar results are proved for economies with production (the last can be compared with works of Fuchs and Mas-Colell).

1. Introduction The goal is to prove a theorem of the Debreu (1970) type which includes

the extensions of Smale (1974), also referred to as IIA, and also permits the introduction of production and equilibria where a consumer does not neces­sarily consume every commodity. The conclusion will be of the same type; the number of equilibria is finite and each is stable for perturbations of the economy under generic conditions.

Production has been already introduced into this theory by Fuchs (1972) and Mas-Colell (1973). However, our results don't use implicit or explicit convexity assumptions on either the utility o production functions.

Section 2 extends our results of IIA to include finiteness and stability of price equilibria for almost all economies where commodity vectors of the price equilibria might be on the boundary dP of commodity space. This corre­sponds to the highly reasonable possibility that at equilibrium, some consumers don't consume every commodity. Such equilibria are excluded in Debreu (1970) and IIA. In section 3, similar results are obtained for rather general consumption sets replacing the positive orthant as commodity space. Section 4 introduces production sets and puts them into a space. Section 5 is devoted to a statement of the general theorem including production. The last section gives a proof.

2. The goal of this section is to prove the finiteness and stability of price equili-

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120 S. Smale, Global analysis and economics IV

bria for a pure exchange economy where commodity vectors may lie on the boundary of commodity space. In other words it is allowed that an agent at equilibrium may consume a positive amount of only some commodities.

The previous papers in the subject including Debreu's (1959) original one and my account IIA did not take into account this possibility. As we shall see here, however, the same conclusions apply in the more general context. Let us be more precise.

Commodity space P for us will be the closure of the 'open' commodity space P. Thus

P=x = (xl *')eJ?V>0, P=x = (x\...,x')eR'\x>^0.

Utility functions will be C1 maps ut: P-* R (compare IIA) which satisfy a strong no critical point hypothesis as follows:

A coordinate subspace of R' is a linear subspace obtained by setting certain coordinates equal to zero. We recall that a critical point of a function is just a point where the derivative is zero (or equivalently all the partial derivatives are zero).

Hypothesis. Strong no critical point hypothesis. For any non-trivial coordinate subspace aid R', the restriction of ut: P -» R to a, n P has no critical point.

Let U be the space of all C2 functions on P which satisfy this hypothesis and impose the Whitney topology (see IIA) on U.

In this section an economy is a point (r, u) of (P)m x (U)m = S. Here r = 0"i. • • •» I'm) is an initial allocation with r, eP and u = (uu ..., um) is an m-tuple of utility functions with each u, e U.

The space of states (of an economy) is !f = (Py x S+, where

S+ = peR'\p = (p1 A P ' £ 0, £(A = 1. A state is a pair (x, p) where x — (xlf..., x j is an allocation and p is a price system.

A price equilibrium of an economy (r, u) is a state (x, p) which satisfies

(El) Z*,-Ir,; (E2) for each i, x, is a maximum for u, on the budget set

Bud = xEPlp(x)£prfi.

Theorem. For an open dense set 0 of economies in (P)m x (17)", the set of price equilibria is finite and this set is continuous in the parameters (r, M) of the econ­omy in C.

Suppose that (x, p) is a price equilibrium for the economy (r, u). Let a(x) -« a = (a , , . . . , a*), where a» is the smallest coordinate subspace containing x . Then (E2) implies

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S. SmaU, Global analysis and economics IV 121

(F2) for each i, xt is a maximum for u, on the set Bud,, - x e P r. <x,|p(x) £ p(r,).

In turn (El), (F2) imply the following conditions (using the strong no critical hypothesis on the ut):

(Gl) 2>««Ir,; (G2) p(xt) = prj), i= 1 m-1;

(G3) for each i there is a number kt > 0 such that A,DW,(*i)l., = A , -

Here the vertical bar means restriction and the price system p is interpreted as a linear map p: Rl -* R, p(y) = £-i/>y.

We will say (in this context) that (x, p) is an extended price equilibrium or (x,p) e Ett(r, u) if it satisfies (G1HG3). Note that first this is consistent and generalizes the definition in II A, and that as we have just seen.

Proposition 1. Let (x,p) be a (classical) price equilibrium for an economy (r, u). Then (x, p) is an extended price equilibrium, (x, p) e E%x(r, u).

We now prove the finiteness and stability theorem for E„(r, u). For this fix a = (at O with <t, coordinate subspaces of R1 and consider the subset £„(/•, u) of elements of Etx(r, u) which satisfy (G1)-(G3) for this choice of (otj a j . This will be sufficient for the study of Etx(r, u) since there are only a finite number of choices of a and

En(r, u) = [JE,(r, u). a

Define a map

* . . . - * . : ft ( J ' W x S * - n ( 5 + n a , ) x S +

by 4><(x,p) = (AiDuMI . , , . . . , KX>um(xJ\^,p) where A, > 0 is chosen to normalize DujOtj)!., (by our hypothesis this last expression isn't zero). Define

<*«= 0>i, ••••>'»./>)£ JJ(S+ noc,)*S+\ Xa "p\„

some Xt > 0, each i.

Then of course (x, p) satisfies (G3) if and only if <f>,(x, p) e A,. Proceeding as in IIA one obtains:

Proposition 2. Let Yu be the set ofuin (l/y such that <f>,t u is transversal to Au. Then Y, is open and dense in (£/)".

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122 S. Smale, Global analysis and economics IV

Let S?t = (x, p) e y\xt e a, and then define m m

Zr,« = (*./>) e PJZx, = £r„ />(*() = P(rt), i= 1 m-1 and

r . - ( r , * ) e ( i » r x ^ J J 6 l r , . . M

Proposition 3. (a)r„ w a submanifold of (P)mxy„ of dimension m/+£dim a,-m. (b) 77^,: (T5)" x y , - » y , restricted to T.is regular.

The proof goes again as in HA [but for (a) first set x( = 0 instead of r j . Similarly we have:

Proposition A. Let ue Ya. Then Zt%u= (r,x,p) e f j ^ u(x,p) e Aa is a submanifold of T, of dimension ml.

Arguments of IIA apply to yield an open dense set 0 c S so that the con­clusion of the main theorem holds for extended price equilibria, for fixed a, when(r, u)eGi.

We then have immediately that local uniqueness and finiteness for the price equilibria themselves. But as A. Mas-Colell first pointed out to me, this doesn't quite give the stability for the price equilibria. There may be a discontinuity whenever the following situation is encountered.

Condition. There exist distinct (x,p), (x, p) e EtI(r, u) such that MX*I) = u,(xt) for some i, or Etr, u) r> Ef(r, u) # <f>, some a / /?.

The final step in the proof of the theorem is the remark that the set of (r, u) in 6 x c g for which this condition fails is open and dense in 0,.

We end this section by noting that if each utility u, satisfies the condition below, then the finiteness and stability hold for almost all r (without varying u) - compare IIA.

Condition. D2«,|a( „ f(xt) is negative definite on the kernel of Du,|„( „ ,(*,) each xt and coordinate subspace a, of R'.

3. The goal of this section is to prove the usual finiteness and stability statements

of price equilibria in a pure exchange economy with general consumption sets Xt. These ideas are used also for economies with production.

A consumption set Xt for the ith consumer will be described in terms of C functions, r £ 1, /,*, k = 1, . . . , N,, and M, as follows:

Xt = xe *'|/*(*) ^ 0, k = 1, . . . , Muft\x) = 0, jfc = M,+ l,... ,Ar (.

The following regularity condition on the/j* will be assumed throughout.

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S- Smale, Global analysis and economics IV 123

Hypothesis. If x e Xt and B is the set of k such that ft\x) = 0, then the vectors D/i*(x) are linearly independent as k ranges over B.

A special case of Xt is the usual commodity space P (section 2) where Nt =» M, =/.and/,*(*) = **•

For simplicity in the following we will exclude the equality constraints or take M, = N,. However, these could be included with very little change in the wording.

To see the structure of the Xt better, fix x, e X,. Let BJx,) « B, c 1 A be the subset of indices such that k e B, if and only if ft (x^ = 0. Furthermore let p, = xe Xt\ft

k(x) = 0 for k e B,. Then /?, is a closed set and a submanifold of R' in a neighborhood of x,. This fact follows from the non-degeneracy hypothesis on the /,* and the implicit function theorem. The dimension of fit is equal to / less the number of elements in B,. The tangent space of /?< at x, is given by

T.,(fid= fl KerDyftx,).

That a function ut: Xt~* R is C1 means that u is the restriction of a C1

function defined on an open set containing X, (compare IIA). We also use the following no critical point hypothesis:

NCP Hypothesis (1st form): Let x, e X,, 0, be as above. Then x, is not a critical point ofw, restricted to f, (unless dimension)?, = 0!).

NCP Hypothesis (2nd form): Du,(x,) is not a linear combination of the Vf,k(xd,keBt.

Let C\Xt, R) be the linear space of C2 functions «,: X, -» R endowed wish the Whitney topology (compare IIA) and denote by Ut = Ut(X^ the subset of C2(X„ R) of those functions satisfying the NCP hypothesis.

In this context, we assume the Xt are fixed consumption sets for the m agents i = I , . . . , m; fix the fi as well. An economy is given by an initial allocation r = (ru ..., O , an element of the Cartesian product Y\i^t> aQd a ° »»-tuple of utility functions u = uu ..., O e Yii^i- (We have suppressed the X, and /(*.) The space of economies is thus the Baire space Q = Ylt^ix Yl^f

Perhaps it would make more sense to allow greater latitude for the initial allocation r = (rlt..., r^) in the definition of economy. As Mas-Colell has pointed out this becomes important in the case of equality constraints in the consumption sets. In his paper (1973) he develops this direction.

The space of states of an economy (r, «) in this context is S? = Y[<*i * S+ o r

the set of pairs (x,p) where x is an allocation, x = (x , , . . . , x j x, e X, and p is a price system. A price equilibrium of (r, u) is a state (x, p) which satisfies

(El) Lx, = £r,;

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124 S. Smale, Global analysis and economics IV

(E2) «, on the budget set Bvd = xeX,\p(x)Zp(rfi

has a maximum at xt e Bud.

Theorem. There is a dense open set 6 of economies in Yit^ix FL^i w'tn tne

following property. If (r, u) e G then each price equilibrium for (r, u) is locally unique; the set of price equilibria is continuous in (r, u) e C. Furthermore if the set

* 6 l M > i - I / i

is compact, then the set of price equilibria for (r, u) is finite [assuming (r, u) e <P]. Here a price equilibrium is said to be locally unique if it is the only one in some

neighborhood in the space of states. The proof of the theorem of this section is similar in part to the proof of

the theorem of the previous section. Note that from our critical point condition on the utility functions, it follows

that Xi in (El), (E2) satisfies p(x() = prt). Thus in (E2) we may replace the inequality sign by an equality sign.

Let (x, p) be a price equilibrium for (r, «); let B(x) = B = (B^ Bm) and fj(x) = p = (/Jj,.. . , pm) where the B,, /?, are as in the earlier part of this section. Then xf e p, and from (E2) it follows that xt is a maximum of u, on p, r\ (Budget set) or that x, is a maximum of u, on

x<=p,nX,\p(x)=P(r,). It follows from a version of the Kuhn-Tucker theorem [Intriligator (1971)]

that there are Xt £ 0, /i* 1 0, for k e B(, not all zero such that for each i = 1, . . . , m, p = -i,Dw,(jr,)+£»£B( fi*Dft"(xi).

Lemma. Neither side of this equation is zero.

Proof of lemma. Suppose the lemma were false. Apply the right hand side as a linear map to 7",,(/?,) to obtain A,Du^J = 0 on /J,. But by the no critical point hypothesis, this implies that A, = 0. Therefore the Dffod are linearly dependent as k ranges over Bt. This contradicts the regularity hypothesis on the/,*. The lemma is proved. The same argument shows that the A,, /if are unique.

We have that if (x, p) e & is a price equilibrium for the economy (r, u) with B„ P, defined as above then (x, p) satisfies:

(Fl) 2>«-Ir,; (F2) pixt) = prd, ' = 1 m-\; (F3) /,*(*,) -=0, k e Blt each i = 1 m;

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S. Smale, Global analysis and economics IV 125

(F4) for each i = I , . . . , m, there exist A, £ 0, p\ £ 0, not all zero, for k e B„ with

p = A ( D H ^ ( ) + I ^D/,*(X,)-fti, Fix B = (Blt..., BJ), denote by \Bt\ the number of elements of Bt, let

S(v) = vl\\v\\ for veRk and S(Rk) = the unit sphere in Rk. Define a map

*.» = <I>B-- n ^ x n ^ x n *<*s+ - n *"" xos+)m+i

by [(A,), (Ml), i*d,P] - K/.V,)), 5(AfDH^()+I>jD/(

k(x^,p]. Here the quantities indexed by / in parentheses are meant to run from 1 to m\

Let A = 0 x A where 0 is the origin in Y[,R "" and A the diagonal in (5*)"+1

so that I J l(/l) is precisely the set of (A, n, x, p) which satisfy (F3) and (F4).

Proposition. There is a dense open set 0 of (U)m, such that if ueS, then &: B 's transversal to Ox A.

If B, is empty for each i = 1 m this proposition is the same as Pro­position 2 of the previous section. But then the B, contribute to make ^„,« even more likely to be transversal.

To finish the proof one defines yB = (x,p) e y\xt e /?, and lets Ir< B, rB be defined analogously to I„ „ r, of section 2.

The analogues of Propositions 3 and 4 are true. One needs only to remark that the restriction of the projection

is an imbedding; and this amounts to the uniqueness of the A„ /** in (F4). The proof of the theorem follows.

4.

The goal of this section is to define a space SF of production sets Y of a single producer. Toward this end let G be the space with the Whitney topology of C1

real valued functions from Rl to R. Fix a positive integer N large enough to account for the technology of the producer in what follows (e.g., take N = /). and denote by GN the Cartesian product of G with itself N times.

Next define 3~ to be the set of g = (g 1 , . . . , g") e G" with the property: for each x e R', let Ax = k\g\x) = 0; then the vectors Dg\x), keAx are linearly independent.

Note that T is open and dense in G". lfgeG" let (the production set) Yt - xeR'\g\x) £ 0.

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126 S. Smalt, Global analysis and economics IV

Finally say that g,hsf are equivalent if Yt = Yh and let it be the quotient space of $~ with the quotient topology.

Proposition. & is Hausdorff and a Baire space (a countable intersection of dense open sets is dense).

The proof of this was developed in a conversation with A. Weinstein and M. Hirsch. One merely shows that the quotient map 3~ -» it is open and closed. In any case we don't use the proposition.

We remark that equality constraints can easily be added to define Yt (compare section 3) and the rest of the results in the paper hold in that generality.

5.

For some background on the private ownership economies we discuss here, one can see Debreu's book (1959).

In the rest of the paper, we take as fixed components of our economies the consumption sets X( of section 3 as well as their representation by the functions /,*. Also supposed given once and for all are numbers 0,„ ^ 0, i = 1 , . . . , m, a = 1 n, Y,fit* — 1> where 0ta is to be thought of as the share of the ath firm owned by the rth consumer.

An economy (suppressing the Xt and 0^) is a point e = (r, u, g) in (/?')" x YliUtx&y~ & w ' t n ^ U* defined as in section 3, &* as in section 4 and n producers represented by g = (gu .. .,g„).

The state space of an economy e = (r, u, g) is

s'.-fU.xflr.X'5*, i - i «= i

where Y. - Y*. - yeRl\&yj $ 0, k = 1 N).

A state (x, y, p) e Sr*t is called a price equilibrium for the economy e provided: (Ei) 2>i = 2>.+2><;

(total consumption = total production + total initial goods); (E2) xx is a maximum for u, on the budget set of the ith, consumer,

Bud = x e X,\p(xt) £ P(rd+Z0.M i

(E3) y, is a maximum for p restricted to Yt (profit maximization).

We can now state our main result. Main Theorem. There is a dense open set 6 in 8 with the following property.

IfeeO then each price equilibrium is locally unique; the set of price equilibria is continuous in eeO. Finally ife e 0 and the set of states satisfying (El) is compact, then the set of price equilibria is finite.

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S. Smale, Globed analysis and economics IV 127

6.

The proof of the Main Theorem (previous section) is just an elaboration of the proof in section 3. Here we give some indications. Suppose (x, y,p) is a price equilibrium; let p„ Bh etc. be defined as in section 3. Then property (E2) of section 5 implies the following for each i = 1 , . . . , m,

(E2a) p(xd = p[rd+10„py.);

(E2b) there exist non-negative numbers not all zero, A,, /if, for k e Bt such that /^A.D^xJ+E/ i jD/rt* , ) .

ktBi

Let A, « k\gi(yj *= 0). Then from (E3) of section 5, we obtain for each « = ! , . . . , « . (E3a) «*0g =. 0, k e A* (by definition!); (E3b) there exist oil £ 0, not all zero, for k e A, such that

P= Z o M O v ) . * M .

Let A=(A1 An), and B = (Bx Bm). Say that (x, y, p) e E AtB (r, u, g) if (El), (E2a), (E2b), (E3a), (E3b) are all satisfied. From what we have seen every price equilibrium is in one of the EAt B(r, u, g). There are clearly only a finite number of possible pairs (A, B). We restrict attention to one of these sets EA , BO", «• g), and claim the conclusion of the Main Theorem (section 5) holds for this set. But this argument goes as in section 3, expanding the domain and range of the map \I/Ui B, by adding factors to take care of the oj, etc. No new ideas are involved but the number of symbols needed increases. One should make the remark that the derivative conditions and transversality properties used on g depend only on Yt and not the particular representative chosen.

References Debreu, G., 1959, Theory of value (Wiley, New York). Debreu, G., 1970, Economies with a finite set of equilibria, Econometrica 38, 387-392. Fuchs, G., 1972, Private ownership economies with a finite number of equilibria, mimeo.

(Paris). Intriligator, M., 1971, Mathematical optimization and economic theory (Prentice-Hall,

Englewood Cliffs, N. J.). Mas-Colell, A., 1973, On the continuity of equilibrium prices in production economies: The

linear activity model case, mimeo. (Berkeley). Smale, S., 1974, Global analysis and economies IIA; Extension of a theorem of Debreu,

Journal of Mathematical Economics 1,1-14.

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Journal of Mathematical Economics 1 (1974) 213-221. © North-Holland Publishing Company

GLOBAL ANALYSIS AND ECONOMICS V Pareto theory with constraints

S. SMALE University of California, Berkeley, Calif. 94720, U.S.A.

Received March 1974

1. Introduction We consider the problem of optimizing several functions subject to constraints

defined by inequalities. For one function, this is a problem of maximization akin to the Kuhn-Tucker theory. The point of view here, however, emphasizes calculus and ignores convexity. The results of sections 2 and 5 extend the theory of our paper 'Optimizing several functions', or OSF, [Smale (1973)] to the case with constraints.

The problem is motivated by mathematical economics. In particular, the abstract results of section 2 are applied to a pure exchange economy. The con­straints in this case are given by the condition that an economic agent doesn't possess a negative amount of any good. In other words, allocations x = (x,, . . . ,xm) of resources to m agents put the commodity vector xt in the space

P = (xi xl)eR'\x'2 0.

In this way we get first-order conditions for Pareto Optima and price equili­bria in sections 3 and 4. In case these allocations give each agent a positive amount of each good, these results coincide with those in in [Smale (1974a)].

In section 4, a complement to the 'Basic theorem of welfare economics' is given. This 'basic theorem' gives conditions as to when a Pareto Optimum is supported by a price system. Section 4 gives conditions for the uniqueness of this support­ing price system.

As is usual in these papers, the analysis doesn't depend on the utility functions, but only on the preference relations defined by the utility functions.

2. We consider the following problem: 'Optimize' real C2 functions uiy... ,um

defined on an open set W c Rl subject to constraints given by conditions of the

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form gp(x) ^ 0, f! = ! , . . . , » . Equivalently, find the Pareto Optima of the function « = (M, u„) on the set W0 = xeW\ gfx) 2s 0, P = 1 n. Here the functions g,: W -* R are supposed C' and sometimes C2. Our analysis applies equally well to the case of an arbitrary manifold for W.

In this section the 'first-order' theory is studied. We say that a C1 curve 4>: (a, b) -» Wis admissible (relative to the given constraints gf) provided:

d (a) — u,(<f>(t)) > 0 all / e (a, b) and / = 1 , . . . , m, at

(b) for each p = ! , . . . , « ; if gp(<t>(t)) = 0, then there is a neighborhood J of /0 in (a, b) so that gfi(<f>0)) is non-decreasing for / 6 J.

Note that (b) implies that whenever gf(<t>(t)) = Othen

An admissible curve can't leave W0. We say that x e 0 provided no admissible curve passes through x. Thus in the

case of no constraints, 6 is the same 6 as in OSF, the Pareto set (first-order). Of course we are concerned with the part of 6 which intersects W0.

It is important to remark that if x e W0 is a (classical) Pareto Optimum for the restriction of the u,: W0 -* R, then x e 9.

Theorem A. Suppose ut, gf are functions on Was above. Let xeO; then define B = P e 1 , . . . , n | gf(x) = 0. Then there exist non-negative numbers Xt, \it, i = 1, . . .,m,Pe B not all zero, such that

(♦) £A,D«((x)+ £ H,Dg,(x) = 0.

For the converse we need a non-degeneracy condition on the functions gt. Non-Degeneracy Hypothesis on the constraints gt,..., g„: For each x e W, let

Bx = the set of fi for which gf(x) = 0. Then for each x e W, the set of Dgf(x) for ft e Bx is linearly independent.

Theorem B. Suppose that the functions ut,gf are defined on W as above and that the constraints gt satisfy the Non-Degeneracy Hypothesis. For xe W let B — Bx be the set of P such that gf(x) = 0; suppose there exist non-negative numbers A,, \if, i = 1,. . .,n,Pe B not all zero such that

(.) X < £>«,(*)+ £ »,Dg,(x) = 0.

ThenxeO.

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S. Smale, Global analysis and economics V 215

We give first the proof of Theorem A. If *) fails, then all the vectors Du,(x), Dgfi(x) lie in the same open i space in R? (compare OSF). Let rj be a non-trivial element in the cone spanned by these vectors, and t -> 0(f) a curve in W such that <b'(t0) = r\. Then Di/jWoWf) > 0, each i, and Dgf(<f>(t0)Xr]) > 0, each fi e B, so <l> suitably restricted is admissible; so x $ 9. This proves Theorem A.

Before proving Theorem B we point out that the Non-Degeneracy Hypothesis on the constraints together with (*) implies that at least one of the A, is not zero.

Towards the proof of Theorem B suppose that <f>(t) is curve through x with say <f>(t0) = x. Apply (*) to <f>'(to) to obtain

X Xt D«,<x) <f>Vo) + X n, Dg,(x) 4>'(t0) = 0. fitB

Since X, Dui(x)<f>'(t0) and \it Dgf(x)<f>'(t0) are all non-negative, they must all be zero. But by the remark in the preceding paragraph some A, are notO; hence for that /, DMJ(X)(0'('O)) = 0. This implies that <p is not admissible. Hence xeO. This proves Theorem B.

Remark 1. In case x in W has the property that gf(x) > 0 all P, Theorems A and B coincide with the first-order proposition of OSF (the no constraint case).

Remark 2. If m = 1, and one is maximizing a single function, Theorem A amounts to some kind of version of the Kuhn-Tucker Theorem [Intriligator (1971)]. In fact, if the Non-Degeneracy Hypothesis on the constraints is met, one may take Xt = 1 and the condition (*) becomes

DM,(*)+X H, Dg,(x) = 0 , n, > 0,

3.

We apply the abstract theory of the previous section to the case of a pure exchange economy.

We recall the setting of this pure exchange example. Let P = x e R1 ] x = (JC1, . . . , x1), x* > 0 denote commodity space; suppose u,: ?-* R is a C2

function representing the utility function of the ith consumer, i = 1 , . . . . m. Fix the total resources reP, P the interior of P. Then the state space of this pure exchange economy is

W0 = x = (xl,...,xJe(P)m\YJxl = r.

We have also u, defined on W0 by ufa) = u/x,); thus we have obteined the situation of section 2 with «f: WQ -* R,i = 1 m. The constraints are given

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by xf ^ 0, each i = 1, . . . , m, k = 1, . . . , (. Thus x), the amount of the kth good allocated to the ith consumer, is non-negative. We can put ourselves literally in the situation of section 2 by defining W = (xj xm)e Rl | £ x , = /. Then W => W0, and we can assume that the functions u, are extended in some way to W. Of course M,outside of W0 is irrelevant for most things.

An allocation x e Wo >s Pareto Optimal (in the usual sense) if there is no yeW0 such that u£y) ^ u£x), all /, and u/y) > up:), for some/

As in section 2, admissible curves are defined in W and 6 is the set of x with no admissible curve passing through x. If x is Pareto Optimal then x e 0.

To fit in more simply with tradition we make a monotonicity assumption throughout on the utility functions (this could be easily relaxed or abandoned using negative prices).

Monotonicity: The utility function u,:P-*R of the /th agent satisfies D w ^ e P e a c h x e P .

Theorem. Let x e WQ. Then xeO if and only if there are positive numbers pk, k = \,...,(, £Q>*)2 = 1 and X, > 0, i = 1 m, such that p* S* A, Du( (*i)]f oil i,k, equality holding when xf # 0.

Here D« X*)]* " '^e fcth component of the vector Du,(Xj) /n /?*.

Corollary 1. Ifxt e ^ a//1 //re« xefl if and only if there are /» = (/»',..., p*), Xt> 0,as in the theorem with

p = A, Du^x,), all i.

This is a classical condition and this Corollary is in III, as Proposition B of section 1.

By the remarks preceding the theorem, the theorem exhibits a necessary condition for a (classical) Pareto Optimum.

The reader may illustrate the theorem with points on the boundary of the Edgeworth Box.

The rest of this section is devoted to the proof of the theorem. To put our present picture into the language of section 2, define the constraints

gy.W-+Rbyg*,x) = xk,,k= \,...,A&ndp= \,...,m. Thus

Wo = xeW\gkf(x)>0, a l l* ,0.

Lemma. The Non-Degeneracy Hypothesis on the constraints of section 2 is satisfied.

This follows easily from the hypothesis that every component in R1 of £ x, is not zero.

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5. Smalt, Global analysis and economics V 217

Theorems A and B of section 2 apply to yield that for x e rV0, x e 9 if and only if there exist non-negative A,, /if, with the X( not all zero such that

lA,D»,(*) + 5 > D r f ( * ) - 0 .

Here the indices (/, k) on /if range over the set such that x) = 0. We apply this equation to an element v e (R')m, » = ( r , , . . . , DJ, £ i>( = 0.

Choose t> = (0, . . . , 0, v, 0 , . . . , 0—v, 0 . . . 0), with v e R' in the »th place and - v in the kth place. Then

for all S e Rf. Note that given A: there is somei such that x\ # 0, since ]T *, € P. Tentatively

define for given A:, p* = A, Dui(A:j)]i for some such /. By the previous equation pk is well-defined, i.e., is independent of the choice of/.

Since at least one of the Xt is not zero and Du£x,)]k > 0, each /, A;, by the monotonicity hypothesis, it follows that some pk > 0. (It turns out eventually that each pk > 0.)

We now normalize/? = (p1, ...,/>*) by multiplying by a positive constant so that £ (p1)2 = 1, keeping the same notation for />*. Multiplying the X, and p\ by the same constant keeps our equations in force. We keep using Xlt /if for the new numbers.

Finally note that in case xf = 0, our above equation yields that pk = A, DQiixdY+ri or pk—XtT>u^x^]k > 0. The converse goes similarly and the theorem is proved.

4. We add prices explicitly to the treatment in the previous section. The notation

of HI is followed; however the u,: F -* R has no bar as in the previous section. While we recall the basic definitions one can see III for more details.

An economy (r, «) is given by w-tuples of initial allocation, r = (r t, . . . , rj, r, e F, r+0, J f j e F , and utility functions, u = (u x , . . . , «„)• The states of the economy are points (x, p) e £f = (J5)* x S*, with x = xl,..., xj, (xt e F)t an allocation and a price system p = (p1 />'), Pl > 0, £0>')2 = 1- A P"ce equilibrium of an economy (r, w) is a state (x, p) such that £ x, = £ r„ and on the budget set Bud = xt e F \ pixt) = p(r,), H, has x, as a maximum.

Theorem A. Jf(x,p) is a price equilibrium for the economy (r, u), then there exist X, > 0, i = l,...,m, such that

p* ~£ Xt Dufajf, oil i = l,...,m, k = 1, . . .,€,

with equality holding whenever x * # 0 .

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Recall DufaD]k denotes the fcth component o/D*xj) in R1.

Corollary. In the theorem suppose each xteP (at the equilibrium each agent has a positive amount of each good). Then there exist Xlt..., Xm, all positive, with

p = Xt DH|(Xj), I = I , . . . , m.

For the proof of Theorem A let x be as in the theorem, with x = (xt,..., xm), xt e P. Now let fit be the set of indices k such that x\ = 0. Denote by «t the co­ordinate subspace of R1 defined by setting all the kth coordinates equal to zero, where* e/?,.

It follows from the budget maximization condition of price equilibria that xt is a maximum of u, on the set

xe«,nP\p(x)=p(rd.

Thus xt is a solution of the problem: Maximize u( on the set of y e R1 subject to yl > Q,p(y) = p(rd-

It follows from Theorem A of section 2 (for example) that there exist non-negative numbers A,, j*f, for k e fit and a such that

X, D«^i ) e I /i? Dgfrxd = op,

where gf(*,) «= x\. It follows that, since g) and p satisfy the constraint condition at xt, Xt> 0.

From this and the monotonicity hypothesis, it follows that o > 0. Thus we may suppose a = 1. The rest follows as in the proof of the theorem of section 3.

We will say that a state (x, p) is a first-order price equilibrium if it satisfies the condition of Theorem A.

Theorem B. If(x, p) is a first-order price equilibrium for the economy (r, u) then xeQ (/.«., x is first-order Pareto Optimal'). Conversely ifx e 9 (of section 3), then there exists a price system p such that (x, p) is a first-order price equilibrium.

Remark 1. Note that under convexity hypotheses, first-order conditions are complete conditions. Thus in this case Theorem B can be translated into classical terms.

Remark 2. Theorem B generalizes Propositions A and B of m , section 4, where it is supposed that each xt is in the interior, P, of P.

We pose the following problem: Let (r, u) be an economy, with r = ( r , , . . . , r j an initial allocation, u = (M,, . . . , uj an n-tuple of utility functions,

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S. Smalt, Global analysis and economics V 219

u,: P -» R. Let 0 be the Pareto set for the map W -»/?*, x -* (w,(x),..., um(x))t w i t h l F - x 6 ( P r | I x . - = I r , .

Problem: Let xe 9; when is there a unique price equilibrium (x,p) of the economy (r, «) ? In other words, when is there a unique price system supporting the Pareto Optimum x ?

There is an extensive literature on the subject of existence, see, e.g., Debreu (1959) and III. We thus consider the question, given xe 9 when is there at most one/> with (x, p) a price equilibrium ? For the answer we introduce a condition:

No isolated community condition: We say that at x e 9 (or any allocation for that matter) there are no isolated communities provided there is no non-trivial partition of the agents 1 , . . . , m) such that agents which are partitioned into strictly different classes have no common commodity at x.

In other words, let S, <= I , . . . , m) be the partitioning, Smn S, = 4>,<t # 0 and u 5 , = 1 , . . . , m\ then when ieSa,jeSf, a # p it follows that either JCj or Xj is zero.

Theorem C. Suppose xeQ satisfies this condition, there are no isolated com­munities at x. Then there is at most one price system p such that x,p) is a first-order price equilibrium.

The proof is straightforward using the earlier parts of this section.

Corollary. A Pareto Optimum with no isolated communities has at most one supporting price equilibrium.

We recall the assumptions on the utilities that are used here:

(1) Each u-,: P - R is C.

(2) For each xeP, Du,x) e P.

5. We consider second-order theory with constraints. This section was rewritten

after Smale (1974b) and Wan (forthcoming) to sharpen and complete the results of the first version, and also to extend Corollary 2 of Wan.

Suppose that we are given an open set W in some Cartesian space /£* (one could take W to be a manifold as well). The problem is to optimize C2 real valued functions « , , . . . , wm on W subject to the constraints that the C2 functions gx, ..., g„ be non-negative. In different words, find conditions for a point x* e W0,

W0 = x e W | g.x) > 0, a = l «,

to be a local Pareto Optimum of the functions ux um restricted to W0.

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A point x e W0 is called a strict optimum for u^,..., uM on W0 provided: if y e W0 and w^) > ut(x) for all i, then y — x. Thus, being a strict optimum is a stronger condition than being an optimum. A point is called a local optimum if it has a neighborhood N in W0 so that the point is an optimum for the functions restricted to N. A local strict optimum is defined similarly.

Let drVo = xErV\g.(x) = 0, « = ! , . . . , / » .

Theorem. Suppose x e d W0 is a local optimum for the functions ult.. ,,um on fV0. Then there exist numbers A , , . . . , k,*, / i , , . . . , /i, ^ 0, not all zero such that

XA iD«(W+I/i.D^W = 0. (1)

Furthermore, let it be given xedW0 and A, Xm, pl , . . . , /*„ ^ 0 not all

zero satisfying (/). If the bilinear symmetric form,

ZXiD2ui(x) + ZfiaD2gtl(x),

is negative definite on the linear space,

v e /?* | P A , grad w,(x) = 0 all i and v-fi, grad gt(x) = 0, all a,

f Aen * w a local strict optimum for u , , . . . , um restricted to IV0.

We give the proof. The first part of the theorem is a consequence of Theorem A of section 2. Now, given the functions ut,..., um, gt,..., gH on W define * : W -» /T + " by ft*) - («,(*) «„(*), * ! ( * ) , . . . , *„(*)). Then:

Lemma. A point xs8W0isa strict optimum for « , , . . .,umonW0 if and only ifx is a strict optimum ofty on W.

The proof is an easy consequence of the definitions. Now one applies the theorem of Smale (1974b), or, equivalently, Theorem B

of Wan (forthcoming), to the function ij/ to obtain the proof of the second part of the theorem.

Remarks

(1) It is no loss of generality to suppose that ga(x) = 0, a = 1 n. If this weren't true, one could reduce the number of constraints appropriately.

(2) The theorem clearly has implications for Pareto Optima and price equilibria in economic theory; but I have not detailed these conditions, even for a pure exchange economy.

(3) If the gradients of the ga at x are all linearly independent, then some kt is not zero.

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(4) Forn = 0, the result coincides with Smale (1974b) and essentially Theorem B of Wan. For m = 1, it is Wan's Corollary 2. In the original version of this section, the theorem was stated with stronger hypotheses.

References

Debreu, G., 1959, Theory of value (Wiley, New York). Intriligator, M., 1971, Mathematical optimization and economic theory (Prentice-Hall,

Englewood Cliffs, N.J.). Smale, S., 1973, Optimizing several functions (OSF), to appear in the Proceedings of the Tokyo

Manifolds Conference. Smale, S., 1974a, Global analysis and economics, III, Pareto optima and price equilibria,

Journal of Mathematical Economics 1,107-118. Smale, S., 1974b, Sufficient condition for an optimum, to appear in the Proceedings of the

University of Warwick Topology Symposium. Wan, Y.H., forthcoming, On local Pareto optimum, to appear in the Journal of Mathematical

Economics.

313

Dynamics in General Equilibrium Theory

By STEPHEN SMALE*

I find myself something of an outsider in writing a paper like this; and for a mathematician to write a paper with no technical mathematics is a real (but very constructive) confrontation with the prob­lem of communication. What I hope to achieve here is to convey some of my ideas on the relation between dynamics and the traditionally static, economic equilibrium theory.

Since I have been working in mathe­matical economics, I have been struck by the number of attacks on general equi­librium theory, on mathematical eco­nomics and even economic theory in general. Coming from a radical back­ground, I have much sympathy for some of the arguments brought forth.

I first want to say a few words on the subject of mathematics in economics and economic theory in general. The role of theory per se hardly requires defense; theory can give a deeper understanding of any subject, subtle relations are seen, inconsistent ideas are exposed, new horizons are revealed.

A criticism commonly made of economic theory is its failure to make predictions of crises in the country or to anticipate cor­rectly unemployment or inflation. One must be cautious in the social sciences about looking toward physics for answers. However, some comparisons with the phy­sical sciences seem profitable in connec­tion with the above criticism. In those sciences, where theory itself is in a far more advanced state, limitations can be

* Dcpt. of Mathematics, University of California, Berkeley.

seen in a similar way. For example a given individual human body functions according to physical principles; however no physical scientist would predict a heart attack. The physical theory gives under­standing of aspects of what goes on in the human body only under very idealized conditions. The physical theories even­tually play some role in the education of medical doctors, who can then say some things, some times about a patient's sus­ceptibility to a heart attack, preventive measures, and cures.

The economy of the world or even a nation is a very complex phenomenon, like a human body, involving a number of factors, both economic and political. It is no more reasonable to expect eco­nomic theorists to predict a nation's economic future than for a theoretical scientist to predict the future health of an individual.

Questions about the need for mathe­matics in economic theory have been raised. Indeed, the successes of mathe­matics in economics have not been nearly as impressive as in physics. Yet the notion of money and prices already intro­duces mathematics into economics; and the mathematics becomes deeper with the equation of equilibrium, supply equals demand. When one considers the equation, supply equals demand for sev­eral interdependent markets, the mathe­matical problem already takes on some sophistication.

What is special about "general equi­librium theory" as opposed to economic theory in general? For me the importance

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of general equilibrium theory lies in its traditions which are deeper than any other part of economic theory. These traditions, which of course derive from actual economic history, explain why equi­librium theory has played such a central role and possesses such depth in content. Many of the procedures and mathe­matical methods used in other parts of economics grew out of developments in general equilibrium theory. Also equilib­rium theory plays an important role in communication within the economics pro­fession. Since most economists are knowl­edgeable in equilibrium theory, they can understand new ideas more readily when presented in that context. Also, since equilibrium theory has been studied so much, new ideas introduced there show weaknesses and also strengths most quickly.

After all of this is said, equilibrium theory will eventually stand or fall, de­pending on its truth as an important idealization of actual economic systems or as a model with values of justice, of efficient distribution and of utilization of resources. As a normative theory, I find great merit in its decentralization features (Schumacher's popular book, "Small is Beautiful" expresses some of my senti­ment on decentralization). There are also, without doubt, basic failings in the theory which are well-presented in the economic literature, and there are some weaknesses which I wish to discuss presently. In fact these problems can be seen and under­stood especially clearly due to the well-developed structure of the theory; and one can use the body of general equilib­rium theory as a tool in developing alter­nate models. To me it would seem overly difficult to construct and communicate successfully any alternate economic theory without having first studied very thor­oughly equilibrium theory.

I would like to give some reasons why

I feel equilibrium theory is far from satis­factory. For one thing the theory has not successfully confronted the question, "How is equilibrium reached?" Dynamic considerations would seem necessary to resolve this problem. Another weakness is the reliance of the theory on long range optimization.

In the main model of equilibrium theory, say as presented in Gerard Debreu's Theory of Value, economic agents make one life-long decision, opti­mizing some value. With future dating of commodities, time has almost an artificial role. The model is reminiscent of John von Neumann's game theory. I like to make an analogy between "Theory of Value" and the game theoretic approach to chess. The possible strategies are laid out to each player in advance, paths in a game tree, or a set of moves, one move to each position that could possibly occur. Each player makes a single choice of strategy. The strategies are compared and the game is over. Of course, chess isn't played like this. And in a situation more complicated than chess, where life-long consumption plans replace strategies, I don't believe economic agents act that way either.

In fact even the very best chess players don't analyze very many moves and cer­tainly don't make future commitments. Their experience together with the envi­ronment at the moment (the position), some rules of thumb and some other considerations lead to decisions on the playing board.

My personal economic decisions are of a similar nature, from buying a book to buying a house; from a decision to travel to decisions about my job. Between one economic decision and another, there has been a real passage of time, circumstances have changed, and the new decision takes place in this new environment.

Long-run optimization would be im-

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practical, even if it were emotionally acceptable, because of barriers of com­plexity. Complexity keeps us from analyz­ing very far ahead. The amount of time involved in making a decision is an important factor, for a chess player or a purchaser. Dynamic models based on some kind of behavioral strategy could meet these objections.

Sometimes static theories pose para­doxes whose resolution lies in a dynamic perspective. Let me give an example from the theory of duopoly of the Cournot (or Nash) equilibrium solution. Under classi­cal hypotheses on the profit functions, consider such an equilibrium (rh r*) where n is the rate of production of duo-polist number i. Then agent l's rate of production maximizes profit among all such rates with r* fixed. On the other hand the solution is unsatisfactory because there is another state, say, (r\, r'-), where each duopolist with reduced production is taking a higher profit. In the actual dynamic world it is unlikely that the duo-polists would stay at the Cournot solu­tion, knowing that they would both be better off at nearby states.

It doesn't require explicit cooperation for these agents to move off the Cournot solution. In fact with flow of information and implicit threats in the context of a passage of time, one can argue that the duopolists will move to an optimal state from a Cournot state. But this resolution requires a real passage of time, that after each market move, the opposing duopolist has another opportunity to move. One can readily think of examples of duopoly where an increase of advertising is with­held by one agent knowing the other agent would match an increase and both would be worse off. James Friedman has written on this topic.

I feel that dynamics could also play a role in the resolution of Kenneth Arrow's paradox in the politics of social choice.

Politics and elections in particular are actually part of a dynamic process, ballot­ing being just a stage. The process looked at as a whole involves a number of moves such as a candidate's speech, a political ad, revising a position on some issue, etc. After each action of a candidate, other candidates have the option of taking an action of their own; voters' opinions evolve. For this reason, I would think that a dynamical model of the political process would give much better perspective than a static model of simple voting. In relation to this, the work of G- Kramer comes to mind.

We return to the subject of equilibrium theory. The existence theory of the static approach is deeply rooted to the use of the mathematics of fixed point theory. Thus one step in the liberation from the static point of view would be to use a mathematics of a different kind. Further­more, proofs of fixed point theorems tra­ditionally use difficult ideas of algebraic topology, and this has obscured the eco­nomic phenomena underlying the existence of equilibria. Also the economic equilib­rium problem presents itself most directly and with the most tradition not as a fixed point problem, but as an equation, supply equals demand. Mathematical economists have translated the problem of solving this equation into a fixed point problem.

I think it is fair to say that for the main existence problems in the theory of eco­nomic equilibria, one can now bypass the fixed point approach and attack the equa­tions directly to give existence of solu­tions, with a simpler kind of mathematics and even mathematics with dynamic and algorithmic overtones. In the last part of the paper we elaborate on this point.

Behind my own work on the questions of dynamics in economics, lies certain foundational work in the equilibrium theory in terms of calculus. The early

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dpelopment of mathematical economics, boluding the 19th century and even up w;World War 11, was largely in terms of calculus; it was no doubt the influence of game theory, and associated fixed point theorems that gradually reduced the de- ppdence on calculus. In Theory of Value &#;5959, Debreu wrote of the work of von Napmann and Oskar Morgenstern which freed mathematical economics from its traditions of differential calculus and compromises with logic. He wrote of the fadical change of mathematical tools from calculus to convexity and topology.

But then in his paper on a finite num- ber of equilibria, Debreu returned to aalculus tools; my own work, "Global Analysis and Economics," has been to try to'systematize the use of calculus in equi- librium theory. This can be justified on Several grounds. First, the theory is brought closer to the practice. With cal- culus, one has in the derivative a linear approximation. It is these linear conditions that are so basic to practical economic studies. Comparative statics depend on derivatives; the same is usually true for stability conditions; dynamic questions are more accessible via calculus, When general equilibrium theory is developed on calculus mathematics, not only is theory brought closer to practice, but greater unity is achieved. Furthermore, recent work on approximation by differ- entiable functions in economics gives further justification to the use of calculus.

Finally before moving to the construc- tive side of the question of dynamics in equilibrium theory, it is worth making a remark on the nature of goods. I t seems from our experience that it is important especially in modeling dynamics to put goods into two ideal classes, either com- pletely perishable or completely durable. The theory seems different for the two kinds. For example, Walras equilibrium seems suited to the perishable, continually

GLOBAL ANALYSIS 291

endowed class of goods, while for durable goods, the kind of equilibrium found in the fundamental theorem of welfare economics seems more appropriate. We hope that the rest of the paper makes this point clearer.

We discuss now the results of our paper, "Exchange Processes with Price Adjust- ment." This is a model of a market of durable goods; a particular example from personal experience of such a market is a weekend "mineral bourse" where agents with minerals and/or money meet to exchange, buy, and sell fine mineral specimens for collectors. Here one sees a truly dynamic process of exchange and price adjustments which converges to an equilibrium toward the end of the weekend of the "bourse."

An early work in some of the same spirit was done in 1962 by Frank H. Hahn, Hirofumi Uzawa, and Takashi Negishi. This approach is called a "non- tatonment," because in place of the Wal- rasian Tatonment, a sequence of actual trades takes place over time. On the other hand, our model differs from the 1962 work in that it is nondeterministic, the dynamics proceeds without an ordi- nary differential equation, long-run opti- mization is dropped, and a large body of examples is constructed.

The main result of our paper can be stated as follows:

"In a pure exchange economy, an ex- change price adjustment process, respon- sive to transaction costs, and which doesn't stop unless forced to by market conditions, converges to a price equilib- rium. There exists such processes starting from any state of any (pure exchange) economy.''

In the paper, mathematical content is given to all of the phrases used here, and the result is proved. Here we give a brief explanation of some of the terms used.

A "state" of an economy means a set of

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292 AMERICAN ECONOMIC ASSOCIATION MAY 1976

data characterizing the economy at a given time. Our use of the word "state" is akin to its use in physics. For example in a pure exchange economy, a state will consist of an allocation of the resources, or equivalently the set of goods of each agent and a price system.

A state in general will change over time, e.g., by exchange and price adjustments. Thus a "process" as used in the main result, means a passage in time of a state. Or equivalently, a process is a path (over time) in the space of states of an economy.

This process, to qualify as an exchange process, must satisfy economically justi­fiable conditions for exchange which are embodied in the following axiom. The exchange axiom for the process asserts that: (a) the total resources of the eco­nomy are constant (there is no produc­tion) ; (b) exchange takes place at current prices; (c) an exchange increases satis­faction of the participating agents; and (d) some exchange will take place pro­vided that it is possible consistent with (a), (b), and (c).

For the process to qualify as a price adjustment process (as in the quoted main result), we demand that it satisfy a price adjustment axiom denned in terms of a short run version of demand. A usual excess demand approach requires long-run optimization for the agents while our spirit is closer to that of behavioral strate­gies. At given prices and goods possessed one defines the infinitesimal demand of an agent to be the direction his preferences take him when restricted to his budget set.

The price adjustment axiom asserts that prices adjust in the direction of some weighting of the infinitesimal de­mands of all the agents.

A Walrasian price equilibrium depends on the traders' endowments. Thus if one allows a real passage of time, say an actual exchange to take place, and several such, this initial endowment becomes for­

gotten. Thus if one allows a "nontaton-ment" kind of time passage, one must replace a Walrasian price equilibrium by a different notion of price equilibrium.

As stated in our main result above, a price equilibrium is a feasible allocation and price system where, for each agent, satisfaction is maximized on his budget set denned relative to his wealth at equi­librium. Equivalently, a price equilibrium is an optimal allocation together with a supporting price system, as studied in the correspondence of the fundamental theorem of welfare economics.

A detailed mechanism of price setting and transactions is not developed, but it seems likely that the model is consistent with doing this explication.

Next we discuss some problems and results on the classical Walrasian model from the point of view of dynamics and algorithms. We prefer an alternate, well-known interpretation of the Walras model to that given in Debreu's Theory of Value.

Suppose that the goods are perishable, with labor a main example. One might envision a situation where each day an agent starts his economic activity with a fixed endowment of labor, or fish which won't keep. The next day he will have a new endowment of the same, but none left from the day before.

The consumption variables are the amounts of commodities consumed each day, of an agent. Thus both the endow­ment and consumption bundles in com­modity space will be interpreted as the rates of endowment (fixed over time) and consumption respectively. A com­pletely satisfactory dynamics (which isn't available) for this problem would con­struct and analyze paths over time in the space of states, that is commodity vectors for each agent and price systems (or sets of price systems). These paths should obey economically justifiable axioms of exchange and price adjustment, and

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VOL. 66 NO. 2 TOPOLOGY AND GLOBAL ANALYSIS 29J

probably should lead, at least under some economic conditions to a Walrasian equi­librium, starting from an endowment allocation and any price system. At the most satisfactory level, these paths should be given interpretations in terms of indi­vidual agent's actions in price offerings and purchases. In my view, a behavioral strategy for agents would be more desirable than decisions based on long run optimization.

Martin Shubik and I have worked on the problem in this setting without any definite success. On the other hand, it seems as if we might eventually obtain such a model with convergence provided a condition such as gross substitutes is satisfied. I should emphasize that I am not talking about any "tatonment" in this kind of dynamic, but rather an actual process, where agents are adjusting their goals, consumptions, prices over time to arrive at balanced budgets where the value of the rate of consumption equals the value of the rate of the fixed endow­ment for each agent.

I would like to turn now to some work carried out in my article "A Convergent Process of Price Adjustment and Global Newton Methods," which has more suc­cess on the mathematical side of the above problem.

One way of looking at this work is to first alter the Scarf Algorithm from find­ing fixed points to solving a system of equations, especially the system, supply equals demand in many variables. We define an ordinary differential equation, called a "global Newton," which is a ver­sion of (the altered) Scarf's Algorithm. Under rather general hypotheses (com­parable to those needed in the execution of Scarf's Algorithm), solutions of the global Newton converge to the set of solutions of the original system (e.g., supply equals demand). Combining this fact with methods of numerical analysis,

one obtains a different but analogous algo­rithm to that of Scarf. Morris Hirsch and I have implemented this effectively on a computer and are developing the algorithm from a numerical analysis point of view. It applies to systems of n nonlinear equa­tions in n variables, without hypotheses on the system of nonzero Jacobian, con­vexity, or monotonicity.

Let z(p) be the excess demand as a func­tion of prices p=(pi . . . pi) so that p* is an equilibrium if z(p*) = 0. Then the global Newton takes the form

(1) Dz(p)^=-\z(p), at

sign X = sign Determinant Dz(p)

Here Dz(p) is the matrix of first partial derivatives of z. If one takes X= 1 and uses Ruler's method of discrete approximation to (1) then one obtains Newton's method for solving z(p) = 0. Using equation (1), one can obtain a proof of the existence of economic equilibrium without using fixed point theorems or algebraic topology.

Consider the problem of representing a process of price adjustments by (1). Recall that the classical "tatonment" process has an embodiment in the equation

(2) dp dt - * ( * )

Arrow, Leonid Hurwicz and Herbert Block have shown that solutions of (2) converge to economic equilibrium under hypotheses on s such as gross substitutes. These hypotheses are substantial and are strong enough to imply the existence of a unique equilibrium. On the other hand Scarf subsequently showed that under classical properties on preference rela­tions, almost all solutions of (2) could oscillate for all time.

Now (1) can be considered as a modi­fication of (2), which involves more subtle intermarket relations and which will con-

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294 AMERICAN ECONOMIC ASSOCIATION MAY 1976

verge when (2) doesn't. In particular (1) will converge in the Scarf example.

Generally speaking, (1) converges starting from almost any initial price system on the boundary of the price simplex. One could also formulate the equation in quantity space and look for an interpretation of the process in terms of budget balancing actions on the part of the agents. The interested reader could pursue these topics further in the papers cited.

REFERENCES K. Arrow, Social Choice and Individual

Values, New Haven 1963. and L. Hurwicz, "The Stability of the

Competitive Equilibrium T," Econometrica, 19SS, 26, 522-52.

, Block and L. Hurwicz, "The Sta­

bility of Competitive Equilibrium II," Econometrica, 1959, 27, 82-109.

G. Debreu, Theory of Value, New York 1959. , "Economies with a Finite Set of Equi­

libria," Econometrica, 1970, 38, 387-92. J. Friedman, "A Non-Cooperative Equilib­

rium for Super Games," Rev. Econ. Studies, 1971, 113, 1-12.

H. Scarf, "Some Examples of Global Insta­bility of the Competitive Equilibrium," Int. Econ. Rev., 1960, 1, 157-72.

, The Computation of Economic Equi­libria, New Haven 1973.

S. Smale, "Global Analysis and Economics, IIA-VI," /. Math. Econ., 1974-75, /, 1-14, 107-17, 119-27, 213-21.

, "A Convergent Process of Price Ad­justment and Global Newton Methods," preprint, Berkeley.

-, "Exchange Processes with Price Ad­justment," preprint, Berkeley.

Journal of Mathematical Economies 3 (1976) 1-14. © North-Holland Publishing Company

GLOBAL ANALYSIS AND ECONOMICS VI

Geometric analysis of Panto Optima and price equilibria nader classical hypotheses

S. SMALE Cowies Foundation, New Bavtn, Com. 06520, US^t.

Received December 1974

L

The main goal is to understand the structure, especially local, of the set of Pareto Optima and price equilibria of classical economic systems in a differenti-able setting. We use very little of previous papers in this series, but do use calculus of several variables in a systematic way.

One result obtained is the structure of a submanifold on the set 6 of Pareto Optima. The same result is found for the set of price equilibria A. Then the Fundamental Theorem of Welfare Economics is given in a strong form.

Theorem. The map «> from A~* 0 which assigns to a price equilibrium, the corresponding allocation defines a diffeomorphismfrom the set of all price equilibria to the set of all Optimal Allocations.

A diffeomorphism is a differentiable function between manifolds with a differentiable inverse. In particular <p is one-to-one and onto; thus for example an optimal allocation has a unique supporting price system, and this association is smooth over all optimal allocations.

A local-analysis of 6 and A is made, which is used in work in progress on dynamic processes in economics.

The above results are obtained in models with production, but we have stopped short of analysis on the boundaries of the consumption sets. The analysis is an 'interior analysis', as far as consumption sets go and' we stay away from singu­larities of the production submanifolds.

In section 3 our attention is devoted to the Walrasian price equilibria (emphasis on an initial endowment). A little study is made of conditions for such an equilibrium to be catastrophic in the sense it is discontinuous in the parameters of the economy. Under these conditions the parameters of the economy are taken to be the endowment allocations.

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2 5. Smalt, Global analysis and economies VI

Some of the results here were proved under different conditions in Smale (1974a, b). In this paper, however, we don't make any genericity hypotheses or use transversality theory, at least in any explicit way. Convexity conditions are used instead. On the other hand the kind of arguments used would seem to make the convexity conditions more a matter of a convenience than a matter of principle.

Conversations with Gerard Debreu and David Fried have been very helpful in working these ideas out

2. We review our setting of a pure exchange economy. Preferences of the Jth

consumer are supposed to be represented by a C2 utility function u,:P-*R where commodity space P is taken as

P-(xl,...,xt)eRl\xl>0 each / .

L e t 5 - \y e Bf |2jfy>a- IandS+ - S n P. Define gt: P -♦ S by

*K*)-gradul(x)/||gradul(;t)||,

so that gfa) is the unit normal to the indifference surface of the preference relation at x. Then we suppose throughout that

0) gi(x)€S+, all xeP (differentiate monotonicity).

If Vx - v e Rl | vg£x) - 0, then the derivative, Dgfcc): R' -* Vx restricts to Vx to map Vx -* V„ as a linear map, yfa). It is easily seen that y^x) is a symmetric linear map. We further suppose

(2) 7<00 has only negative eigenvalues (differentiable convexity).

Here (2) is the same as the condition that D2u£x) on Ker Dufe) is negative definite [Smale (1974a)]. It is also equivalent to convexity together with Debreu's (1972) hypothesis of positive Gaussian Curvature. In particular (2) implies that u,-1(c, oo) is a strictly convex set for every real number c

The space of states of this pure exchange economy with m agents is then

W-xe(P)m\x - (* ! , . . . , x j , x,eP, J> , « j ,

where je.P is the fixed vector of total resources. Then x e W'aPartto Optimal (or simply optimal) if there is no y e W with u/(yD £ ufad all /, strict inequality one i. Also x is called a strict Pareto Optimum if uyD £ «<«(*i) all iimplies that y-x.

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S. Smalt, Global analysis and economics VI 3

As in my previous paper (1974a) we let 6 be the set of states x « (x, , . . . , x„)efV which satisfy the first-order condition: 9 => xe W \g,(xi) doesn't depend on «. For completeness we give a proof of the well-known fact:

Proposition 1. The set of Pareto Optimal points coincides with B. Also so does the set of strict Pareto Optimal points.

Proof. Strict Pareto Optimal implies Pareto Optimal, and Pareto Optimal implies the first-order condition. See for example Smale (1974b). Now suppose x — (xlt.. . , x j e W satisfies the first-order condition, with say^X^i) - ; > e S + . Consider a state y = (yl3.. .,yj e W with a^vj £ U((x,) all /. Now let II be the orthogonal projection of R' onto the oriented line through p. By the differentiable convexity hypothesis it follows, that n(yj £ i7(x<) each i with strict inequality in case yt # x,. But both x, y e W so £ x 4 « £71 and thus £Z7(;t() = X^O'i)' Therefore x « j», and we have that x is strictly Pareto Optimal. This proves Proposition 1.

Remark. Of course the above proof works with milder convexity hypotheses on the «(.

Corollary. The map u: W' -» Rm defined by [u(x)]t — «|(x,) restricted to 8 is one-to-one.

The following has already been stated by me (1974b), but no proof was given.

Theorem. The set of Pareto Optimal points 9 is an (m—l)-dimenswnal sub-manifold in W.

Our proof relates to the set A of price equilibria and the 'Fundamental Theorem of Welfare Economies'. Towards this end define a space of states ^ - ( i > ) - x S + , a n d

A - (x,p)ey\gl(xd « / > , ! * , - s.

Proposition 2. A is a submanifold ofS? of dimension m—l.

The proof of Proposition 2 contains the major part of the argument of this section and proceeds as follows. The following Lemmas 1 and 2 are easy con­sequences of the calculus, and the implicit function theorem.

Lemma I. A0 - x,p)eSr* \ £ x , - s is a submanifold of y of codbnen-sion (i.e., dim £"-dim A0) I with tangent space at (x, p) e A0 given by

T,JAo) - (5c, p) e (R')'x^ I £ * , - 0,

where p^ — v e R1 \ vp — 0.

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4 S. Smalt, Global analysis and economics VI

Lemma2. For each i - 1 , . . . , mt At « (x,p) *& \gfad - p is a sub-manifold of y of codmension /— 1, and at (x,p)eAt its tangent space is

T,,Mi> - (*.» s (**>"*J*11 DgjCxjXXJ - W.

Recall in Lemma 2, that Dg,(xi): -R1 - /r1 is the derivative (as a linear trans­formation) of g,:P-*S+ at x„ and TJS+) - p*.

Note that A is the intersection /l - \T-o Ai- Thus for the proof of Proposi­tion 2 it is sufficient to show that the A, have normal intersection at a given (x, p) e A. This means that the linear subspaces TxJiA^ intersect normally, or that

M

dun H TM.,(Ad » m - 1 .

For the moment let T - fl TXt,(Ad- Then

T - (*, Pi e (**)- xp>- \ £ f , - 0, Dgfcdxi - ?.

Further define

j - t f 6 * - I f -c^ , . . . , /?„ ) . £/?,-<),

and^: T-» J by ftx) - (p-x"t,. • -,p-xj. Note that since £p-x",» p £ x , - 0, <p is well-defined.

From what we have said, Proposition 2 is a consequence of the following

Lemma 3. The map $:T-* A is a linear isomorphism.

Proof. Let0€d. We will show that there is a unique X 6 Tsuch that # S ) — fi. For each i, X, can be written uniquely in the form 2, - xj+jf?, with p-x| * 0 and x' e Ker DgfaJ [remember that g^Xi) — p, and by our differentiable convexity hypothesis on the preferences, DgfaD restricted to p1 is an iso­morphism; thus Ker Dgtfri) and / ^ provide a direct sum decomposition of R'].

Towards solving <pX) - fi for X, let X? satisfy p-Xf - fo. Then j>£3J? - 0, s o t h a t ^ e / r 1 .

Since 7( - Dg<(X|) restricted to p 1 is symmetric with negative eigenvalues for each /, so is 7,"' and Erf 1 ** w U- Thus there exists a unique £ sp1 satisfying 2>f ^ + 1 5 ? - 0. Let X\ - 7f ^ Then £ * , - 0, D g ^ i ^ i - P. x s T and ifti) — p\ This finishes the proof of I emma 3, and Proposition 2.

We now write T — TxJiA) as the tangent space of the manifold A at x.

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5. Smalt, Global analysis and economics VI S

Proposition 3. Fundamental Theorem of Welfare Economics) The set 9 is a submanifold of W, and the mop of A into W which sends x,p) into x is diffeo" morphism alofA onto 9.

One form of the Fundamental Theorem of Welfare Economics [Arrow-Hahn (1971), Debreu (1959), Intriligator (1971)] asserts under more general conditions that the map <x,: A -» 9 above is onto. In other words every optimal allocation is supported by a price system. Of course Proposition 3 implies Theorem 2. See also section 4 for the case with production.

The proof of Proposition 3 uses Proposition 2 and the 'first-order theory'. More precisely, define a: WxS+ -» W by <t(x,p) - x. If (x,p)eA <= Wx. S+ c Sf, a(x,p)e9. Let otj: A -» 9 be the restriction of a. Define fl:W-* WxS+ by B(x) - (x;si(*i)); so B(x) e A if xe9. Let 0 t : 0 -* A be the re­striction of B. Then on A, fii • «f is the identity. Also fi is an imbedding (see below). Therefore one can conclude that on A, atj is an imbedding. The rest follows.

For x e W define Kx to be the kernel of the derivative Du(x): TJiW) -» Rm, of u:W-*Rm. Thus Kx is a linear subspace of TXW) - x e (*')- j £ =c, - 0,

Proposition 4. Kxisa transversal to 9 or/^n TJi.9) - 0 in TJ.rV).

This proposition implies that TJ^W) has as a direct sum decomposition AT,® Tx(9). For the proof note that if it 6 AT, then ie,eKer Dw,(x,), or xtgi(xd - 0. Thus in the proof Lemma 3, x' - 0, all i. Then x\ « 0, xt » 0, and* - 0. Proposition 4 is proved.

An imbedding of a manifold is a C1 map which is one-to-one, and the derivative is one-to-one at each point

Corollary. The map u: W-+ Rm restricted to 9 is an imbedding.

The corollary, besides Proposition 4 uses the corollary to Proposition 1. Define a submanifold AT, (an afBne one) of If by considering Kx to be contained

in W with its origin at x. In the Edgeworth Box (fig. 1), AT, is the tangent line to the indifference curves at xed, lying naturally in the box. Formally AT, — yeW\y- x+x, XeAT,.

Then from Proposition 4, AT, and 9 intersect transversally in W at the point x.

Proposition 5. For each x in 0, AT, n 9 — x. Also there is a neighborhood ff(9) of 9 in W with the property that if re N9), there is a unique xe9 such that re AT,.

Proof. Let yeRxn9. Since yeKx, (yt-x^-g((x,) - 0, and ufrb < ufad, using our convexity hypothesis. Therefore y i 9.

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6 S. Smalt, Global analysis and economics VI

The second part follows from a simple version of the tubular neighborhood theorem of differential topology. See Golubitsky-Guilemin (1973).

Corollary. For each endowment allocation r in N(ff) there exists a unique Walras price equilibrium.

The proof follows from Proposition 5 and the observation that for x e 6, p — gi(xx), (x,p) is a Walras price equilibrium for the endowment allocation r if and only if r eRx.

Compare Balasko (1974).

u,-constant

(^constant

Fig.1

Remark. Suppose that each u,:P-*R satisfies the boundary condition «," '(c, oo) is closed in Rl each c. Then it follows from the existence theory [see Debreu (1970)] that \J„,KX - W.

Remark. One can say some things about u: W -* Rm from the point of view of singularities of maps [see Golubitsky-Guilemin (1973)]. If .T in W is not in 6, then the derivative Dux) of u at x is non-singular (i.e., onto) or x is a regular point of u. This is a consequence, for example, of the rank proposition in Smale (1974b). Furthermore at each x e 8, u is a. fold, the simplest kind of singularity. This follows from the fact that the second intrinsic derivative £ x , D2ut(xt) is a non-degenerate form on the kernel of Du(x) [Smale (1974b) and Golubitsky-Guilemin (1973)].

We begin by defining a space of Walras equilibria with fixed total resources, s e P. Keeping the situation of the previous section as to conditions on the

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S. Smalt, Global analysis and economics VI 7

preference relations and notations, let Q, re(P)m \ p , - s, S?, =» Wx S+. Sf, constitutes a space of states of the pure exchange economy; Q, will be the space of parameters of the economy. An element of Q, is an endowment allocation (or 'endowment reallocation') keeping the total resources always the same. The methods used in the sequel permit one to make other choices for the parameter space.

Define

I - (r,x,p)eQ,xS',\p-xt-p-rl, gfcd - p, / - l , . . . , « .

Then (r, x,p)el means exactly that (x, p) is a Walrasian price equilibrium relative to the endowments given by r « ( r x , . . . , /„).

Proposition 1. I is a submanifold ofQ,x£ft with dim I — dim Q, and the tangent space TrtM / I ) of I ax (r, x, p) is given as the set of (r, X, p) e (R*)' x (*Tx/>x which satisfy p , - 0, p , - 0,

p-(r,-Xi)+/>-(rj-£j) - 0, i-l,...,m,

(the mth equation here is redundant)

Dj,(x,)(*,) - p, / — l , . . . , m .

Proof. Let 4>t: Q,x y , -* R be defined by 4>ir, x,p) « p-Xt-p-r,. Then it is easily checked that <f>, is regular (has no critical points) and so ^"'(O) is a submanifold. Similar reasoning with the other equations reduces the proof to checking that all the submanifolds defined by each of the equations have transversal intersection (compare to the proof of Proposition 2 of section 2). Thus the proof is reduced to showing that the following system of linear equations in (r, jc,p)e (/?')" x (/*)" xp- has at most an (m—1) /-dimensional space of solutions for given (r, JC, p) e Q, x y„

p i - 0, p , - 0, ?(*<-' i)+P(*<-^) - °.

Vgfcdxi - P-

To prove this, define a linear map

4>: (R1)" x (R*)m x p 1 - R' x # + ( * ) — l * 0^)",

by sending (r, x, p) into

( p . . I* i . P(xt-r^+p.(xt-rd]7^1, Dg/x^icJ-pir.,).

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8 S. Small, Global analysis and economics VI

If j> can be shown to be surjective (onto), a simple counting of dimensions gives us what we need. Thus let <xl eR1, <x2 eRl, P, eR, i 1 , . . . , m—l, and 6, ep*-, i - 1 m, be given. Solving the following system of equations in (?> x, p) will finish the proof.

(1) P i - « i . (H I * i - « a .

(2) ?•(*,-/<)+/>•(**-?<) - fi„ f — 1 m—l.

(3) Dgfadixd-P - Slt / - 1 , . . . , m.

Towards finding the solution, write a2 « Oj+otj, with otj-p — 0, Oj e Ker Dgi(x t). As in section 2, let y, - Dgj(*j)|,.i.. li'-P1 -* P*~- Choose p~ so that ( S r r ^ + Z r r 1 * ! - «i «o4 «i - rr'Q+Sd- Then I x - «',. Let xf - «5, 2 t - *i+iEJ, x, - Xi for / > 1. Then (10 and (3) are satisfied.

Now one can easily choose flt i » 1 , . . . , m— 1, so that (2) is satisfied and finally fm satisfying (1). The proof of Proposition 1 is finished.

In a similar fashion one can define

Z* - (r,x,p)eQxS'\'Exlm,£r„p-xt <-pr„ g&d-p.

The following is proved in the same way as Proposition 1.

Proposition 2. Z* is a submamfold of Qx& with dim I * « dim Q, and the tangent space TriX,JZ*) ofZ* at (rtx,p) Is given as the set of(r,X,p~)e(R')x (J*0" x / ^ *hich satisfy:

P i -1*«. p(r,-xd+p(rt-xd - 0, / - 1 , . . . . m,

(the mth equation here is redundant)

Dg<(*<)(*i) A i - 1 , . . . . m .

Corollary. [Debreu (1970)] Except for a set of initial endowments of Q of measure 0, the number of price equilibria is discrete. Furthermore with Debreu's boundary condition (see Remark at the end of section 2), one has the exceptional set closed and finite replaces discrete.

For the proof of the Corollary, one simply applies Sard's theorem to the projection QxSf'-* Q restricted to Z* as in Smale (1974a).

Returning to the situation of Proposition 1 one may think of fig. 2.

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5. Smalt, Global analysis and economics VI 9

Let Ug3: Q, x ST, - Q, be the projection and TI:Z-* Q„ its restriction to Z. We consider the question: for what (r, x,p)eZ is II catastrophic? In other words, find a condition on (r, x, p) such that the linear map Di7(r, x, p) be singular as a map from one linear space to another of the same dimension. For non-catastrophic (or regular) (r, x, p) e Z, one knows by the implicit function theorem that the price equilibrium (x, p) varies continuously with the parameters of the economy r. On the other hand if (r, x,p) is catastrophic, then a small change in the parameter r of the economy could produce a large jump in prices. In fig. 2 the big dot on IT is an example of a catastrophic point

The following gives an analytic criterion for (r, x, p) 6 1 to be catastrophic. Remember (r, x, p) e I if and only if (x, p) is a price equilibrium for the endow­ments given by r.

4*

I Q ,

Fa. 2

Proposition 3. The point (r, x,p)eZ is non-catastrophic if and only if the linear map hrjttr: Tx(0) — A is an isomorphism where hr^r « A£*.*+^* with

h^.,x) - (Pgl(xlyxl)(x1-ri), ..., Dgm(xJ(ZJ-(xm-rJ),

and

4>*(x) - (pxlt...,pxjj.

The map ^ was studied in Lemma 3 in section 2.

Proof of Proposition 3. First we examine analytically what it mesas for a point (r, x, p) in Z to be catastrophic. Fixing (r, x, p) in Z, we have in Proposi­tion 1 equations on (f, 3c, p), describing 7*Pt,t,(I). Then from the definitions, (r, x, p) is non-catastrophic if and only if given any r with £ r , «» 0, there exist (jf, p) with (r, x", p) e rr*,^Z). In other words, can the equations of Proposition 1 always be solved for (x, p)? Referring to Proposition 3 now, if if 6 T^ff), then £x"( = 0 and Dff/xjx, is independent of i, and say is p.

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10 S. Smalt, Global analysis and economics VI

Thus the condition for (r, x, p) to be non-catastrophic amounts to solving for x, the equations,

P-(*t-r i)+/>*i - p r „ / « l , . . . , m ,

and£^-r, — 0. The equivalence with the condition of Proposition 3 can now be seen. Proposi­

tion 3 is proved.

The following gives an affirmative answer to a conjecture Debreu made to me in connection with his recent work on the rate of convergence of the core.

Corollary. If the endowment allocation r is a regular economy then the map ifr:0->A defined by x -> (*i(xi)-(*i-#,), . . . . gm(xj(xm-rj) is a dfffeomorphism for xina neighborhood of a Walrasian price equilibrium (relative tor).

The proof of the corollary is obtained by simply differentiating the map \jf, applying Proposition 3 and then the implicit function theorem.

Remark. Proposition 3 yields some perspective on what can cause catastro­phic jumps in prices in the framework of general equilibrium theory. By Lemma 3 of section 2, <px is always an isomorphism; thus it is the effect of h*^,t which causes jumps. If JA^.,1 is small then there are no catastrophes. This term h*^^ can get big for two reasons. One is that xt—r, becomes large or that r gets far from 0 (compare Proposition 5 of section 2). The other is that the curvature of the indifference surfaces becomes large from the expression Dgfai). One should keep in mind that all of this is an interior analysis. Every consumer owns at least a little of each commodity.

4.

Our model of an economy with production goes as follows. To each of m consumers is associated a consumption set X, an open set in Rl, i » 1 , . . . , m. We suppose m £ 1. On each Xt is supposed a preference relation which is represented by a C* utility function u,: X, — R. Define gf: X, -»• S+ by y^x) [gradujC^l/dgradviCOH] for each x in X,. We suppose throughout that gt satisfies the differentiable monotonicity and convexity hypotheses of section 2.

It is also supposed that there are n producers, and to each is associated a technology which is represented by a closed submanifold Y, in Rl, a. « 1 , . . . , n. Note that we are not making the very restrictive hypothesis that Y, be a hyper-surface.

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S. Smalt, Global analysis ami tconomtcs VI 11

For our economy with production, but no prices explicitly yet, define the space T of attainable states by

x-x,y)eX*Y\Y.xt-Yj.+s.

Here Xand Y are Cartesian products X =- n,Xt, Y - n,Ym, xteX„y.e Y. and s e R' denotes the endowed resources of the economy. The defining con­dition of T of course just relates the total consumption to the total production.

Proposition 1. x is a submanifold o/Xx Y.

Proof. One needs to check that in Xx (R*)m, the submanifolds defined by the 7. and the condition £ x , — £.y«+J all intersect in general position. Thus fix x =» (* , , . . . , x j and .y - (yt,.. .,yj, xt6 Xiandy.e y.with£x, » £>.+*. Then it is to be shown that the set of (x,y) with x^R', y.eTfm(YJ and £x"j >- Y.?* bas dimension equal to /m+£ dim Y.-l. For this, let y, e T,.(YJ> a «- 1 , . . . . n, be given. Then the xt satisfy the single vector equation £ x , ™ Y,y, with exactly lm—l degrees of freedom. This very easy proposition is proved.

Define functions u,: T -»• R, i — I , . . . . m, by U|(x,,y) — u^xj, where arfx,) is the value of the individual utility at xt. There should be no serious confusion using the M, for two slightly different meanings. One may now define the notion of admissable curve and infinitesimal Pareto set 9 in T as in Smale (1974b). That is (x, y)ex belongs to 0 if and only if there is no curve <p: (— 1,1) -»• T with <f>(0) - (x, y) and d/dr («^(0) > 0, aU /, /.

Proposition 2. (First order) A point (x, y) of x is infinitesimal Pareto (i.e., (x, y)eff) if and only if

(») giCO <* some constant vector say pin S+, and

(b) p 6 Nu( YJ (i.e., this vector is normal to Y, at yj.

The proof uses the first-order condition, e.g. Smale (1974b). Thus (x, y) e d if and only if there exist Xt £ 0, i - 1 , . . . , m, not all zero such that £-1, Du^x, y)(x, y) - 0 all (x, y) e ^ ( r ) . Supposing (x, y) e 0, in this equation, one can take y - 0, so that £ x , = 0. Then as in Smale (1974b), A, grad «,(xj is in­dependent of i, and one has (a). For (b) fix i and t and take x, — 7, with all the other components of x and y zero. This leads to gfad-y, — 0 orp-ym — 0 for all J, 6 Tjji YJ. The converse is similar.

Before proceeding any further we introduce a convexity type of hypothesis on the technology submanifolds Ym.

Hypothesis on Ym. For each peS+ the real valued map ft: Y,-* R which sends y in Y,to p-yhas exactly one critical point and that critical point is a non-degenerate maximum.

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12 S. Smalt, Global analysis and economies VI

We assume this in all that follows. Suppose that for p e S+, y* is the maximum given in the hypothesis. Then

D/»0'*) — 0 and Dz/,0'*) >s a negative definite symmetric bilinear form on the tangent space T^i Yj of Y% at y*.

One can write Dzf,(y*) =» p-Hy, where Hy> is the second fundamental form of the submanifold Y. at >•*.

Let us look at this geometric invariant of the technology a bit. One has defined for any y e Ym, the second fundamental form Hy as, for example, in Hermann (1968). This form Hy is a symmetric bilinear form defined on the tangent space Ty y j with value in the normal space Ny( YJ) of all vectors orthogonal to T,( YJ. One can think of H7 as simply the second derivative of the inclusion Y, -» R' with values projected into T^YJ. For peN,(YJ, we write p-H, as this real valued form on T / YJ.

Our hypothesis above has the following consequence. For ye Ym, let 77,: R! -» TJYJ be the orthogonal projection, and if p e S + n N,(YJ, define fi,: p1 -*ir by Q/p)-w ~p-H/JIjo, w). Then 2 , is symmetric, and from the hypothesis on I""., Q, will have non-positive eigenvalues.

Proposition 3. The Pareto Optimal points in the space of attainable states t of our economy with production coincide with the points of 6, and in fact the points in 9 are strict Pareto Optimal points. The map «: T -* Rm whose coordinates are ult restricted to 9 is one-to-one.

Proof. Let (xm, y*) e 9 and (x, y) e T, with u£x, y) £ ufcc*, y*) all i. We wish to show that (x, y) — (xm, y*). Let p • gt(xf) (keeping in mind Proposi­tion 2) and let 77 be the projection of R' onto the oriented line through p.

Since u / x j £ ufccf), it follows from our convexity condition on the u, and Proposition 2 that £ / ! * , £ £ ^ W ) and strict inequality if x * x*. From I*« - 2 > . + ' , 5 > * - J>*+jitfollowsthat£/7y. 2 £77>-*.Butby Proposi­tion 2 and the hypothesis on Ym, this is impossible unless y« - >"«, each a. The rest follows.

Proposition 4. (Fundamental Theorem of Welfare Economics, simple form) Given (x*, y*)e9 <r T, there is some p*eS+ (which is unique) with the properties

(a) x* maximizes utility u, on the budget set x, GXt\p*-Xt£ p*- x*, (b) y* maximizes profit p*-yton Y,.

Conversely let (**, y*, p*) in T X 5+ satisfy (a) and (b). Then (x*. y*) e 9 and is therefore Pareto Optimal.

Remarks. One could equivalently say that x* e X is an optimal allocation (for this economy) provided there is some y* e Y with (x*,y*)ex and moreover (x*, y*) e 9, and restate Proposition 4 in terms of optimal allocations in X.

332

5. Smalt, Global analysis and economies VI 13

Proof of Proposition 4. Let (x*, y*) s 9. Then choose p* » g j(*j). Then (a) is satisfied (Proposition 2 and properties of «,). Also (b) is satisfied (Proposition 2 and hypothesis on YJ.

The converse follows easily from Propositions 2 and 3. Now we extend the proceeding framework a bit to consider all states of the

above economy together with prices. Thus let 5" = X x Yx S + , and define

A-fr,y,p)ey\gAxd-P, P*N,m I* , - £y.+s.

It follows from Proposition 4, that elements of A can be thought of as price equilibria [or equilibria relative to a price system in the terminology of Debreu (1959)]. Note that an element of A does not depend on some endowment.

Proposition 5. The set A is a submanifold of S? of dimension m—\. Further-more the tangent space TXi,.,(A) is the set of(x, y, p) in TXjJi?) which satisfy

I*< - 2> . D * * ( * I ) * I = P,

Q,.(P) - y.-

Lemma 1. Ax - (*, y, p) e Xx Yx 5+ | p e NJm( YJ. Then A, is a submani­fold of codimension I-1 with tangent space

T^AJ = x, y, p | Q,.(P) - ?..

Proof. The condition that p e N,m(YJ can be replaced by pD<t>,(yJ 0 where #«: Y, -* Rl is the inclusion and p-D^.00 is the map TrJiYJ -* R defined by y, -»p-ym.

The rest of the proof of Lemma 1 proceeds by calculus.

The proof of Proposition 5 is now reduced (by arguments in Proposition 2 of section 2) to showing that the following map is an isomorphism:

*:T,,,JA)^A, (X,y,p)-(J>-Xl,--;P-Xj.

Here TXJtf(A) is defined as in Proposition 5 even though A has not yet been shown to be a manifold.

The proof of this is similar to the proof of Lemma 3 of section 2, and we omit it.

B

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14 S- Smale, Global analysis and economics VI

Theorem. Strong form of the Fundamental Theorem of Welfare Economics) The set ofPareto Optima 6 is a submanifold ofy(0 c T C y) of dimension m — 1, JO is the set of price equilibria A, and the map $: A -* 6, induced by (x, y, p) -» (x, y) is well-defined and a diffeomorphism.

The proof follows from Proposition 5 in the same way as Proposition 3 of section 2 was proved.

Again one may define J^,y as the kernel of the map Du(x, y): TXtr(r) -* Rm

and one can prove as in section 2 that for (x, 7) e 0, the intersection of Kx,y with r ^ i s z e r o .

References Anow, K. and P. Harm, 1971, General competitive analysis (Holden-Day, San Francisco). Balasko, Y., 1974, Some results on uniqueness and stability in general equilibrium theory,

Reprint (Berkeley)-Debreu, G., 1999, Theory of value (Wiley, New York). Debreu, G., 1970, Economies with a finite set of equilibria. Econometrics 38, 387-392. Debreu, G., 1972, Smooth preference. Econometric* 40, 603-616. Golubitsky, M. and V. Guflemin, 1973, Stable mappings and their singularities (Springer,

New York). Hermann, R., 1968, Differential geometry and the calculus of variations (Academic Press,

New York). Irrtriligator, M., 1971, Mathematical optimization and economic theory (Prentice-Hall,

Englewood Cliffs, N J.)-Smale, S^ 1974a, Analysis and economics HA, Extension of a theorem of Debreu, Journal of

Mathematical Economics 1,1-14. Smale, S., 1974b, Global analysis and economics HI, Panto Optima and price equflibria,

Journal of Mathematical Economics 1, 107-118.

334

Journal of Mathematical Economics 2 (1976) 107-120. © North-Holland PuWishini Company

A CONVERGENT PROCESS OF PRICE ADJUSTMENT AND GLOBAL NEWTON METHODS

Steve SMALE Unwtnity of California, Btrktity, CA 94720, U.S^i.

Received October 1973

Section 1 One goal of this paper is to give a relation between such diverse parts of

economic theory as the Arrow-Block-Hurwicz dynamics of price adjustment and Scarf's algorithm for finding economic equilibria. The underlying concept is an ordinary differential equation, which we call a 'Global Newton' one, associated to a system of n real functions,/ , , . . . , / , , of n real variables, xx xn.

The key feature of this differential equation is that, under suitable hypotheses, its solution will tend to a vector ( ;e j , . . . , .t*) satisfying

/i(*f - O - 0 , . . .,/,(*? x,*) - 0, (1)

or in vector notation,

/(*«) - 0. (1')

In fact in this way an algorithm for solving (1) is provided. The differential equation itself has the form

where the sign of X is determined by the sign of Det Df(x). Here D/(x) is the

linear transformation with matrix representation

D/W . (*y and i. is a real valued function of x, unprescribed as yet.

This work was done at Yale in the fall of 1974, and I would like to thank the Cowlcs Foundation and the Mathematics Department there for their hospitality.

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108 S. Smalt, Price adjiatmtnt and Global Ntwton

We give now some background of our work. Suppose given a world of / commodities and corresponding prices each measured by non-negative real numbers. A price system is a vector £ « (pl,..., p), p, £ 0, where p( represents the price of a unit of the ith commodity. Let R, denote /-dimensional real Cartesian space, and

We suppose an economic setting which provides demand and supply functions A5.'M.-0-*iJ l+ I with D, S functions of prices in #+. The condition for economic equilibrium (or price equilibrium) is then 'supply equals demand* or D(p) — S(p) as a condition on the price system p. The derived excess demand is the function Z:R'+-Q -* R* given by ((p) - D(J>)-S(J>). Thus &) is the excess demand for the ith good at prices p. Natural hypotheses on <J are:

(a) t(fy) - ftp) for i. > 0 (homogeneity),

(*>) P'iip) 0 (inner product). Wains Law [for motivation see Quirk and Saposnik (1968)],

(c) if p, - 0, then UP) > °-

Actually can be taken as the primitive economic notion and wfll be assumed to satisfy (a), (b) and (c). Boundary conditions derived from a micro-economic setting, while technically more complex, are similar in principle. We hope to pursue this matter elsewhere.

By (a) it makes sense to normalize price systems or represent them in the space

S'+-peRi+\Z(pd1>-l-By (b) Zip) is tangent to SV at p so ( is a vector field (or ordinary differential equation) on &+. By (c) this vector field <J points in on the boundary w t n a t the Hopf theorem [see Milnor (1965)] yields a price system p* with <Q?*) - 0 or D(p') » S(p'). This is proof of the existence of economic equilibrium. A con­structive, 'dynamic' proof of this existence result is given in section 4.

The question goes back to Walras; is there some process (economic and/or mathematical!) which leads to an equilibrium?

On price space S1* the differential equation

d>/d/ - tip) (3)

is well-defined and the equilibria (i.e., zeros) of this differential equation coincide with the price equilibria for the excess demand function (. Solutions of (3) can be thought of as price adjustments where a positive excess demand for some good raises the price of that good.

336

5. SmaU, Prict adjustmmt and Global Ntwton 109

Do such solutions lead to equilibria? The answer is yes according to Arrow, Hurwicz and Block (1958, 1959)

provided a further substantial hypothesis 'Gross Substitutes' is made on £ (see section 3).

The answer is no in general according to Scarf (I960) who constructs an example of excess demand derived from economically plausible individual demands with the property: almost all solutions oscillate for all time.

Subsequently, Sonnenschein, Mantel and Debreu [see Debrea (1974)] showed that could be pactically arbitrary and still be derived from preference relations of classical types.

We find here a modification of the differential equation (3) whose solutions converge .to equilibria under quite general conditions, in fact in situations with multiple equilibria (where previous methods failed to converge). This modifi­cation grew in part from a study of another development in mathematical economics, Scarf's algorithm (1960).

Scarf found an algorithm for finding fixed points of continuous maps of a simplex, which he applied in particular to the location of economic equilibria. This method i$ a simpiicial method and turned out to be closely related to Hirsch's paper (1963) as can be seen especially dearly in the papers of Eaves and Eaves-Scarf (1975). Furthermore a differential analogue was already suggested in Hirsch's paper and this differential analogue was developed for algorithmic purposes (in a fixed point context) in the paper of Kellog, Li and Yorke (to appear).

What we do here is to look at the Scarf algorithm in the context of eq. (1). By using the differential point of view, a simple derivation leads to the differential equation (2). In the next section we will see how this goes. In section 3, these ideas are applied to the excess demand function <J described above.

For a background on ordinary differential equations written in the style of this paper, see Hirsch-Smale (1974).

Many conservations with Curtis Eaves and especially Herb Scarf were very helpful for this paper. Also subsequent work with Moe Hirsch has had the effect of smooching the account given here.

Section 2 We consider the purely mathematical problem here of solving a system of n

non-linear equations in n unknowns through an associated ordinary differential equation. Suppose the domain M for these functions is given as a bounded open subset of real Cartesian space A* together with its boundary dM which we suppose to be smooth (i.e., a submanifold). Thus dM is contained in M, and M is a closed set.

Let / : M - JP, fix) - (/!(*),...,/.(*)), x - (*!, . . . , JO 6 M,

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110 S. Smalt, Price adjustment and Global Newton

be a map with continuous first and second derivatives (i.e., Cl). Let Df(x): B? -» RT be the first derivative off at x as a linear map so that Dfx) has a matrix representation in terms of the partial derivatives,

We seek a solution of

fx*) - 0, x* € M. (4)

The 'Global Newton' equation is the ordinary differential equation on M given by

DAx)^--V(.x). (5)

where A is a real number unspecified for the moment, but the sign of X is deter* mined by the sign of Det Df(x) (Det A means the determinant of A).

In general one cannot expect solutions of (4) to exist; a boundary condition is necessary to assure such an existence. To see the ideas most clearly, we postu­late a simple boundary condition, but one that can be substantially weakened and still have the results valid.

Boundary condition. For each x e BM, Det Df(x) # 0, and there is a choice, (a) signi(.T) - sign Det Df(x), all xeBM, or (b) signX(x) « -sign Det D/tr), all xeBM, which make -M.x) D/\x)~lfx) point into M at each xeBM.

We specify the sign of X accordingly. Thus we will assure that either (a) holds for all xmM with Det D/(.r) # 0 or that (b) holds; and that -iDfi.x)"lfx) points into M at each x e BM.

Define £ - / " '(0) so that £ is the solution set of (4). We may now state the main result:

Theorem A. Let f.M -» /i" be C1 and satisfy the boundary condition. There exists a canonically defined subset Z of measure 0 in BM such that if x0 6 BM, x0 i Z, then there exists a unique Cl solution to: [t0, ft) -» M of (5) starting at x0 [i.e., <p(t0) — x0] with |d^/d/(/)|| - 1, and rt maximal, rt <J oo. This solution converges to East — r,.

Almost all Cl functions f.M -*IF have the property:

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S. Sntatt, frit* adjustment and Global Newton HI

Gtntricity hypothesis. If x e E, then D/(.r) is noa-singular.

If/satisfies the genericity hypothesis then E must be a finite set. Furthermore:

Theorem B. Iff: M — if* is C1, satisfies the boundary condition and genericity hypothesis, then I of Theorem A is a closed set (of measure 0) and the solution <p(t) starting at any x0 6 dM, x0 $ Z, converges to a single point x' satisfying (*•) - 0.

Before proving Theorems A and B, we note other forms (5) may assume. For example one can simply take X » Det D/(.x) [or -Det Df(x) in case (b) of the boundary condition] to obtain

D/W ^ - - (Det D/W)/(x). (6)

Moe Hirsch has suggested taking B(x) so that with matrix multiplication one has (Det Df(x))£ - Df(x)3(x), where I is the identity matrix. Then any solution of

cLr/dr- -B(x)Ax) (7)

is also a solution of (5) [apply D/(x) to both sides of (7)]. This gives a desingu-larized version of (5).

Note that since

dx d/

one can express (5) in the simple form

d//dr - -If. (8)

Eq. (8) is the 'target space' version of (5), and from it one can derive a geometric interpretation of the method.

In case D/(x) is singular for no x in M, we may choose A - 1 [case (a) of boundary condition] and rewrite (5) in the form

d*/d/ - -Df(x)-lf(x). (9)

Theorems A and B apply, modified only by a different choice of parameteri­zation of <p. Writing dx/d/ as a finite difference, (9) becomes

* . + i - * . - - D / W V O O , 00)

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112 5. Smalt, Prtet adjustmtm and Global Newton

or exactly Newton's method of iteration (which of course is always effective in some neighborhood of £). In this case (assuming the boundary condition) our proof demonstrates the existence of a unique solution x* of /(*•) » 0, and that every solution of (9) will tend to x*. This special case is well-understood classically; see Ostrowski (1973).

For the proof of Theorems A and B, given f:M -* R* with E - / ^ ' ( 0 ) . we define an associated map g:M-E -* S*~l with target space S*"\ the unit sphere in R". Let

then g is a C2 map and we may apply the following proposition [implicit function - Sard, see Abraham-Robin (1967), Milnor (1965)]:

Proposition 1. For any C1 map g:M—E — S*~l (where M is n-dimensional), there is a set A .the ^exceptional set of critical values') of measure 0 in S""1 with the property that if yeS*"1, yiA, then g~l(y) is a Cl non-singular curve in M—E (more properly, a \~dimtnsional submanifold).

One can describe A as the image under g of the set C of critical points of g where .t 6 C if the rank of Hg(x) is less than n-1.

We also know:

Proposition 2. For each x in M—E, the kernel of the derivative Dg(x) [i.e., » e R*\Dg(x)(v) - 0] is the set

Ker Dg(x) - v e Sr\Df(x)(v) - Xf(x\ some X s R.

For the proof one uses standard calculus techniques to obtain

for all x 6 M, v 6 R*. From this formula one identifies Ker Dg(x) as in Pro­position 2.

We remark that from the chain rule it follows that Ker Dg(x) 3 Ker Df(x) and that if D/(.x) is non-singular then Dg(x) is surjective. Also if the corank of D/(JC) is one, then the corank of Dg(x) is one or two.

The tangent vectors at xeg~l(y) to the curve g~l(y) in Proposition 1 are precisely the vectors in Ker Dg(x) in Proposition 2.

For the proof of Theorem A, apply Proposition 1 to the map g and let A be the exceptional set in 5"~l. Let I - x e dM\g(x) e A). Then since the restriction g0:dM -» S"~l of;, is a local diffeomorphism, the measure of I is zero.

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S. Smalt, Price adjustment and Global Newton 113

Note also that if the genericity hypothesis is made on/, then the set of critical points C of; is compact; then A is closed and gZ\A) — Z is also closed.

Now let x0 e BM, x0 # I, y0 - g(x0), so that g'l(y0) » a 1-dimensional submanifold in M—E. This curve has a component y starting at *0; by a simple orientation argument along •/, using the boundary condition, this curve y cannot meet BM in any other point than .r0. Therefore since 7 is a closed set and a 1-dimensional manifold in M—E, y can be parameterized by t in [t0, /,) and as t -* t, the limit points must all be 'at co* in M- E, or in E.

It remains for the proof of Theorems A and B to identify the solutions of the Global Newton equation (5) starting at x0 with y. But this is done by Pro­position 2.

We finish section 2 with a sequence of remarks:

(A) I have been working with Moe Hirsch on the development of this algorithm and its implementation on a computer (in particular HP-65 and POP 11)- We hope to publish our results soon.

(B) There is a gradient version of the Global Newton, which goes as follows: For xsM define a bilinear symmetric i'orm <xMv,w) «• Dfx)(v)'Dfx)l(w). The form a defines a positive (though in general indefinite) metric on M.

Also let $:M — R be the function # x ) - | /(x)j2 . Then consider: dx/dt « -i grad 4K.x), sgn X - sgn Det D/(x) [or -sgn Det D/x) in (b) of boundary condition] where the gradient is taken with respect to a. This equation will agree with (5) whenever A »* 0.

(C) One can weaken the boundary condition of f:M — R*. For xeM let KM — v € R*\D/x)(v) » i/(x), some X fi R. Then one may relax the condition that Det D/(x) # 0 for x € BM by imposing dim Kx - 1 for x c dM, etc

The theory still applies.

(D) One can also prove the theorems in case M has a piecewise smooth boundary. For example let b,:R" -* R, i « 1 , . . . , N, be C functions which satisfy: if *(,(*) - 0 , . . . , * Jx) - 0 then the vectors grad i , , (x) , . . . . grad 6 j x ) are linearly independent. Let M be the region defined by

M - x 6 / ^ ( x ) fc 0, all i - 1 , . . . , iV.

Suppose that M is bounded. For such M an analogue of the boundary condition is:

Piecewise smooth boundary condition. For each x e BM, Det D/(x) > 0 and Do,(;cXD/(xrl/(;0) < 0 for each », such that *,(*) - 0- (Theorems A and B are valid with this boundary condition.)

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114 S. SmaU, Priet adjustmtnt and Global Ntwton

(E) Solutions of our basic equation will cross smoothiy much of the singularity set of/[where Df(x) has corank 1 and/is a 'fold' singularity]. 'Branch points' give trouble to the solution curves of (5). These are points where Dg(x) has corank 2. Also there can be families of periodic solutions of (5) (of course branch points and periodic solutions don't invalidate Theorems A and B).

(F) An abstract setting to the theory can be given for maps/: 3 / -* V when M is a compact oriented connected manifold with boundary and V'a a vector space of the same dimension equipped with an inner product

(G) One may replace (5) by the differential equation

D/(x) £f - -(it/t(x) V.W).

where each sgn Xt is specific as before, and the rest of the section remains valid.

Section 3 Here the results of section 2 are applied to the case where the function/is the

excess demand of some economy. A slightly different context than section 1 is chosen, because the excess demand there is not given as a map from a domain into a linear space of the same direction.

Following a certain amount of tradition in theoretical economics we suppose a distinguished commodity or a 'numeraire' [Quirk-Saposnik (1968)] so that prices of the other goods are measured in terms of this numeraire good.

Thus suppose there are (/+1) commodities, with prices denoted by non-negative real numbers p0, • • •, Pi, where p0 is the price of a unit of the numer­aire good. We work in the domain of price systems (j>0,.. .,/>,) which satisfy p0 > 0. In this domain, by the homogeneity property of price systems (see section 1), one can normalize a price system by dividing by p„. Thus a price system (p0,..., p J can be represented by a unique point

0»i/Po. •••./>(//><>) in #+•

With this interpretation, the space of price systems is R?+. We suppose Cx

excess demand functions Z0(p0 />,), t(p0, ...,p^,..., i,(p0, ...,pi) are given from some economic setting, and that Walras Law is valid. Thus

M--Z £ OP). — i Pa

so that the equations

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5. Smalt, Price adjustment and Global Newton 113

« P * ) - 0 , i - I /,

are necessary and sufficient conditions for economic equilibrium. Thus it may be supposed that the excess demand is represented by a map

i'.R'j. — R' where source variables are normalized price systems and target variables are (excess demand for) commodities numbered one through /. The zeros of this map <J are precisely the price equilibria.

We now state a boundary condition on £, which is slightly more subtle than that in section 1, and compares directly to that of section 2.

Heuristically, the condition says that as p( - 0, <?,(;>) -♦ oo and dZjdpiip) — - oo as in fig. 1. Of course one has to take into account the fact that <J is a function of several variables.

ft

t

Pi

Fif.1

Boundary condition on <J. For p e d#+. let Jr be the set of indices such that Pi « 0. Then the system of linear equations

iir'j--i*. * - i A 0D

has a unique solution (vlt.. .,vj and o, > 0 for / e Jt.

The boundary condition will be satisfied at p for example if the system (11) is dominated by terms of the form,

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116 S. Smalt, Price adjustment and Global Newton

3& j - P f - -if i6j>* »P i

dp,

and hence the heuristic justification above. Now R?+ is not bounded and so we are not yet in a position to apply section 2.

Note that the boundary condition on i is still valid for pt dose enough to 0, i « / , , so that one does not need p, — 0 literally. Now extend the boundary condition to the numeraire good or to all /+1 prices and commodities to obtain the extended boundary condition on <J. Note that going to oo in &+ corresponds top0 going to 0; so the extended boundary condition on <J make the solutions of the Global Newton equation, which start near oo (or outside some bounded set), behave as if they started on the boundary.

Now the hypotheses of section 2 are satisfied for the excess demand and so Theorems A and B apply to yield:

Theorem C. Suppose as above, the excess demand :/?*+-» Rl, defined in terms of the numeraire satisfies the extended boundary condition on £. Then (maximal) solutions of

i *i%> - -Xt„ sgn X - (-1)' sgn Dec (p). (12) j - i *Pj dx \dpjj

starting on the boundary of R+ (or near oo) except for some set I of measure 0 in the boundary ofR'+ will converge to the set of price equilibria.

It can be checked that derived from a 'regular' economy satisfies the genericity hypothesis of section 2.

Theorem D. Ifiin Theorem C is derived from a regular economy in the sense of Debreu (1970) or Smale (1974) then the set Z will be closed and a solution starting in dR?+ -I will converge to a single price equilibrium.

Say that [e.g., as in Quirk-Saposnik (1968)] an excess demand ?0 (, satisfies the gross substitutes condition or G.S. if d£Jdp, < 0 each i and dZJdpj > 0 whenever i # j , i, j « 0 , . . . , /.

If i satisfies G.S. then D(p) is non-singular for all price systems p. Here D(p) ™ <>ZAp)ldPi> i,j ™ 1, •. •, /• This follows from work of L. McKenzie since D<JO) has a 'quasidominant diagonal', see Quirk-Saposnik (1968, pp. 167 and 173). From section 2, the remark after (10) we have:

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S. SmaU, trie* adjustment and Global Newton U7

Theorem E. Let i be as above (Theorem C) and satisfy G.S. then there exists a unique price equilibrium p' and every solution of (12) (with, e.g., I » 1) con­verges to p*.

It remains an interesting question, to what extent and in what situations one can find a valid economic interpretation for (12). It is clearly more subtle than the classical equation ot section 1, p — Up).

Consider the simplest case where <J satisfies G.S. and choose X - 1. Then if the off diagonal terms of dfjdpj are neglected, the equations become the equations of Arrow, Hurwicz and Block.

Our equations contain more explicitly, relations between the various markets in their effect of price adjustment.

We end this section with a couple of remarks. In Arrow-Hahn (1972, p. 303), a similar ordinary differential equation is

discussed very briefly. Also this book has a general account of the Arrow, Hurwicz and Block results and related work.

One can avoid choosing a numeraire by applying the theorems of section 2 to other functions derived from the excess demand. We give two explicit pos­sibilities. First let

J , - ^ « A , * | 2 > , - 1 .

l o - O - e t f l l y . - O . Define

/W - «(P)-I «00.1 0.

where <J is the excess demand of section 1. The price equilibria are solutions of f(p) - 0 and section 2 applies.

Finally, H. Scarf has suggested

/:^i-<*<,; Ap)m(Piii(p) PA&))>

so that ptit are the values of the excess demand, for use of the Global Newton method. He has shown me that D/O) is never singular in the G.S. case.

SecfJoa4 The goal of this section is to present existence proofs of the main theorems

of general equilibrium theory in the spirit of the preceding sections. Giving the existence proofs in this context achieves unity by bringing together existence

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118 S. Small, Price adjustment and Global Newton

theorems, algorithms, and dynamic questions. Also the proofs are simpler than those going through algebraic topology. The ideas are close to those of Hirsch's paper and to Li and Yorke, but we retain the equation approach rather than the fixed point approach. Roughly speaking these proofs work with simpler bound­ary questions than the preceding sections, but the process itself seems more complicated. Results are given in increasing generality.

Theorem F. Let J, » pe Pf\pt £ 0, £ p , - 1 be the price simplex and let A0 - peP,\Jj>, - 0; let <p:Ax — A0 <= Pf be a Cl map, <b(j>) » (^t(Pt, . . .,p,), . . . , <p/plt.. .,pd) such that <bfa>) > 0 ifp, - 0. Then there isp9Ax with (pip) - 0.

Proof. Let E - p e J t | <f>(p) - 0 and 3d, - p« Ax \pt - 0 some». Then for each pe d t - £ , there is a unique I - X(p) £ 0 such that p+JL<bip) is in 3d t . Define h:Ax-E -» 3Jt by A(/>) « p+Mp)4Kp)- Note that A is the identity on the boundary. Let 3, be the d<th face of d,, so d, - p « A, \p, « 0. Then A can be checked to be continuous and in fact will be C1 at a point p such that k(p) belongs to only one d,.

Now by the Sard-implicit function theorems as in section 2, we can choose p0 in just one face (to be a regular value of A) so that h"i(p0) is a non-singular curve. Consider the component of h~l(p0) which starts at j70. This component must lead to £ as one travels along it starting from P0. Therefore £ is not empty. Since E is the solution set of <bip) - 0, the theorem is proved.

Corollary 1. Let z:PJ+—0 -* PJ an 'excess demand function', of class C1. We suppose then z satisfies homogeneity, Walras Law and the boundary condition, *i0>) > 0 if Pi m 0 (all as in section 1). Then there is some p* 6 Ax with z(p*) - 0. There exists a price equilibrium.

Proof. Let <j> be the map <b: At -» J 0 given by (pip) « zip)—<£!-i *fo))p-

Then the following lemma is easily checked:

Lemma, (b^p) > 0 if pt « 0. Furthermore if z-satisfies, z%ip) £ 0 if pt « 0* instead of the stronger boundary condition, then <p satisfies the same.

Apply Theorem F to obtain^* with (pip*) < 0. Then

Take the dot product of both sides with p: This yields by Walras Law that S - i *&') " ° » p r o ^ i Corollary 1.

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5. Smalt, Price adjuttmtM and Global Newton 119

Proposition 3. The theorem is valid under the weaker hypotheses:

(a) ip is continuous (not necessarily C2).

(b) $i(p) £ 0 if Pi » 0 (rather than strict inequality).

Proof. Let a <p be given as in Proposition 3. Given « > 0, using the Weier-strass approximation theorem, one can easily construct a \f>: d t -» A0 satisfying the hypotheses of the theorem with iff uniformly approximating <p on A^ within s. Now let t, -» 0. Choose ^ ( 0 as above with « - c, and let/*40 satisfy ^(0(p)<0 m

0 according to the theorem. Then the pw will have a subsequence converging to p* 8 Jj with (p(p*) - 0.

Then following the proof of Corollary 1 we have:

Corollary 2. Corollary I remains true if the excess demand is merely supposed continuous and to satisfy the weaker boundary condition, zfo) £ 0 ifp( 0.

In ail the above, the excess demand with minor changes could have been derived from a micro-economic setting. We hope to pursue this in a future account

In the very general existence theorems of economic equilibria, e.g., as in Debreu's Theory of Value*, one is given the possibility of production with constant returns and preferences which are not strictly convex. In these cases only an excess demand correspondence is obtained. However there exist theorems which yield continuous functions approximating such correspondences. Debreu has given me a reference for this: Cellina (1969). Thus even in these cases one can obtain existence via the above procedure.

We end by quoting from Scarfs book (1973, p. 29) concerning Brouwer's fixed point theorem:

'The transformation of an analytical question concerning the solution of equations into a geometrical statement about continuous mappings of the simplex is in many instances quite artificial, but it may be necessary for the application of this powerful technique.'

We hope that we have shown in this paper that this 'transformation', at least in some sense and in some ways is not so necessary.

References Abraham, R. and J. Rabbin. 1967, Transversal mappinp and Sows (Benjamin, New York). Arrow, K. and F. Hahn, 1972. General competitive analysis (HoWen-Day, San Francisco, CA). Arrow, K. and L. Hurwicz, 1958, The stability of the competitive equilibrium I, Econometric*

26,522-552.

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120 S. SmaU, Prici adjustmmt and Global Ntwtoit

Arrow, K., H. Block and L. Hurwicz, 1959, The stability of tbe competitive equilibrium U, Ecooometrica 27, $2-109.

Cellisa, A., 1969, Approximatioa of set functions and fixed point theorems, Annali di Matematice, Pura ex Applkata 82,17-24.

Debreu, C., 1970, Economies with a finite set of equilibria. Ecooometrica 38,387-392. Debreu, G., 1974, Excess demand functions, Journal of Mathematical Economics 1,15-22. Eaves, C. and H. Scarf. 1975, The solution of systems of piecewise linear equations, Cowles

Foundation Discussion Paper no. 390 (Yale, New Haven, CT). Hirsch, M., 1963, A proof of the noa-retractibtlity of a ceil onto its boundary, Proceedings of

the American Mathematical Society 14,364-465. Hirsch, M. and S. Smaie, 1974, Differential equations, dynamical systems, and linear algebra

(Academic Press, New York). KeUog. B., T.Y. Li and J. Yorke, forthcoming, A method of continuation for calculating a

Brouwer fixed point, in: S. Karamadiar, ed., Computing fixed points with applications (Academic Press, New York).

Milnor, J., 1965, Topology from tbe differentiable viewpoint (University Press of Virginia, Charlottesville, VA).

Quirk, J. and R. Sapoiaik, 1968, Introduction to general equilibrium theory and welfare economics (McGraw-Hill, New York).

Ostrowski, A., 1973, Solutions of equations in Euclidean and Banach spaces (Academic Press, New York).

Scarf. H., 1960, Some etsmpUi of global instability of the competitive equilibrium. Inter­national Economic Review 1,157-172.

Scarf, H., 1973, The computation of economic equilibria (in collaboration with T. Hansea; Yale University Press, New Haven, CT).

Smafe, S., 1974, Global analysis and economics IV, Journal of Mathematical Economics 1, 119-127.

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JounuU of Mathematical Economies 3 (1976) 211-226. O North-Holland Publishing Company

EXCHANGE PROCESSES WITH PRICE ADJUSTMENT

Stephen SMALE Unhtrsity of California, Berkeity, CA 94720, US^i.

Received November 1975

Section 1

The goal is to show:

Main icsnlt. In a pure exchange economy, an exchange, price adjustment process, responsive to transaction costs, which does not stop unless forced to by market conditions, converges to a price equilibrium. There exist such processes starting from any state of any pure exchange) economy.

Before we present the mathematical model (section 2), we give some discus­sion. Studied here is a non-tatdnnement process in a pure exchange economy with price systems. Our treatment has much of the spirit of works published in 1962 by Uzawa, Hahn, Hahn and Negishi, Morishima and surveyed in Arrow-Hahn (1971, ch. 13). See especially the last reference for the history and discus­sion of the economic side of this problem.

It is also the case that our treatment has a number of new and different features from the 1962 work. For one thing, our economic process is not supposed to satisfy some ordinary differential equation. Such a hypothesis implies too deterministic a setting for a state of an economy moving in time. Economics seems to be different from theoretical physics where the state at one time deter­mines the state at all future times. Economic forces don't seem to have such a Newtonian flavor and one has to be careful about carrying over methods from physics to economics.

Another consideration is that there is no (apparent, at least) economic prin­ciple which gives us a differential equation. At best, an-assumption that an eco­nomic process satisfies a differential equation should be treated similarly to the assumption that preference relations can be represented by utility functions.

Rather than differential equations, our work focuses on cone fields given essentially by non-linear differential inequalities and equalities.In this way we have been able to minimize or eliminate ad hoc hypotheses.

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212 S. Smalt, Exchange processes with price adjustment

The last sentence of our result shows that the processes we work with exist in a great variety of situations. This kind of theorem seems not to have been shown in the earlier models.

What can one say about an economic interpretation of this model? My closest experience with such a market is a "mineral bourse" or large mineral show, and the essence of the economic process of this market seems to fit into the work of this paper. This kind of market lasts two or three days; initially, buyers, dealers, traders (many agents are all three) bring mineral specimens of some value and/or money. When the show starts, minerals are traded, bought and sold, more quickly at first, and one sees price equilibrium reached in the afternoon of the last day as prices stabilize and exchange slows to a halt The exact equilibrium depends on factors such as which agents first encounter each other.

Of course, money also plays a role in the mineral bourse, but I believe our model here could be developed to include money on that level. Also our model does not deal with the detailed mechanisms of exchange and price adjustment Again this can probably be dealt with in our framework. Both of these seem interesting lines of investigation. The work of F. Fischer (1972] is relevant

We would see some kind of behavioral strategy as. the force behind agents' moves in explicating the model here.

The kind of goods that don't fit into this model are those like labor, or perish­able ones. Our feeling is that the theory of economic processes is best developed in two parts depending on two idealizations of goods into durable and perish­able classes. This paper deals with the durable class and Smale (forthcoming) is concerned more with a way of looking at the perishable class. A theory of production processes may have to do with bringing the two together.

To elaborate a little on the above "main result", a process is given by a path over time in the space of states of a pure exchange economy. The path associates to each time an allocation of resources of the economy and prices for each good.

This process, in order to qualify as an exchange price adjustment process, is supposed to satisfy two hypotheses, an exchange axiom and a price-adjust­ment axiom. The exchange axiom asserts that (a) the total resources of the eco­nomy are constant (there is no production), (b) exchange takes place at current prices, (c) an exchange increases satisfaction of the participating agents, and (d) some exchange will take place, provided it is possible consistent with (a), (b) and (c).

The price adjustment axiom is defined in terms of a short-run version of demand. A total excess demand approach requires long-run optimization on the part of the agents and is less consistent with our spirit, which is closer to that of behavioral strategies. At given prices and goods processed, one defines the short-run demand of an agent to be the direction his preferences take him when restricted to his budget set

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S. Smalt, Exchange processes with price adjustment 213

The price-adjustment axiom asserts that prices adjust in the direction of some weighting of the short-run demands of all the agents.

Classical hypotheses of convexity and monotonicity on the preferences of agents are assumed, as well as differentiability. Also, we require a boundary condition on the preference relation which prevents an agent from trading off every last bit of some commodity.

A Walrasian price equilibrium in a pure exchange economy depends on the trader's endowments. Thus if one allows a real passage of time, say, an actual exchange to take place, and several such, this initial endowment becomes lost in the shuffle. Thus if one allows this kind of time passage ("non-tatdnnement") one must replace a Walrasian price equilibrium by a different notion of price equilibrium. For our purposes, i.e., as stated in our main result above, a price equilibrium is a feasible allocation and price system where, for each agent, satisfaction is maximized on his budget set defined relative to his wealth at equilibrium. Equivalently, a price equilibrium is an optimal allocation together with a supporting price system.

I first spoke on this non-tatdnnement process four years ago and during this passage of time, it has developed into the form presented here. Discussions with many economists have been very helpful in this development. The first ideas (with cone fields) are in Smale (1974), and a later form is in Smale (forthcoming).

SecrJoa 2

Here we give the mathematical formulation of the results expressed in section 1.

A description of the economy with our assumptions goes as follows. Take commodity space P as

P - ( xeA' | x - (** . • • • . A * ' > 0 .

We suppose that preferences for the ith consumer can be represented by a C1

utility function «,: P -» R, i - 1 , . . . , m, with the following properties:

Monotonicity (differentiable version). The gradient of w, at x, grad ufe), is in P, for each x in P.

Convexity (differentiable version). The second derivative D2u^x) (as a bilinear symmetric form) on the space

oeR'\vgndu^x)mO

is negative definite. We are using the inner product on Rl in the expression vgndufa).

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214 S. Smalt, Exchang* pnxxsses with price adjustment

Boundary condition. For each e e R, ut'l(c) is a closed subset of Rl.

These conditions have been much discussed in some recent literature [see Debreu (1972) or Smale (1976a)]. Possibilities for their relaxation will be dis­cussed (section 5).

While the three assumptions are expressed in terms of a utility function, these properties depend only on the underlying preference relation. Let

5 + - xeP\ Ml - 1,

where ||x||a - £ (x1)2. If ««: P -* R a a utility function satisfying the above conditions, define g,:P-+S+ by gfc) » grad «,0r)/||grad u&x)\\. Then g, can be defined by the indifference surfaces of ut directly as the unit normal, and we will use the gt which have a utility representation as above in the follow­ing.

The economies we consider will have fixed total resources denoted by re P. An economy means the data (r, gu..., gj) as described.

The space of (feasible) allocations is then

Wr- W- xe(Pym\x-(xl,...,xJ, xteP, 2 > , - r ,

using vector notation. (In section 3 where prices are ignored, W is the set of states of an economy.) The Pareto optimal points in FT form a submanifold of dimension m— 1 which is denoted by 9 [Smale (1976a)].

Prevailing prices will be elements of 5+ and so we speak of states of our economy as elements (x,p) of Wx S+, where x » ( x l f . . . , x j is an allocation and p — (p1,..., p') is a price system. Here x, denotes the vector of goods of the ith agent andpj is the price of one unit of they'th good.

An economic process in this context is a Cl map 0: [a, b) -* Wx S+, where <b(t) - (x(t), p(t)) is the state of the economy at time t. Such a process can be thought of as a continuous sequence of exchanges and price adjustments. We have chosen b < ca and left the interval open ended. If one used a closed interval [a, b], we would be in a somewhat similar situation except that con­vergence as / - b would be a foregone conclusion. We prefer not to have such a built-in hypothesis.

The exchange axiom on the process, [a, b) -* Wx 5+, t -* (x(t), pit)), takes the following mathematical form:

Exchange axjon

(a) p(t)'x'tf) — 0, each t e [a, b) and i — 1 , . . . , m (or exchange takes place at prevailing prices). Here x\ or xj(f) is dx^di (f) and we are using the dot pro­duct

352

5. Smalt, Exdianft pneatts with priet adjustment 215

(b) If x'f * o, then x't -glx^ > 0, for each t and i [if agent number j exchanges some goods, he increases satisfaction; note by the chain rule, x\gfjc^ > 0 if and only if d/dr "<*,(')) > 0]-

(c) Some x[ + 0 if possible, consistent with (a) and (b) (some agent will trade if the market situation permits).

To spell out (c), at t, some x[ is not zero if there exists a non-trivial solution in it - (x, x j , x, e R', of the following system of equations:

p(tyxt 0, i - 1 , . . . . « ,

xjgjixfc)) > 0, if ic, * 0 , each/

Implicit already in our notion of process is the condition that the total resources are constant or that £ x£t) — r.

For a process to qualify as an exckange/price-adjustment process in addition to the previous exchange axiom, it must satisfy a price adjustment axiom. Toward this end, define the short-run demand d£xitp) of agent / with goods x( eP at pricesp e 5 + by

d^x„p) = *,glxd,

when xr:R'-* Rl is the orthogonal projection of Rl along p onto the linear subspace p L of vectors orthogonal to p. We are only concerned with the direc­tion of d^xt,p), not the magnitude. Another way of expressing dt(xbp) is to consider the budget set of agent / with goods x, at prices p,

J - yetflp-y-p-x,.

Restrict the utility function u, to B and take the gradient at x„ grad «il«C*i). Then there is X > 0 so that

grad ut\Axd - kd^x„p).

Thus d, is the direction of increasing satisfaction fastest at given prices. The price adjustment axiom says that prices change according to forces generated by these short-run demands. Precisely, say that a process t — (x(t),p(t)) satisfies the price adjustment condition if:

Price adJMtmtat axjoa

There is some < > 0, and for each te[a, b), there is a set of numbers lt > «, /, < 1/a, i - 1 m, withp'W - I IMtuP).

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216 5. Smaie, Exchange processes with prict adfustmau

Here the « imposes a slight uniformity. Roughly speaking, if these demand forces are in the same general direction, the prices adjust correspondingly, while if the infinitesimal demands are in opposing directions, no constraint is imposed on p'(t).

It is easy to construct an example of an oscillating curve where xt) doesn't converge to anything. However, these examples can't happen near the Pareto optimal set or if a hypothesis is made reflecting some idea of transactions costs. We make this precise by giving mathematical content to the condition on the process "responsive to transaction costs". First the condition is given in terms of utility functions. The process [a, b) -» ff x S+ , t -* (x(t), p(t)) is responsive to transaction costs if either

(a) Um t — b x(t) r\9 ** <p, where 9 is the set of Pareto optimal points, and lim,.^ x(t)a the set of all limit points *(/,) for all sequences tt -»b\

or

(b) there is 5 > 0, so that £ [«,(*,(/,))-!<,(:»:,(/<,))] £ 6 £ I W r J - ^ / J H for ail pairs f0> h e fo *) w ^ *o < h-

This says that (away from 9), relative to the size of the transaction, there is a total gain in satisfaction, perhaps small.

An equivalent way of expressing this condition is, off of some neighborhood of 9, the sum of the angles between x&t) and the indifference surfaces through x&t) are bounded uniformly away from zero as r -+ b.

Note that we have not defined the notion of transaction costs itself and that this condition for the process depends only on the preference relations and not their particular representation by utility functions.

We add that if the process is assumed to satisfy an ordinary differential equa­tion on Wx S+, this condition could be eliminated.

Finally, note that this condition does not involve prices, just the exchange process.

What should the phrase mean, "process doesn't stop unless forced to by market conditions"? How should this condition be modeled?

Suppose, for example, that an exchange/price-adjustment process (x(t), p(t)) converges to (x0, p0) as t tends to b and yet a non-trivial exchange/price-adjust­ment process can start at (xo,p0)> Then surely the process has stopped pre­maturely or is incomplete.

But there is an easy example of an exchange/price-adjustment process (*(/), pit)) such that x(t) — x0 while pit) doesn't converge and yet a non-trivial exchange process can start from x0,p0 for any pQ in the limit set of pit). Thus in this case exchange is possible for any limiting price system and so the process should be considered incomplete. These considerations motivate the following definition:

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S. Smale, Exchange processes frith price adjustment 217

An exchange/price-adjustment process [a, b) -» WxS+, t-+x(t), pit)), is said to be incomplete if

(a) (x(t), pit)) converges to (x0, />„), and a non-trivial exchange price adjust­ment process starts from (x0,p0);

or (b) x(t) converges to x0, and for any limiting price system p0 e lim p(t),

non-trivial exchange can take place at (x0, p0) in accordance with the exchange axiom.

A process which is not incomplete will be called complete. Thus "complete" formalizes the condition "stops only when forced to by

market conditions". Recall that a (non-tatdnnement) price equilibrium is a pair (x,p)e WxS+

which satisfies the conditions gfaD — p for i » 1 up to m. The set of price equflibria forms aa (m— l)-dimensional submanifold A of WxS+ [Smale (1976a)].

Now we can give a mathematically precise restatement of the main theorem of section 1.

Theorem. An exchange price adjustment process [a, b) -+ WxS+, t -* WO). pit)), complete and responsive to transaction costs, must converge to a price equilibrium as t tends to b. There exists such processes starting from any state of any pure exchange) economy.

The proof will be given in section 4. Meanwhile, a simpler version of this result (no prices) will be stated and proved (section 3).

Section 3

We consider here the main theory of the paper for the case of a pure exchange economy, but no price systems. The situation is somewhat simpler than with prices and yet many of the basic ideas are present In the next section, prices will be added. The notation and the hypotheses on the economy set down in section 2 will be used. Here complete proofs of the results will be given.

Thus W will denote the set of states (for the purposes of this section) of a pure exchange economy with fixed total resources re P. An economic process is given by a map

[a, b) - W, t - x(t) - (x,( /) , . . . , xJLt)).

This process satisfies the exchange axiom (compare with the previous section), provided:

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218 S. Smak, Exchange processes with price adjust****

Exchange axiom. For each t in \a, 3),

(a) x',g^xj > 0. if x\ * 0, i - 1 m. (b) Some x\ # 0, if there exists a non-trivial solution in x" - (x„ . . . , x j ,

X, 6 £ ' of the following system:

*r* i (* i>>* * *«*<>, / - l , . . . , m .

This means (a) for someone to trade, he must improve his lot, and (b) if trade is feasible consistent with (a), some trade will take place.

Built into the construction is the condition that £ xt » r. A curve t -* x(t) satisfying the exchange axiom will be called an exchange

curve. An exchange curve t — x(t) is called incomplete if Um(_» x(r) « x0 and a non-trivial exchange curve starts at x0. An exchange curve is called complete if it is not incomplete. Responsive to transaction costs is the same as in section 2.

Theorem. Suppose [a, b)-* W, t-+ x(t) is an exchange curve responsive to transaction costs and complete. Then as t tends to b, x(t) converges to a Pareto optimal point of W. Furthermore, given any economy, as in section 2, and any non-optimal state, there exists a complete exchange curve, responsive to trans­action costs, starting at that state.

Toward the proof of the theorem, we elucidate the exchange axiom. For each x e W, a cone CJ of vectors will be defined (it can be thought of as a set of tangent vectors to W at x). This cone depends on the economy and x but not a process (i.e., an exchange curve). Let

C ; - x e ( * r i * » ( 2 i *J ,

x , e * ' , I * , - 0 , X,-gtxd>0 or X, - 0, each/.

It can be seen from the definitions that:

Fact. A curve x: [a,b)-* FT is an exchange curve if and only if x'(/) e CJ(r) each t and x'(f) + 0 if C?(t) * 0.

This (cone) CJ plays a key role in this paper. It is neither open nor closed.

Proposition 1 (a) CJ is a convex cone. (b) CJ — 0 if and only ifx is Pareto optimal.

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S. Smalt, Exekang* processes with price adjustment 219

(c) The function £ uf is a strictly increasing function along an exchange curve at all non-optimal points.

(d) If t - x(t) is an exchange curve and x(t0) is optimal, then x(t) = x(t0) for all / £ f„.

The proof of (a) is routine and simple. For (b), one uses the characterization of 9, the set of Pareto optimal points by the first-order condition [see, e.g., Smale (1976a)] : x e 9 if and only if all the g^'i) coincide. Let

C „ - 2 e ( * ' ) J l * , - 0 ; x,*(*i) > 0, each/.

Then it is easily seen and known that x is Pareto optimal if and only if Cx is empty. Since CJ => C„ if CJ - 0, then Cx is empty and so x is optimal. On the other hand, let x be optimal; then all the normalized gradients gfad coincide. So if X e CJ, and x » (x , . . . , x j , then all the non-zero x, lie in the same open half space; and £ x, = 0 implies in fact that each x, is zero. This proves (b).

For part (c), let / -* x(t) be an exchange curve. Then the condition that x<(0'ffi(Xf(0) £ 0 all t, implies that u, a non-decreasing along the curve, each i. On the other hand, if x[t) ft 9, x't-g,(x^) > 0 for some /'; thus at that t, a, and £ ut are strictly increasing. This proves (c). Then (d) follows. The proposition is proved.

Proposition 2. Suppose the exchange curve x: [a, b) -* W is responsive to transaction costs. Then lim,-.» x(t) exists.

Proof. Consider first the case that the curve / -*• x(t) stays outside some neighborhood of 9, and let a, p belong to the limit set (lim x(t)\t -* b. Suppose a ft P, and let d — ||<x—0||. There is an increasing sequence tj such that | W J J ) - a | | < d/4 and ||x(r2l+1)-/?|| < d/4 for all i. Thus IWf„+ , )-*(i„) | | > d\l for all /. From the transaction cost condition there is some k with uk(xk(t)) ~» oo. From the boundary condition and total resource constraint this is impossible. Thus at - P, and in this case lim,..* x(t) exists.

Next consider the case that lim,..* x(t) r\9 ¥• <p, and let a € 9 satisfy 2 e lim x(t), with at = (att,..., O - Suppose p # a is also in this limit set Let tk -* b be an increasing sequence such that x(t2J) -* a, xfoy+i) -* p. For any i, since utxlt2j)) — i/,(<x,) and ufefa)) is non-decreasing, it follows that ut(ad " uAPd- Since a is optimal, then p also is; p e 6. But u on 9 is on* to one [see, e.g., Smale (1976a)]. Therefore a - /?, proving Proposition 2.

Propositions. (ExtensionProposition). Given any x0 e ff, <x0 6 C£,, a0 ?* 0 ifCx0 * 0, there is a function a which assigns to each xeW, a(x) e CJ, <x(x) 9* 0

357

220 5. Smaie, Exchange processes with price adjustment

ifCtitO and a(x0) = <x0. Furthermore, >(x)i! £ s0'f!+l for all xeW,tt is continuous and C1 off of 9. In other words, x is a vector field on W, extending ct0 with the properties mentioned.

Proof. By an easy version of the Whitney extension theorem [see Golubitsky-Guillemin (1973, p. 17)] there is a Cl function p on W, with values in [0,1], which is zero exactly on 9, and is 1 at x0 if x0 $ 9.

If fl can be defined on the set W—9 satisfying the properties of Proposition 3, then we are finished by defining

<x(x) - px) 0(x), for xeW-0,

cc(x)«0, for x e 0 .

Now P can be constructed locally as in the proposition by a constant extension and a partition of unity using a locally finite covering [as in Golubitsky-Guillemin (1973) for example] yields a globally defined 0 since the cones C* are convex. Just note that in this process, if <x(x) e CJ, <z(x) # 0, then <x(x) e Cf for y near x.

We are now ready to prove the theorem of this section. Let be given t — x(t) as in the first part of that theorem. Then by Proposition

2, xt) converges as t -* b to a point, say x0, in W. Suppose that x0 it 6. Then CJ, it 0 by Proposition 1. Let non-zero <x0 e C]£ and apply Proposition 3 to obtain an extension a to W. Solve the ordinary differential equation x' » <x(x) to obtain a solution / -* x(t) with initial condition x(0) — x0. Since x'(t) » at(x(t)) e CJ„ x(t) is an exchange curve (non-trivial) starting from x0. But this couldn't happen since our exchange curve was assumed complete to begin with. Thus x0 e B, and the first part of the theorem is proved.

For the second part of the theorem, given a state x0 $ 9 of some economy, we take any non-zero a0 in C£. By Proposition 3, extend <x„ to a as in the preceding paragraph. Solve the ordinary differential equation x' — <x(x) with initial condition x(0) = x0. This yields an exchange curve t — xt) with x(0) — x0. Along any non-trivial solution curve £ u, is a strictly increasing function by Proposition 1. Also limx(r) is not empty by the boundary condition and total resource limitation [x(t) stays in a compact set]. Under these conditions, it follows from properties of differential equations that if y 6 limr__ x(f), then a(y) « 0. Thus ye 9, and as in Proposition 2, x(r) converges to y. By a re-parameterization as in Schechter (1975), x can be assumed defined on the interval [0,1), thus finishing the proof of the theorem.

See also Schechter (1975) for related theorems.

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5. Smalt, Exchang* processes with price adjustment 221

Section 4

We give the proof of the theorem of section 2 (and thus the main result of section 1).

The first part of this proof depends on the construction of a field of cones over rVxS+ corresponding to the exchange and price adjustment axioms. More precisely, for each (x,p) in Wx. S+ define

Bx,, * e CJIx,-p - 0, each /.

Let Axr - Bxp - 0 if BXt$ * 0. Otherwise let Ax<t - Bx%r - 0. Here C? has been defined in the previous section. Note that / -* (*(/), p(t)) satisfies the ex­change axiom of section 2 if and only if x't) 6^x(t) i ^o for all /.

Define for each « > 0,

4,,,(«) - \pe#\p-p - 0, p - £ Wxup), « < / , < -C i-i «

and

V** - U *>«.»«• •>o

Then / -» (x(t), pt)) satisfies the price adjustment axiom of section 2 exactly if there is some e > 0 andp'(0 € Dxit)i MO(e) for all t.

Write <p — (x,p) and let

and

t > 0

Proposition 1 (a) IT,,, J'J a convex cone. (b) 7,,, » 0 if and only i/(x, p) is a price equilibrium. (c) If DXtf contains 0, but isn't 0, then Ax>f # 0. (d) (x, pje A if and only if Dx>, = 0.

Note especially (c) which asserts that if p' - 0 (prices are not changing) and (x, p) is not an equilibrium, then AXtf ^ 0 so that exchange takes place in accordance with the exchange axiom.

For the proof, the first-order conditions [see, e.g., Smale (1976a)] assert that for x,p) e W x 5 + , (x,p) e A if and only if g,(xj - p for all /. But Dx.r = 0 if and only if all the dfae„p) - 0 or equivalently gt(xd - p, all /. This gives (d).

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222 S. Sutale, Exchange processes with price adjustment

Of course if Yx?f = 0 then DXtP » 0 and so (x, p) e A. Thus for (b) it is suffi­cient to show that if (x,/>) e A and Dx%, 0 then Axp - 0. But if (x,p) e A then x is optimal [the fundamental theorem of welfare economics, e.g., Smale (1976a)] so C* » 0 (compare section 3). This implies (b).

The proof of (a) is direct and we omit it It remains to prove (c). By the hypothesis of (c), we may write £ I(d,x„ p) » 0

where ltd^x,p) ** 0 for some i. Define X, « l^xitp), each /, and x =* (x l f . . . , xJ . Then x * 0 and £ x, » 0. Since dt » itwg£xi), it follows that if x, # 0, then x,'£,(xj) > 0. Therefore X is a non-zero element of Ax%r We have con­structed in x an infinitesimal trade. Proposition 1 is proved.

For x e Wlet Dx be the cone in Rl generated by x j , i — 1 , . . . . m. Thus

D'x - [p e Rl\p - I /,*(*«), /»£ 0.

Proposition 2. Let t -»(x(/), /Kf)) 6* on exchange price adjiatment process with x(r) -» x0 aj f -♦ 6. Assume \imt~hp(t) doesn't exist. Then every pa in the limit set, lim/KO satisfies

(a) p0eD'X9,and (b) />0 is not a vertex of D'^.

One may restate (b) as (b') ifp0 - g£(x0)i)for some i, then ^ ( x 0 ) ^ =- £ , /# , «*o)y) ', £ 0,*/(x0),) * *,((*o)«).

We give this proof in outline with details to be filled in by an interested reader. First, let

4 ,00 - inf X0>,-*)2,

where pb q( are the coordinates ofp and q, respectively. It can be shown from the price adjustment axiom that if a\pit)) is positive,

then it decreases as t increases. Next one shows that near p0 in Proposition 2, p' must lie in a field of uni­

formly pointed cones. Putting these two pieces of information together yields a proof of Proposition 2.

The rough idea in the following proposition is to show if p e D'x then Ax%t # 0, via the route that p e &x if and only if 0 s Dx%t and then apply Proposition 1(c). However, these statements are not exactly correct We proceed:

Proposition 3. Ifp e D* andp is not a vertex of !/„ then Ax%r j* 0.

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S. Smak, Exchange processes with price adjustment 223

Proof. We need to sharpen the argument of Proposition 1(c) slightly. Let J be the set of j such that p — gj(xj). If / is empty, then let p - Ylg^x,),

I, £ 0, where not all the /, are zero. Apply rtr to obtain 0 — YJMxt> P) ™ £ *<• Since some x, is not 0, one can finish as in Proposition 1(c).

Suppose now / is not empty and let JeJ. Since p is not a vertex,

P - *•/*/) - E *«*X*i). *i > 0. iaf

Here / is a non-empty indexing set with the property if / e I, then gfad # £/(*/)• Application of *, yields

0 - £ M*(*i.J») - £ *«.

with x, not zero and again one finishes as in 1(c). This proves the proposition.

Proposition 4. (Extension Lemma). Let (x0, p0) e fVx S+ and (JT0, n0) e Y*o.*r Then there exists an extension (X,«) of(X0, «0), (Xxrp), Jt(x, p)) e ?*„, with X, n continuous inx,p,it being Cl and X being C1 off of the set where X — 0. T7msX(x0,pQ) - 2"O,K(XO./>O) " *<>•

JVoo/ We construct first n, then Jr*. Since 0,,, is a cone, if it is constructed locally, a partition of unity will finish the construction. Thus let (x*, p*)e W*S+. If (x*,p*) - (x0,p0), let n(x*,p*) - x(x0,p0). Otherwise let >t(x*,p*) be an arbitrarily chosen element of Dx,tf^ In either case we can write *(x*, p*) — Y.Wx?,p*) with /, > 0. For (x,p) near (x*,p*), let nx,p) - JJMtbP)-This gives a desired local extension and hence a global extension for Jt.

Now let

Q-(x,p)erVxS+\AM,,-0.

We will show that Q is a closed set, or that the complement of Q is open. Let (x*,/>*)#Qand(ic',p')e^.^;letybethe(non-empty)setof/withJcl' ¥• 0.

Consider for x, p near (x*. p*) the following system of equations for x, 6 R',

(i) 3c,-0, for /*/ ,

(U) *Vp - 0, for each /« / ,

(iii) £ x, - 0,

(iv) X,y,(xO>0, for / s / .

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224 £. Smalt, Exchange processes with price adjustment

By linear algebra, since (ii) and (iii) have a solution for p » /?*, a solution will exist for p near p*. Furthermore, this solution will be close to the original x[. Therefore, (iv) will be satisfied for x near x*.

This proves our assertion that Q is a closed set. Next, construct p: WxS+ -* [0, 1) which is zero exactly on Q, continuous

and Cl off of Q. The proof is finished just as in Proposition 3 of section 3.

Now to the proof of the main result Let r -+ (x(t), pit)) be an exchange price adjustment process, complete and responsive to transaction costs. Then by Proposition 2 of section 3, we have convergence x(t) -» x0. If p(t) converges also, say to p0, then (x0, p0) e A. Otherwise, application of Proposition 4 in the same way as the extension lemma was used in section 3, would violate completeness

Suppose on the other hand p(t) doesn't converge to a point. Let some price system p0 be in the limit set. Apply Proposition 2 to see that/>0

6 '»0» DU* that p0 is not a vertex. Now apply Proposition 3 to see that A.x^^ is not zero. Finally, application of Proposition 4 shows that completeness is violated. This finishes the proof.

Section 5

We indicate briefly and tentatively how a discrete version of the main result could go. This version allows exchange at different prices at the same time and has some other advantages; perhaps one could eventually dispense with differ­entiability using this approach.

Think of the real numbers R as literally time and let / be the set of positive integers; some bounded map a:J-* R+, such that oc(q+1) £ <x(q), for each qeJ. A process is then some a as above and a map / -*• fVx S+ where x(q) -* x(q+1) is interpreted to be an exchange at prices p\q) at date a(q), for qeJ.

To qualify as an exchange price adjustment process, the process must satisfy the following two axioms.

Exchange axiom (a) p(qy(xAq+l)-xq))-Q. (b) ««<*<(?+ D) £ «<(*<(*)) with strict inequality if x,(q+1) * xq). (c) Given?, x(q),p(q), then x(q+1) ?* x(q) provided there is some x(q+1) ?*

xq) satisfying (a) and (b).

Price-adjustment axiom. There is some « > 0 such that p(q+l)-p(q) — ^Ifd^x^q^piq)), e < I, < 1/e, each /'. The process will be called responsive to transaction costs provided either (lira x(q)) n 9 * 4> or there is a <5 > 0 such that Z O i f o f e + l ) ) - * , ^ ) ) * aiW*+l)-*fo)|| for each?.

Completeness is defined analogously to the definition of section 2.

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S. Smalt, Exchange processes with price adjustment 223

One also defines the notion of "localizable", or behaviorally oriented for the process to mean that given any 6 > 0, one can interpolate a new exchange/ price-adjustment process by adding intermediate trades and adjustments so that each move is less than <5.

Then with the added hypothesis of localizable from the previous paragraph, the main theorem of section 2 would seem to be true in this context also.

Return now to consider the problem of improving on the main result of section 1. There exist numerous examples to show no hypothesis can be dropped and the conclusions remain valid. However, there are definite limitations in our hypotheses. Some have been discussed previously.

The boundary condition on the preference relation, even though it has been used frequently in mathematical economics, is hard to justify on economic grounds. On the other hand, there has been enough work in difierentiable mathematical economics with mild boundary conditions, so that this limitation does not seem to be serious. Schechter's (1975) developments in this direction are the most complete to date.

If one were to abandon the convexity hypothesis of the preference relation, I believe with slight modifications the theory would go through. The convergence would lead, however, to an allocation which is locally optimal and to a state which would not be a price equilibrium, but only an extended price equilibrium in the sense of Smale (1974).

Probably one could also replace a price system out of equilibrium with some kind of cone of possible prices.

References Arrow, K. and F. Hahn, 1971, General competitive analysis (Holden-Day, San Francisco,

CA). Debreu, G., 1972, Smooth preferences. Econometric* 40, 603-616. Fisher, F., 1972, On price adjustment without an auctioneer. Review of Economic Studies 39,

1-15. Golubitsky, M. and V. Guillemin, 1973, Stable mappings and their singularities (Springer,

New York). Hahn, F., 1962, On the stability of pure exchange equilibrium, International Economic Review

3,206-213. Hahn, F. and T. Negisbi, 1962, A theorem on non-tatonnemcnt stability, Econometric* 30,

463-469. Morishima, M., 1962, The stability of exchange equilibrium: An alternative approach. Inter­

national Economic Review 3, 214-217. Scbechter, S„ 197S, Smooth paieto economic systems with natural boundary conditions,

Thesis (University of California, Berkeley, CA). Simon, C. and C. Titus, 1973, Characterization of optima in smooth Paieto economic systems,

Journal of Mathematical Economics 2, 297-330. Smale, S., 1973, Global Analysis and economics I, in: M. Peixoto, ed., Dynamical systems

(Academic Press, New York). Smale, S., 1974, Global analysis and economics HA, Journal of Mathematical Economics I,

1-14.

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226 5. Smalt, Exchange processes with price adjustment

Smale, S., 1976a, Global analysis and VI, Journal of Mathematical Economics 3, 1-14.

Smale, S., forthcoming. An approach to the analysis of dynamic processes in economic systems, in: C. Schwddiaiier, ed., Equilibrium and disequilibrium in economic theory (Redd, Dordrecht).

Smale, S., 1976b, A conversant process of price adjustment and global Newton methods, Journal of Mathematical Economics 3,107-120.

Uzawa, H., 1962, On the stability of Edgeworth's barter process. International Economic Review 3,218-232.

364

CoUoques Internationaux du C.N.R.S. N" 259. - Syitemes dynamiquej et modele* loonomiquei

SOME DYNAMICAL QUESTIONS IN MATHEMATICAL ECONOMICS

Steve SMALE University de Californie - Berkeley

RESUME Cette courte note met a jour mon article de l'American Economic Review [3].

Un theme particulier de cet article ett divelappi ici, a savoir le lien entre la nature des biens et la notion d'tquiiibre a retenir.

Let me start by posing what I like to call "the fundamental problem of equilibrium theory" : how is economic equilibrium attained ? A dual question more commonly raised is : why is economic equilibrium stable ? Behind these questions lie the problem of modeling economic processes and introducing dynamics into equilibrium theory. A successful attack here would give greater validity to equilibrium theory. It may be however that a resolution of this fundamental problem will require a recasting of the foun­dations of equilibrium theory. One might well keep in mind some historical perspective from physics, making an analogy between Walrasian equilibrium theory and Newtonian mechanics.

How did Relativity Theory respect classical mechanics ? For one thing Einstein worked from a very deep understanding of the Newtonian theory. Another point to remember is that while Relativity Theory lies in contra­diction to Newtonian theory, even after Einstein, classical mechanics remains central to physics. I can well imagine that a revolution in economic theory could take place over the question of dynamics, which would both restruc­ture the foundations of Walras and leave the classical theory playing a central role.

In the direction of attacking this fundamental problem, it seems to me important to idealize economic goods into two extreme classes. One one side are the durable goods, and on the other, the perishable, renewable goods with especially labor as an example. To each of these two classes of goods, one can let correspond two basic branches of equilibrium theory. As an illus­tration, Debreu's "Theory of Value" [1] has two substantive chapters,

365

96

Chapter 5 on the existence of (Walras) equilibria and Chapter 6 on "the fun­damental theorem of welfare economics". I believe that one can associate the durable goods most naturally to the models in welfare economics and the renewable goods to the Walras equilibrium theory.

To see these things, it is useful to explicate the conditions for equilibrium. Assume classical (differentiable version) hypotheses on preferences for example as in [2]. Let there be / commodities, m agents in a pure exchange economy. Let price systems be denoted by p = ( p , , . . . ,p ;) each p, > 0, with Zp* = 1. The endowment et of the ith agent will be a vector in R = (e1 , . . . , el) i e> > 0 and an allocation will be an m-tuple

with each xt in R'. The preference of agent i is supposed to be represented by a utility function ut : R'+~* R.

A pair (x , p) consisting of an allocation and a price system is a Walras equilibrium if these equations are satisfied:

(1) The gradient, grad ut (x,) equals Xp for some X > 0, each i= 1 , . . . , m. This is a necessary condition for xt to maximize satisfaction for agent i.

(2) "LXi = Ee<. This is a total resource condition on the allocation x. In other terms, x is attainable or even "supply equals demand".

(3) p . x, = p . et, i = 1 , . . . , m. These dot products give the values and this is a budget condition.

The first two equations by themselves describe the kind of equilibrium used in welfare economics (e.g. Debreu's Chapter VI).

Returning to problem of dynamics, observe that if one is trading a non-tatonment situation with durable goods (or stocks), then the endowment allocation, after some trades will lose its effect and therefore play no role in any equilibrium attained (see [4]). Thus the notion of equilibrium which is relevant is not that of Walras but that of welfare economics.

While in the durable goods market, a commodity vector JC, in R'+ is interpreted as a stock of goods, in renewable goods models, a point in commodity space is more naturally interpreted as a rate (or flow) of endowments or consumptions.

Thus in a model where the endowment of goods is being renewed continually, the endowments e, should play a role in the equilibrium attain­ed and therefore, a Walras equilibrium defined by the full set of equations (1) - (3) is most reasonable.

Perhaps the non-tatonment theory initiated by Hahn, Negishi, Uzawa has developed to handle the dynamics of durable goods of pure exchange in principle. On the other hand, clearly there is no satisfactory model for dynamics of renewable goods and Walras equilibria.

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97 REFERENCES

[1] DEBREU G. - Theory of Value, New York 1959. [2] SMALE S. - "Global Analysis and Economics VI", Jour. Math. Econ., 3, (1976),

1-4. [3] SMALE S. - "Dynamics in General Equilibrium Theory". Amer. Econ. Rev., 66,

(1976), 288-294. [4] SMALE S. - "Exchange Processes with price adjustment", (to appear, Journ. Math.

Econ.).

DISCUSSION

The discussant, Egbert Dierker, points out that, in his opinion, not only the case of perishable but also that of durable goods exhibits a Walrasian character, since the distribution of initial endowments is important for the final outcome. Smale answers that it is useful to study the pure laboratory cases first. Gabszewicz points out that the use derived from a durable good can be treated as a flow. Smale answers that the market for minerals or for houses cannot naturally be described in terms of flows. Bliss asks whether Smale distinction is appropriate. Smale answers that it is essential to know how to deal with labor, a purely perishable good. Production may then play the role of bringing both, perishable and durable goods, together into one model. The problem, however, is how to put stocks and flows into the same model. Harsanyi remarks that the distinction between durable and perishable goods is not that between a tatonnement and a non-tatonnement situation. The distinction rather lies in the fact that resale is possible in the first case but not in the latter.

Fuchs supports Smale's distinction and remarks that the case of perishable goods can be treated by the theory of temporary equilibria. The discussant points that in most models of price formation expectations about future prices play a major role. He asks to what extent agents anticipate price changes in Smale's model. Smale answers that his model had to be altered if individual expectations of price variations are to be taken explicitly into account.

Gabszewicz further remarks that the question of how to connect a given state with a Pareto optimum in the Malinvaud-Drize-de la Vallee Poussin model is closely related to Smale's treatment of the durable goods case. Champsaur and Comet relate Smale's process to Malinvaud's. But the latter process does not converge in finite time. Smale remarks that it is important how equations are defined near equilibria. Fuchs points out that an interest of Smale's model is that the Pareto set is reached in finite time, so one may minimize the time necessary to reach the Pareto set from the initial state. Smale answers that he originally considered processes responsive to time cost.

Guesnerie asks whether it is possible to reach any individually rational Pareto optimum in the durable goods case. Smale answers that it is likely that one can reach a subset of the Pareto surface of full dimension. A result of this kind has been shown by Schecter in a similar model without price adjustment. Kirman explains that in Smale's process an individual may continuously make losses because expectation about price changes are neglected. Smale says that the conditions characterizing his process require an essential change if one wants to handle this problem. Last Selten remarks that a behavioral point of view may be more appropriate than the requirement of full maxi­mization. In order to relate a theory to experiments is should be put in a discrete frame­work. Smale answers that is should be possible reformulate his theory in a discrete set-up.

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STEVE SMALE

AN APPROACH TO THE ANALYSIS OF DYNAMIC PROCESSES IN ECONOMIC SYSTEMS

We propose here a way of studying dynamic processes in economics which is less deterministic than the approach from ordinary differential equations; in this setting economic hypotheses have a more rational mathematical reflection.

The idea is to first explicate as a mathematical set the space of states & of the economic system to be studied. In our models time is continuous and calculus is used extensively. Thus we take if to be a domain of Cartesian space /?". Then a path of the economy is supposed to be a curve in the space of states or a differentiable map <p: [a, 6] —»Sf of an interval into y . Thus q>(t) is the state of the economy at time t.

Next various hypotheses or axioms are imposed on the nature of </>, which represent the salient features of the economic system under con­sideration ; at least these axioms are those whose implications we wish to study in the model. The mathematics now applies to the problem of where these paths lead and to what extent they exist. I have used this ap­proach to study non-tatonnement processes in a pure exchange economy in the direction of work of Uzawa, Hahn and Negishi, as well as a model of duopoly. I will give three examples to clarify what I have said.

Example 1. Edgeworth Box without price systems. Consider the classic situation of two commodities, two traders. Commodity space is P = (x1, x2) e R*\ xl ^ 0, x2 k 0 with x^eP representing the goods of the first trader, x2 e P, the second. The total resources are fixed, say re P, r = (rl, r1), r1 > 0. Then the space of states for this simple system is the Edgeworth Box

W= (x„ x2)ePx P|x, + x2 = r.

Since x, determines x2 uniquely, one may use x, as a single parameter for W, with x, = (x\, xf), 0 £ x g r1, 0 ^ x2 S r2.

We will consider an economic process consisting of a sequence of

G. Schwodiauer (ed.), Equilibrium and Disequilibrium in Economic Theory, 363-367. All Rights Reserved. Copyright© \9VbyD. Reidel Publishing Company, Dordrecht, Holland

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364 STEVE SMALE

small trades and this process is constrained to raise utility levels of each trader. Thus we suppose given C* utility functions ux:P—*R with classical hypotheses of monotonicity and convexity1. Idealizing the process to a continuous one, say that a path <p: [a, b] —» W is ad-missable if

-£-«,(*,(/)) fcO all te[a,b\ at

j '=l ,2 and one has strict positivity if <p(t) is not Pareto-optimal. Here we have written <p(t) - (x, (/), x2(t)). Say that in addition <p is complete if it cannot be extended to an admissible path [a,b+e]—*V,e> 0.

A relatively easy (and semi-classical) result is

THEOREM. Each complete admissible path ends at a Pareto Optimum. If this path does not end on the boundary of W, then this optimum is on an arc characterized by the condition grad u,(x,) = X grad w2(-*i)» some A > 0. Furthermore given any point x in X, there is a complete admissible curve through x.

The same result is valid without convexity and weaker monotonicity provided some mild genericity is assumed and Pareto is replaced by local Pareto. Thus the model shows in an explicit manner how the pro­cess of small trades relates to local Pareto Optimum, under rather general circumstances [2].

The results extend also to many traders and many commodities.

Example 2. Here we add another state variable, a price system to the previous example. This addition brings in close to the 'non-tatonment' processes of Uzawa et al. [1].

Staying with two commodities and two traders for the moment, a price system is a pair of non-negative real numbers, (plt pz). So pt is the price of one unit of the r* commodity. We suppose the price system normalized so that pi + p\ = 1; let S be the set of normalized price systems so that 5 is the part of the circle in the positive quadrant.

The space of states of this economic system ('pure exchange with prices') is YVx S. Suppose the u, are as in Example 1. Consider now a path ft: [a, b] —* W x S in this economy and let

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ANALYSIS OF DYNAMIC PROCBSSBS IN ECONOMIC SYSTEMS 365

0(0 = (*(»). /»(0). <P(t) £ W, pt) 6 S.

Thus we may write q>[t) = (x, (/), x2(t)). Some simple natural axioms on fi are:

(1) —u,(x,(0) £ ° a11 ' e [°. *] ^d o n c h a s s t r i c t inequality in case it utility-raising trading can take place consistent with current price levels2.

(2) Given (x,p)e Wx S, define the infinitesimal demand of the r* consumer (i = 1, 2) by d,(x„ p) = gradM,/fl,(jc,) where uJB, is the restriction of ut to the set B, = xe P\p(x) = p(x), where p(x) =p • x. Then d,(x,,/>) is a vector in J?2 naturally defined at least up to a positive scalar multiple. If (x, p)efVxS, define the demand cone, Dx at (x, p) to be the subset of R1 of all vectors of the form

2>,d,(x„/>), n(>0. i

Axiom (2) asserts that for each /, the derivative p'(t) e Dxl,Uflt). It has the interpretation that prices rise according to the demand.

(3) For i=l,2,p(t)x't(t) = 0. Or trading takes place at current prices.

(4) This is a completeness axiom similar to that of Example 1. Pre­cisely, fi cannot be extended non-trivially to [a, b + e] still satis­fying (1), (2) and (3), s > 0.

Note that £ x,(f) = r IS a consequence of the construction. The result in this case is:

THEOREM. A path satisfying these conditions, 1, 2, 3, which does not end on the boundary, ends at a price equilibrium (relative to the final allocation rather than the initial allocation). Furthermore through each point of W x 5 there exist paths which satisfy conditions 1, 2 and 3.

The price equilibrium in the theorem is a state (x, p) with the property, £ x , = r, and for '=1 .2 , xt is a maximum for w, on the set qeP\p(q)£p(xJ. I believe this is the natural definition of price equilibrium for a non-tatonnement process.

370

366 STEVE SMALE

The theorem can be extended to the case of many consumers and many commodities. It also has extensions to the case of non-convex utility functions. We expect to give a development elsewhere of these ideas.

Example 3. Duopoly (very briefly!). In this example, the space of states Q consists of the scales qu q2, of production for each of the duopolists; thus Q is the non-negative quadrant in R2. An admissible path in this example is a path a: [a, 6] — (? with the property that the profit of each firm is increasing along the path. Conditions on the market which can justify such a path are:

(a) short-term profit maximization is the dominant consideration, and (b) there is a kind of tacit collusion that takes place via indirect

communication of market decisions and threats.

These conditions become especially reasonable in view of our process taking place over a real interval of time rather than being a static process.

Profit functions of each firm on the space of states are derived from natural, non-convex, production functions. One obtains that a path in this economy can naturally lead to a monopoly, or in case the production functions satisfy certain conditions, to either a stable coexistence or a monopoly. The alternatives in the last case depend to a certain extent on the starting point of the path.

The results contrast with other studies of duopoly, starting with Cournot, where a coexisting equilibrium is necessarily reached.

Dept. of Mathematics, University of California, Berkeley

NOTES

1 These conditions, precisely stated, as used here are:

(a) grad «,(*,)e interior P for all xte P. This is monotonicity. (b) For each x,eP, the second derivative LPu^x,) restricted to pairs of vectors

of R2 which are perpendicular to grad u,(x,) is negative definite.

This is a strong convexity condition.

371

ANALYSIS OF DYNAMIC PROCESSES IN ECONOMIC SYSTEMS 367

2 This has the following precise meaning. For some i, one has strict inequality,

4: »i(*i(0) > 0 >" case (x„ x2,p) at

satisfy this condition: There is a vector ve Z?2 with p(v) — 0 and v • gradu,(x() > 0 for J = 1 or 2.

BIBLIOGRAPHY

[1] Arrow and Hahn, General Competitive Analysis, Holden Day, San Francisco, 1971.

[2] Stnale, St., 'Global Analysis and Economics I: (Example 3): Pareto Optimum and a generalization of Morse Theory' in Dynamical System (ed. by M. Peixoto), Academic Press, New York, 1973.

372

On Comparative Statics and Bifurcation in Economic Equilibrium Theory*

By Steve Smale

It is worth-while to consider what Paul Samuelson had to say about this subject in the early 1940's in "Foundations of Economic Analysis" [2, p. 257-258].

..."It was an achievement of the first magnitude for the older mathematical economists to have shown that the number of independent and consistent economic relations was in a wide variety of cases, sufficient to determine the equilibrium values of unknown economic prices and quantities

"It iB the task of comparative statics to show the determi­nation of the equilibrium values of given variables (unknowns) under postulated conditions (functional relationships) with various data (parameters) being specified. ... In order for the analysis to be useful, it must provide infor­mation concerning the way in which our equilibrium quantities will change as a result of changes in the parameter taken as independent data.

"... For few commodities have we detailed quantitative empirical information concerning the exact form of the supply and demand curves even in the neighborhood of the equilibrium point.

Written for the New York Academy of Science, October 1977.

373

"This is a typical problem confronting the economist: in the absence of precise quantitative data he must infer analytically the qualitative direction of movement of a complex system."

Thus the problem is posed: in an economy to study solutions in x of f (x) = 0 as the parameter o changes where f is a system of n equations in n unknowns. This is the problem of comparative statics. A bifurcation is a pair (x,a) where the Jacobian of f at x is zero. Roughly speaking f is the excess demand of an economy and the parameter a could involve either endowments, tastes or technology. We will make this picture precise in the case of a pure exchange economy where x is an allocation-price equilibrium and a is the endowment vector. For example changes in this parameter could be caused by new tax­ation or oil discoveries. The situation in explicit economic models is more subtle because of Walras Law. But let us specify the model in a simple but basic form.

Suppose there are I commodities so that a commodity i. 1 9. i

bundle is a point in R+ = (x ,... ,x ) = x | x > 0. Suppose next that in our simple economic model, there are m traders, each having a preference represented by a "utility function"

A 2 u.:R+ + R of differentiability class C without critical points,

o 1=1,...,m . Thus agent l prefers x to x' in R+ exactly when u.(x) > u.(x'). To each agent is associated an endowment, e. in R. , i=1, •»• ,m .

o 1 i A price system is a vector p e R so that p = (p ,...,p )

and p3 is the price of one unit of the j commodity. The value of a commodity bundle x G R. at price system p is the

374

I . dot product p«x = J p x . We will suppose that price systems

i=l 2 i 2 are normalized, |p] = I(p ) = 1, so that the space of price l-l systems is the unit sphere S . (Usually also the price would

be positive.) A state of the economy is a pair (x,p) in (R+)mXS " where

£ x = (x.,...,x ), x. e R+ is an allocation of the goods to the m traders and p is a price system. The state is feasible if

m m ili-i-iii'i •

Here the vector of total resources of the economy is just the sum of the endowments. The parameter of the economy is the endowment vector e - (e.,...,e e (R+)

An equilibrium state is a feasible state (x,p) where the satisfaction condition is met: For each i, x. is a maximum of u. on the budget set B = x e R+ | p«x = p-ei> . It follows directly that an equilibrium (x,p) for the economy with endow­ments e.,...,e is a solution ot this system of equations:

(1) Exi = Eei

(2) P'X^ = p«e^ i=l,...,m (note the m equation of (2) is redundant in view of (1)).

(3) Si/x^) = p i=l,...,m

In condition (3) g-(x.) is the normalized gradient of u. at x. and (3) is the ordinary first derivative condition for a maximum. For a much more detailed exposition of these matters with further reference, see [*], as well as [1]. The main goal of

375

this note is to give a theorem which assists in presenting a global setting for comparative statics and bifurcation. This theorem states that under a certain convexity condition on the u. , the set of equilibria in (x,p,e) space is a submanifold, the "equilibria manifold" the key condition may be expressed

t in the following manner: For x e R , let K = K . be the

x,i set of v which are mapped into zero by the (total) deriva­tive DU.(x): Rn - R. Thus K is the tangent to the level surface uT (c) at x . Then

o (C) for each i=l,...,m and x e R* ,

9 the second derivative D IL(x) when restricted to K is negative definite. Theorem. Given the above setting with condition (C) satisfied, the set Z of solutions of (1), (2), (3) (the equilibria of the pure exchange economy) is a submanifold of (x,p,e) space, (R +

l) m"S Mx (R*)m . Our sketch of the proof follows [3] or [4], but a closely

related theorem can be found in the work of Balasko or Delbaen. See also [0] for a somewhat related approach to comparative statics.

If Z were indeed a manifold then the tangent T ^(.Z) x,p,e

of Z at x,p,e would have the following description: T = the set of x = (x. ,...,x_), p, e = (e,,...,e ) with - i - t - 8 x. e R , p e R with p.p = 0 and e. e R satisfying the linear equations:

(1») Ex. = Ze. l i

,m (2') p«(xi-ei) + p>(xi-ei) = 0, i=l,. (3») Dgi(xi)xi = p , i=l,...,m

376

These equations are simply obtained by differentiating equations (1), (2), and (3).

Now conversely it can be shown with a short argument using the inverse function theorem that if the space of solutions of (l1), (2') and (3') has the right dimension, (m / ) , then £ is a submanifold of dimension mi (or empty!)

Proposition. The linear space T of solutions of (l1), (2'), (3' has dimension m I .

For the proof of the proposition, consider the linear sub-space K* of (R1)01 * T (S1_1) x (R*)m of (x,p,e) satisfying Ee. = 0, x.-p = 0, i <_ m-l, and ir e. = 0 for i <_ m-l where it :R •* p is the orthogonal projection onto the space perpen­dicular to p. By simple counting one can see that the codimen-sion of K* is * + m-l + (m-l)(8-l) = mB . The proposition now follows easily from the assertion that K* n T = 0. To see the truth of the assertion, let (x,p,i) 6 K* O T. Let Y^p -* p be restriction of Dg-(x). Then y. has all its eigenvalues negative by virtue of (C) and hence is non-degenerate. Then Ex. = Ei. = 0 . Since x.»p = 0 , y~ (p) = x. for each i . Then EY" (p) = Ex. - 0 and ly~ is an isomorphism. The last uses the fact that Y. comes ultimately from a symmetric form. Thus p = 0 and the rest follows easily.

What does this theorem have to do with comparative statics and bifurcation theory? In fact both of these problems revolve around the study of the map n: E +(R*)m which sends (x,p,e) of E into e . To apply comparative statics at (x,p,e)

377

this map must be a local diffeomorphism at (x,p,e) or the deriva­tive DTT:T »• (R ) m must be an isomorphism. This derivative sends (x,p,e) e T into e, so that a study of the linear equa­tions defined by (l1), (21) and (31) enter. The comparative statics is described by the inverse of this derivative.

Finally, the bifurcation of equilibria occur exactly when Dir at (x,p,r) becomes singular.

The situation is on the surface similar to catastrophe theory. However, because economics does not give us a potential function, (elementary) catastrophe theory does not apply and the situation is closer to the general study of singularities of maps.

Observe in the above situation that changes in the endowment parameters don't necessarily affect a price equilibrium. For example if (x*,p*) is a price equilibrium at an endowment vector e , then (x*, p*) is also a price equilibrium for e where p* • (e-e$) = 0 and Xe* = Ze. . In fact there is an ml - (m±0 parameter family of such e . It is reasonable to ask for an effective family of parameters for equilibria of a pure exchange economy. This can be done as follows.

Consider a point (x,p,w.,s) in the space (R*)m xS x(R~)mxR . Here (x,p) is a state as before, w. is some kind of incoruc param­eter, s is the total resources or a supply parameter. The "effective" equilibrium manifold £ will be the subset of (x,p,Wi,s) in (R*)m x S1"1 x (R*)m x R* satisfying:

378

s

w ^ , i=l,...,ir.

p j i=l,...,m .

Theorem. Z is a submanifold of dimension m+l .

The proof is similar to that of the preceeding theorem. The comparative statics situation is similar to that before except that the w. , and s are effective parameters.

Further if

a: (R*)m x S*"1 x (R*) m * (R*)m x S*"1 x (R*) m x R*

is the map defined by

o(x,p,e) = (x,p,wi= p.eili_i , s = E e ^

then o_1(£) = Z .

References

[0] Balasko, Y., Equilibrium Analysis and Envelope Theory, preprint Ecole Normale Superieure, Paris Nov. 1976.

[1] Quirk, J. and Saposnik, R. , Introduction to general equilibrium theory and welfare economics, McGraw-Hill, N.Y., 1968.

[2] Samuelson, Paul, Foundations of Economic Analysis, Atheneum, N.Y. 1971.

[3] Smale, S., Global Analysis and Economics VI, Jour Math. Econ. 3 (1976) 1-14.

[4] , Global Analysis and Economics, preprint written for for "Handbook of Mathematical Economics" (Arrow-Intrilligator).

Zx_.

P'* ±

g.(x.) 6 i i

379

Econometrica, Vol. 48, No. 7 (November, 1980)

THE PRISONER'S DILEMMA AND DYNAMICAL SYSTEMS ASSOCIATED TO NON-COOPERATIVE GAMES'

BY STEVE SMALE

A new way of looking at repeated games is introduced which incorporates a bounded memory and rationality. In these terms, a resolution of the prisoner's dilemma is given.

THE GOAL HERE is to give a natural way of introducing dynamics into game theory, or at least for non-cooperative games. Perhaps the main idea in this treatment of dynamics is the way the past is taken into account. We suppose for both mathematical and model theoretic considerations that the agents only keep some kind of summary or average of the past outcomes (or payoffs) in their memory. Decisions are based on this summary. This kind of modeling reflects the fact that there exist substantive bounds to the storing and organizing of informa­tion. We give an axiomatization of bounded memory and rationality, with both institutions and people in mind.

On the other hand, the hypothesis used in this treatment leads to a tractable mathematics. Differential equations on function spaces which contain little geometry are replaced by a dynamics on a finite dimensional space. And yet dynamics takes the past into account as a kind of substitute for the theory of delay equations.

The perspective in this paper is that of no finite horizon and no discounting of the future. There is always a tomorrow in our plans, and it is as important as today. Also there is a history, a beginning of history, but no end. Decisions are based on the effect of past actions of agents, not on promises or binding agreements. However communication is certainly not precluded.

Solutions in our games are asymptotic solutions. To be important for us, they must meet the criteria of stability. This criterion is well-defined by virtue of the dynamical foundations of the models.

The first section deals with an example, the repeated prisoner's dilemma, in the language of an arms race. Here a class of strategies, "good strategies," is given where the solution is Pareto optimal, stable, and a Nash equilibrium. Thus at least asymptotically, we have a rather robust resolution of the prisoner's dilemma.

We show how good strategies with optimal solutions might bifurcate into strategies with the worst solutions.

In Section 2, we introduce the dynamics in a rather abstract setting for non-cooperative games. In particular we axiomatize a way of taking the past into account, with limited memory, in our decision making.

In Section 3, some theory is developed. Some ideas in this subject go back to A. Cournot [6] (see also [8]), and of course

J. von Neumann. Von Neumann and Morgenstern write [21, p. 44]: "We repeat

1 This research was partially supported by NSF Grant MCS 77-17907.

1617

380

1618 STEVE SMALE

most emphatically that our theory is thoroughly static. A dynamic theory would be more complete and therefore preferable."

Some of my own considerations which led me to this problem are in [19]. Our work has so many points of contact with previous work in game theory and

related subjects that I cannot begin to do justice to the literature. It is said [9] that there have been more than 2000 papers written just on the prisoner's dilemma.

Fictitious play (see [5, 6,13,17,18]) is perhaps closest to what we do and yet there is a substantial difference in spirit and methodology. On the probability-statistics side there is work of David Blackweli, James Hannan, and Herbert Robbins which bears some relationships.

The prisoner's dilemma figures prominently in several books by Anatol Rapo-port. In [16, p. 20], Rapoport estimates that at least 200 experiments with the prisoner's dilemma have been reported.

Iterated games, especially the prisoner's dilemma, recently have been studied in work of Aumann [3], Brams, Davis, and Straffin [4], Grofman and Pool [10], Kurz [12] and Radner [15], among others.

It is worthwhile to call attention to the earlier paper of Aumann [2]. We are especially in accord with his justification of an infinite number of plays as realistic (p. 295).

A background reference to the problem of bounded rationality of Herbert Simon, Roy Radner and others is [13].

Finally, I would like to give thanks to many economists and game'theorists who have given me brief but helpful remarks on my work as it was in progress.

1. Our goal in this section is to give a resolution of the prisoner's dilemma via

iteration. To begin, we describe the (static) prisoner's dilemma with simple explicit numbers as payoff. Each of two players can play easy (E) or tough (T). The payoff to player number 1 is given by Table I.

T A B L E I

Choice of numbeT

T

Choice of number 1

Since T dominates E, i.e. 3 > 2,1 > 2, the rational choice for player number 1 is to play T.

The game is symmetric, so the payoff for player number 2 is shown in Table II. Thus player number 2 plays T also and the (Nash) outcome is (1,1) which is inferior for both players relative to the outcome (2,2).

0

2

1

3

381

382

NON-COOPERATIVE GAMES 1619

T A B L E II

3

2

1

0

2nd player's choice

T

E T 1st player's choice

I like to consider an interpretation in terms of the arms race where T means to arm, E to disarm. Then the Nash equilibrium (1,1) is two armed camps; the optimal outcome (2,2) is then interpreted as two unarmed states. In this inter­pretation it is natural to iterate the same and eventually to consider a continuum of choices between E and T.

Since we will be taking averages, our state space S is defined to be the convex closure in 2-dimensionaI payoff spaces of possible outcomes. Thus 5 is the convex closure of the points (0,3), (2,2), (3,0), (1,1), or the shaded region in Figure 1.

FIGURE 1

The optimal outcomes are those points on the segment jointing (0, 3) to (2,2) and (2,2) to (3,0). But (2, 2) is the optimum with a special "fairness" property. The (1,1) outcome is the classic Nash equilibrium for the static game.

We now suppose the game is iterated. Then repeated play starting at time 1 yields a sequence of points JCI, xi,..., xt e

5 (in fact each x, is a vertex). After time T (I use T for time as well as tough), the history of the game (outcome) is given by xu •. *T- Let

#r = (*i *r)keS. In our simple example here, we average in the most naive way; thus define

1 T qT:£fT->S,qAxi xT) = -= I xt = xT-

I i-\ So XT is the average income (a vector) up to date T. (See Section 2 for a class of

averaging processes which could be used here.) We suppose that both players make their decisions as to their (T + l)st move

based just on xT. ltAx (for actions or alternatives) is the set E, T), then a strategy

mtSbm

1620 STEVE SMALE

for player number one is a map S —* Ay. A strategy for both players is a map 5:S -* A where s = (sit s2), A = A \ x A2, and At = E, T. Here S is considered to be the space of averaged outcomes. Thus if 5 has been chosen, the game is denned for all time once the starting point x\ is given. More explicitly, say (x\ xT) e yT is given, with Ts= 1. Then xT+, = fI(s(xT)) where xT = (1/T) Z,1, x, and the payoff II: A -*S is given by the tables at the beginning of this section.

Thus the (full) dynamics is given by the map:

*r.yT-+yT^ ( T = I , 2 , 3 , . . . ) .

<M*i, ...,xT)~ (XI, . . . , xT*i),

xT*\=n°s"qT(xi,...,xT).

Most importantly, there is also a canonical representation of this dymamics on the space 5. There is a map 0T-S-*S so that this diagram commutes:

*T

" T

Simply define

r i + /7°5(x) /3T(X) = Y^\ ' xeS-

This dynamical system /3r: 5 -» 5, which is time dependent and discrete, defined on the space of averaged outcomes 5, is central to our approach. We believe that in constrast to the dynamics on "the function space," *I>T-^'T-*S/'T+U this /3r is amenable to a good analysis and a main object of this paper is an effort to demonstrate this.

A solution is a pair (s, x) where s: S -» A is a strategy and x e 5 is stationary for the dynamics defined by s (both tfiT and fiT). Thus if x = xi, then xT+i =xT for T = 1 2

The solution (s, x) will be called globally stable provided given any x i e S, and xY-i = M*V) , r = 1, 2 , . . . , then xr- X as T-♦oo.

We devote the rest of this section to describing a class of strategies with several desirable properties.

A strategy for player number one is a map st: S * E, T. It will be called good provided it satisfies the following conditions (1), (2), (3) below. Let x = (x ,x ) = (a, /3) e S. Thus a is our (as the first player) average income to date. We know that by playing T (tough) all the time we could have a ^ 1. Thus if a < 1, our playing E (easy) has been exploited by our opponent to our disadvantage. Thus

(1) Si(a,p) = T if a < l

(i.e. play T if our average income is less than one).

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NON-COOPERATIVE GAMES 1621

Always a + 0 * 4 so if 0>2,a is correspondingly less than 2. Thus 0 > 2 implies our opponent has been exploiting us. So (2) J , ( a , 0 ) - r if 0>2.

Finally by playing tough too much we may expect to end up in the nonoptimal static Nash equilibrium solution of the prisoner's dilemma, namely (a, P) = (1,1). Thus we must play E (easy) often times. In fact we will justify:

Si(a, 0) = is if 0 *£ a and, moreover, (3) there is some open set U\ in 5 containing the segment

(y, y) e S| 1< y < 2 on which sl = E.

Condition (3) is an expression that we should play a little on the easy side. These conditions (l)-(3) assert that to play a good strategy, I should not let

myself (my easy play) be exploited, but otherwise I should play on the easy side. The word good is justified by the following theorem.

THEOREM 1: Suppose player number 1 plays a good strategy. Then Urn infx\-3> 1. Furthermore play number 2 can do no better than by playing a good strategy in that Urn supir^l. If both players play a good strategy (s*, s*) = **, then x* = (2,2) is a solution and is the unique x* such that (s*, x*) is a solution. In fact (s*, (2, 2)) is globally stable.

REMARK 1: In the last sentence (s*, (2, 2)) is a Nash solution and it is a Nash solution even if the players use a strategy which takes into account the complete history. See Section 3 for elaboration.

REMARK 2: The rate of convergence to the solution will depend on the size of the neighborhood U\ in (3).

REMARK 3: There is a great deal of robustness in this result as indicated by the stability of (s , (2,2)). I regard this as an important advantage of our perspective.

Except for the last two sentences, the proof of Theorem 1 is immediate. Before we give that proof, we give an example to show just how delicate condition (3) is, and why playing on the "easy side" is crucial.

Example of strategies s,: S -* E, T, i = 1,2. We take si = 52 = <7B, —e < u- < e, where

<r» = E on (x,y)eS\(i[(x + y-3)2-l]+(y-x)<0, a„ = T otherwise.

One can easily check that for any M > 0, <rM is a good strategy. But iin<0, then (j, (1,1)) is the solution and is globally stable. (See Theorem 2 below.) So n = 0 is a bifurcation and the solution changes from the best to the worst as n passes from positive to negative.

384

1622 STEVE SMALE

Suppose instead of (3) players play a little on the tough side, i.e. (3) is replaced by (3'): (3') sl(a,p)=T if «*£0, and moreover there is some open set U in S containing the segment (y, y) e S|l < y < 2 on which Si = T.

THEOREM 2: Ifs\ ands2 satisfy (1), (2), and (3'), then there is a unique solution (5, (1,1)) which is globally stable.

The proof is similar to that of Theorem 1. We now give the proof of Theorem 1. The proof goes by this lemma.

LEMMA: Suppose in the situation of Theorem 1, both players play good strategies. Given e>0 there is T0 such that ifT>T0 then for any initial xi € S,

(a) | /M*)-x |<e, allxeS, (b) j fV+ir*3-«. (c) PAx) € N, (A), the e neighborhood of

A = (x\x2)eS\xl = x2, foranyxeS, (d) xVetf.(J).

Here and below we use superscripts to denote coordinates in S <= R2. The proof of the lemma is straightforward. Since

PTW-—X + T + V

(a) is immediate. Because $i(x) or s2x) is E, for any xeS, we know that tp\x) + <p2(x) > 3. Thus (b) follows. If x € N.(4) n ([/, n U2) where Uu U2 are as in (3) of "good strategy," then /3T(X) lies on the segment between x and (2,2). Thus 0T(X) 6 N,(J) since A/«(4) is convex. If x£N,(J) n (C/i n l/2), we know by (a) that /3r(x) and x lie on the same side of A; furthermore 0r(x) lies closer to A than x since both are playing good strategies. This yields (c). A similar geometrical argument gives (d).

To finish the proof of Theorem 1, given any S > 0 let V, be the disk of radius 8 about (2,2). Choose e > 0 such that

N.(A)uVscUlnU2n(x\x2)eS\xl+x2^3. With T0 as in the lemma and T> T0, xT eNm(A) and if xTt V* xT+i lies on the segment between xT and (2,2). Thus there is Tt s> T0 such that if T > Tu xT e V«+.. The rest follows.

REMARK: The above arguments and theorems clearly apply to more general prisoner's dilemmas with different numbers used. But what about the non-symmetric case? Here one can use the literature on fairness and bargaining theory and the above analysis to obtain similar results.

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NON-COOPERATIVE GAMES 1623

REMARK: The case of continuous choices can be developed in the framework of the next sections.

REMARK: I have not investigated the problem of this section, finding "good" strategies for games with more than two players.

2.

We start by giving some little study of averaging processes. Suppose 5 is a closed convex subset of R", which is to be considered as a space of "outcomes" (of some system to be introduced later). Time will be discrete, with a starting value, and infinite. Thus one considers a sequence of outcomes x i, x2,..., xT,. ., each xT in 5.

Now we assume that the agents keep track of only some summary or average of the past history. In fact we will suppose that for each date T there is an averaging function qT which goes from 9T = (xu •, *T)|*J e S to 5 itself. The collection

qT:yT-*S,T= 1,2,...

satisfying qi(x) = x for all x e Sf\ =* S, will be called an averaging process. It is useful to find some reasonable class of averaging processes. Besides the

ordinary averages qrixi,.... XT) = (1/5H) £/_i xt (Section 1), it is important to allow, for example, processes which weigh the recent past more heavily.

For this we consider briefly the problem: how is the past taken into account in decision making by institutions or individuals. And certainly we will idealize drastically.

To help motivate the following, imagine an agent with a computer which holds and deals with handily one or two points in S <= R" (large n); a point in S represents some kind of average of past history. Then a new event happens, represented also by a point in S. What kind of averaging process allows the computer to find the new average from only the old average and the new event?

An example is a stock broker who every morning wishes to scan 2 or 3 indicators of all the stocks listed on Wall Street. These indicators could be weighted averages of the actual daily closings, of the daily changes in the same, dividends, etc. A naive way to do this would be to have his computer hold all of the data for the past decades (i.e. the daily closings, each day, etc.) and then each morning, have the computer calculate the new weighted averages from all the past data and new closings. But considerations of limited memory and bounded rationality make this approach less feasible. Meeting both these problems would be the use of an averaging process which has the property that the new average can be deduced from the old average and the new outcome.

Even there is an analogous situation in a less precise sense in my own brain. Somehow the summary of my past history as of this morning (I don't even remember what I had for dinner 2 weeks ago) combines with today's events to

386

1624 STEVE SMALE

produce a new summary in my brain by tomorrow morning (this would better use continuous rather than discrete time).

We formalize these considerations by imposing an axiom on our averaging process, far:SfT•* S\T = 1,....

MAIN AXIOM (Weak Version): qT(xu..., xT) and xT*\ determine <7r+i(*i, • • •. xT+i) continuously. In other words, for each T, there is a continuous function fT:S*S-*S so that flr+i(*i. . . . , XT+I) = /T+I(<?T(J:I, • •., xT), XT+I)-

Note, by induction (recalling Oi(x) = x), this implies that qr is continuous each T.

MAIN EXAMPLES: Important examples of r satisfying the above are given as follows:

Fix k = 0, or 1, or 2, etc. Let

q(rk)(*i *T)~ V - £<'"*• X ;

Thus for k = 0, we have the averaging process of Section 1. The larger k is, the more heavily the recent past is taken into account.

We pass to the setting of a non-cooperative game with n-players. Associated to the fth player for i = 1 , . . . , n, is the space of actions A,. We will assume that At is a domain in some (real) Cartesian space with boundary piecewise smooth in some reasonable sense.

State space 5 or outcome space for us will be a closed convex subset of R". The basic datum is a map 77: A * S called the payoff where A is the product

A = III". i A,. And we assume that 77 is continuous (and, starting with Section 3, of differentiability class C2).

If the agents i = 1 , . . . , n take actions a j € A i , . . . , an e An respectively, then the outcome is /7<(ai,... ,a„) = /7,(a) and 77(a) = (77i(a),... ,/7„(a)). Thus 77,(a) is the income or payoff given to agent i when (ax,..., aH) are n choices.

One example is the classic Cournot model of duopoly. Another class of examples is generated by finite games. One takes for A( the subset of Cartesian space given as the convex closure of the finite set of strategies, placed so that no one is in the convex closure of the others. Then one extends the payoff of the finite game in the natural multilinear way (all formally just as in the case of mixed strategies, but with the interpretation that plays represent pure strategies). Then S is the convex closure of all finite outcomes defined by the payoff. One could interpret the new game as the extension of the old game where a continuous choice is now permitted.

We compare the situation here with that of Section 1. The spaces S are the same, but now A,=[0, IJ or AE + (1-A)T with A between 0 and 1. Thus a continuum of choices is allowed between Easy and Tough. The map 77: A •* S is the natural extension of E, T2-*S of Section 1.

387

NON-COOPERATIVE GAMES 1625

A memory strategy (sometimes just strategy) is a continuous (and, starting with Section 3, C2) map s:Sav-*A where Smv is mathematically S, however inter­preted not as a single outcome, but as the image otqT:SfT-+S:

Then a strategy s: S«„ -* A has the form s = (s i , . . . , 5„), where s,: S00 + A, is the strategy of the ith player. A memory strategy s together with the given payoff II:A-*S defines a dynamical system, aV»erT, ar:S*r-»#r+i» all T, by ar(jti xT) - (xi, • , xr+i) where xT+i = 77(s(<jT(xi xT))).

PROPOSITION: Assuming the main axiom (weak version) <XT induces canoni-cally a discrete time dependent generally irreversible) dynamical system on the space of averaged states 0T' Sae -» Sav, all T, such that this diagram commutes:

Vr

« T

"T

rT*\

« T * 1

s PROOF: Just let Prix) =/r(x, <p(x)) where «p:s -*S is the composition

* n Sav-+A^»S.

Then for (xi XT)Z&T, let qTxx xT) = x; then QT+I(*I, • • •, *r, XT+I) = /T(X, XT+I) and xT+i = 9>x). It all follows.

That the dynamics 0T'-S-*S can be defined on 5 is central to our work here. It is a time dependent, discrete, dynamics on a fixed finite dimensional space S, amenable to geometric analysis on the mathematical side, and on the modeling side it reflects bounded memory and rationality. Sometimes we write $T ™ P'T to denote the dependence of 0T on the memory strategy s.

Next we give a strong version of the main axiom on averaging processes. Supposing the weak version, we know that via/r+i, qr(x\,..., XT) « xT and xT+\ in the convex space S determine <?T+I(XI, . . . , xr+i) *» xV+i. It is not unnatural in this context to interpret "determine" as lying on the segment between f T and xT+\. This motivates the following condition on the averaging process far :!?T-*S\ all r>0 .

MAIN AXIOM (Strong Version): There existnumbers kx, A2, A3 , . . . with 0< A< < 1 so that£T+i -AT+ixr + (l -AT+i)xT+i, all T « 1 , 2 , 3 , . . . . We take At - 1 .

In other words, there exist AT as above such that the fT: S x S •* S of the weak version of the main axiom can be written /T(x, y) = ATx + (1 - AT)y.

PROPOSITION: / / the Strong Version of the Main Axiom is satisfied for an averaging process [qr], then for each T « 1 , 2 , . . . , there are numbers at(T) > 0,

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1626 STEVE SMALE

i = 1 , . . . , T, such that i f - i ai(T) = 1 and T

qr(x\ *T-)= I MT)xf. i - >

Moreover the at(T) are uniquely defined and in fact the at(T) are of the form at(T) = StfT for functions i * Sh T-*yT.

Thus the qT is a true averaging in the usual strict sense of the word, with the weights given by atT).

PROOF: Let yT = Il(r-i A, and St = (1 - A,)/y(, if i > 1, and 5, = 1; then as stated,

takea,(T) = S(yr. Note first that yT > 0, S, 3= 0, and a,T) > 0. Since

1 1 T T / l l \ « , = - - — , Ial(T)=l( )yT + yr = l.

ji y . - i i = i . - 2 \ y , y i - i /

Now directly one sees that

a i ( l )=l . <Ji(2) = 5 iy 2 = 72 = ^2.

1 1 a22) = ( l _ l ) y 2 = 1_A2

Vy2 yi /

More generally, T+l

AT+IOTUI, . . . , X T ) + (1-ATVI)XT+I= I *ITT+2JCI i - 1

or T T+l

AT+I I 5(yTJC(+(l-Ar+i)xT + l = I SiyT+2x,. i-i ( - 1

Checking coefficients of jt, on both sides leads to the formula for a,(T) and the proposition (for example, A-rvifi.yr = &iyr+u i<T + l, etc.).

Note that if A( = 1, what happened at date / is obliterated from memory.

Also note that in the main examples of averaging processes, listed above,

a,(T) = ik/lik.

Here Si = ik and I 1

, T _ yr _ ; - i ~yr-r™.k'

I / ; - i

We make a final remark: A little more general framework for averaging processes could be useful in some situations. Namely one could take qr'-^r-*

389

NON-COOPERATIVE GAMES 1627

X, T = 1,2,... when Sfr is as above, but X is some different space. Then in the weak version of the main axiom, fT:X*S-+X, etc. Section 1 could be interpreted this way where S = (1,1), (3,0), (0, 3), (2,2) <= R2 and X is the convex closure of 5.

3.

We develop the ideas set up in Section 2. Thus in particular we suppose given throughout: (a) a C2 map, the payoff IJ:A-*S, where A = Ai x . . . x An is action space, and S is outcome space in R"; (b) an averaging process qr-^T -* Sau, T = 1,2,... satisfying the strong version of the main axiom, with Sav = S and 0<A T <l .Tobe chosen is (c) a memory strategy s: Sm -* A of class C1.

Denote by <p = <p,:5«„ -»S the composition <p = 77°s. One might keep in mind the example of these ideas in Section 1 (an example

without the continuity). We say that xeS is stationary for the dynamics defined by the memory

strategies s: S -»A if &'T(x) = x, all T.

PROPOSITION 1: A point xeSis stationary for s if and only ifx is a fixed point of <p,. Also this condition is equivalent to PT(X) = x, for some T.

PROOF: pT(x)=fT(x,<p(x)) = *TX + (l-XT)<p(x), each T. (So 0T(x) = x for some T if and only (1 - Ar)x = (1 -\T)<p(x).) All parts of the proposition now follow.

REMARK: If under the weak version of the main axiom, fTx, x) = x all x, then x a fixed point of <p implies x is stationary.

If 5 is a memory strategy and x 6 S, we will call the pair (s, x) a solution for the game defined by 77: A •* S provided x is stationary for s. Sometimes in this case, x will also be called a solution (relative to s); thus if x is a fixed point of <p,: S * S, (s, x) is a solution. Note also from the proposition the notion of solution is independent of the averaging process.

COROLLARY: Suppose S is bounded. (Recall S is assumed always to be a closed set.) Then given any memory strategy s:S-*A, there is a solution ($, x).

PROOF: The Brouwer fixed point theorem yields a fixed point x of (p,:S-*S.

For a solution to have any important meaning for models, it should meet a stability criterion.

We say that a solution (s*, x*) is globally stable if for any x e S, the following sequence converges to x*: x, £2 = Pi(x), £j = Piixi), It will be called locally stable if there is a neighborhood N(x*) of x* in S such that the above is true for any ace N (x*).

For the next theorem, we will need a further condition on the averaging process.

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1628 STEVE SMALE

DIVERGENCE CONDITION ("The Past Isn't Overwhelming"): The sum 2T-I (1 - A T ) diverges.

PROPOSITION 2: The "main examples" of averaging processes Section 2) satisfy the divergence condition.

PROOF: First note (by the end of Section 2)

l-\T = aT(T) = -r-. I/"

But since Tk+l^T',j=£k,

(k + l)Tk*1^ I /* and so

Tk 1

Therefore

T k + \ TT diverges. Q.E.D.

THEOREM 1: Assume the divergence condition and let (s, x*) be a solution for II-.A-fS.If all of the eigenvalues of the derivative Dip, x *): R" -»• R" have absolute value less than 1, then (,s, x*) is locally stable. If \\D(p,x)\\ < 1 for all xeS, and S is bounded, then s, x*) is globally stable.

Here D<p(x*): R" -» R" is the derivative, considered as a linear transformation, of ^:S-*5cR". One can think of D<p(x*) or D<p(x) as the matrix of partial derivatives. The norm on |AP(X)|| is the ordinary operator norm; Le. ||£><p(x)|| = maxH-Mcir \Dipx)v)\.

EXAMPLE: Let memory strategy s: S -»A be constant. Then the game reverts to the static game and considerations of usual game theory apply. (The past is not taken into account.) But now note that the derivative Ds(x) is zero for all x. Therefore by the chain rule, the derivative D<p,(x) is also identically zero. Finally note, for any memory strategies r:S-*A which is sufficiently close with derivatives (close in the C1 sense) to s, that |lD<pr(x)||< 1, all x. Thus the previous theorem applies to yield the following corollary.

COROLLARY: For a memory strategy r:S-*A, sufficiently Cl close to a static strategy, there is a unique solution which is globally stable.

391

NON-COOPERATIVE GAMES 1629

For proof of Theorem 1, choose a norm on R" such that ||£><p(x)||< 17 < 1 (see [10]). Choose a coordinate system on R" such that x* = 0. By Taylor's theorem there is r > 0 such that for x e £>r(0) = x € R"| ||x||< r, \\(p(x)\\=£ TJ|JX||. Since

PT(X) = ATX+(1-IT)<P(X) and

AT + rj(l -AT) - 1 -(1 -AT)(1 - i f )< 1, it follows that

II^TU)N(Ar + r,(l-Ar))IUII for ||x||<r. Thus N(x*) = D,(0) is invariant under (3T-

For the convergence it is sufficient to show

l/Br...020i(x)|-»O or

(2) fl (AT + T , ( 1 - A T ) ) ^ 0 orby(l) T - l

f[ ( l -( l-Ar)(l-tj))-0. T - l

For (2) we will use a lemma (see Titschmarsh [20]).

LEMMA: Let b„ be a sequence of numbers, 0*zbn< 1. Then IlC-i (\-bH)-*0as N-KX) provided £ b„ diverges.

Now X (1 - Ar)(l - n) = (1 - TJ) 2 (1 - A T ) diverges according to our divergence condition. Therefore by the lemma, (2) follows. This proves the first part of Theorem 1.

For the second part of the theorem we use the following known lemma:

LEMMA : Suppose i/r-.S + SisaC1 map of a compact convex subset of R" such that \\D»lf(x)\\ <lonS. Then there is some (3,0 < 0 < 1 with U(x) - 0(y)|| »s 0||x - y||, for all x, y e S and a unique fixed point x* of iff.

This is easily proved. The idea is that </r shrinks the segment between x and y. Since |D^(x)B<l for each X€S,H0T(X)11<AT + (1-ATHIO*CX)II<1. a n d t h e

lemma applies with & = PT- The rest of the proof goes just as before. Q.E.D. We now pass to questions of Nash equilibria. There is a natural definition of

Nash equilibrium for memory strategies: namely, a solution (s, x), where s:S-* A, is a memory strategy and xeS is a Nash solution provided the following condition is met. For each / = 1 , . . . , n, there is no other solution (s't x') with (x')' > x1 and s) = */ for / * 1. In other words, for each 1, player 1 cannot improve his/her income (in the sense of solutions) by choosing a different memory strategy.

An example of a globally stable Nash solution (s,x) for memory strategies is given by a static Nash equilibrium a e A, so that s a is constant and x = 77(a). Here Nash equilibrium is the ordinary game theoretic notion and the stability follows from our earlier remarks.

392

1630 STEVE SMALE

We wish to prove a theorem which asserts that a Nash solution for memory strategies is also a Nash equilibrium when "supergame" strategies are permitted; i.e., even if an agent is allowed to take the whole past history into account, that agent cannot improve (asymptotically as always) his/her averaged payoff. To that end we define a (joint) continuous supergame strategy to be a map s': U r - 1 SfT -* A with these natural continuity properties: First the restriction ST = S'I&T is continuous, each T, and second, if xT is a convergent sequence in S, then S'T(X\, . . . , XT) converges in A as T-*oo.

Note that our memory strategy s.S^A defines a continuous supergame strategy s':uSfT-*A by s 'rUi , . . . , xT) = s(qT(xi,... ,xT)). The continuity conditions of s' = sV are checked easily using the strong version of the main axiom on averaging processes.

A (joint continuous) supergame strategy s' and an initial jc t€5 define a generated sequence of outcomes xi, x 3 , . . . , € 5 by XT*\ = 77 ° s'rixi,..., XT).

An extended solution is a pair of sequences (sT, xT) where sT\ is a continu­ous supergame strategy and [xT\ a generated sequence from sT and some JC I such that xT converges to some xeS (we define the solution this way to allow x above which may not be stationary).

A solution (s,x) for a memory strategy s:S-*A can be considered as an extended solution (sT, XT) by sT = s ° <fr. xT = x, all T.

An extended solution (sT, xT) is a Nash equilibrium (for the "supergame") provided that for each /, other extended solutions (rr, yT) such that r'T = S'T, j * i, have the property y ' « x' where xT -* x, y-r -* y-

THEOREM 2: Let (s, x)bea Nash solution for a memory strategy. Then (s, x)isa Nash equilibrium considered as an extended solution.

Thus (s, x) still has the property that even if agents take the whole past into account, not just the averaged past, they cannot do better (asymptotically).

Now to the proof. Thus let (5, JC) be a given Nash solution for a memory strategy s: S -» A with (ST, XT) the corresponding extended solution. Consider another extended solution (rT, yr) such that yT * y, r'T = S'T = s' « qr, j * '• It is sufficient to show that y' ^ JC'.

To this end we will define a new memory strategy s* such that (s # , y) is a solution and s" = s1 for j * i. Since (s, x) is Nash, yl^x' and the proof will be finished. Thus how is s":: S -> At defined?

In fact rT(yu ., yr) converges as T-»oo to some aieAi by the continuity hypothesis of supergames strategies. Let s*" be the constant function at. It remains to show that (s0, y) is a solution or that 77j#(y) = y.

Now y r + ! = 77 ° r T (y! , . . . , yT). Take the IimiJ of both sides as T-+ 00 to obtain y = n(bu b2,.--, b„) where b, - at and if / * i,

Ui — 11111 j \HT\yi> • • • > JTl) — ' T-»oo

since

bi = lim s'(qTy 1, • • •, yr)) = *'(y) T-*<x>

lim qT(yi, . , y T ) = y. T-»oo

393

NON-COOPERATIVE GAMES 1631

Therefore y = 77 (s#(y)), finishing the proof of Theorem 2. We have already found calculus conditions for stability of solutions for memory

strategies. What about calculus conditions for Nash equilibria? We proceed in that direction by giving necessary first order conditions. This gives a little introduction to the kind of calculus of variations needed to analyze memory strategies more generally. This calculus might have use in other optimization problems with bounded memory or rationality.

Let Q be the function space of maps s: S -* A\s is C2. Consider the composi­tion i//,

QxS^A^SaR",

where E is the evaluation map, E(s, x) = s(x). Thus tlf(s,x) = IIs(s) and iJ/ = (4>u • • , <I/H) where , = 77/»E. Denote by Z, the set of solutions for memory strategies in Q x S, so

Z = (s,x)zQxS\<Hs,x) = x.

Lets*,x*)sZand

Z=Z(s*,x*) = (s,x)eZ\s' = s*',j*i.

We will suppose a mild nondegeneracy (or "transversality") condition so that I is a submanifold of QxS and so that Zi are submanifolds of Z. See [1] for details of this and also as a reference for the rest of this section. We emphasize as usual that II: A -*S is fixed datum while the memory strategies may vary.

The following is essentially a restatement from the definitions.

PROPOSITION 3: A solution (s*, x*)eZ is a Nash solution if and only if the restriction ifft\Z has (s*, x*) as a maximum, each i.

COROLLARY: / / (s*, x*) is an interior Nash solution (or even locally Nash), then this first order condition is satisfied.

D*l>i\Zt(s*, x*) = 0, each i = 1 , . . . , n.

Here interior means that x* e interior S and s*(x*) e interior A. The symbol D in (3) is the (total function space derivative.

Note in this form, one could easily write down second order sufficiency conditions, and constrained conditions (without "interior") as well.

But let us examine this first order condition more carefully. We will prove the following theorem:

THEOREM 3: Let (s*,x*)bea solution for memory strategies. Suppose besides the previously mentioned nondegeneracy assumption on Z, Zt, we suppose also the following nondegeneracy hypothesis: the determinant of

DIl,*ix.)Ds*x*)-I:RH + Rn

394

1632 STEVE SMALE

is nonzero. Then if s*, x*) is an interior Nosh solution:

diag Dn,>(l')Ds*(x*) - 7)",D77I.(x.) = 0.

This calls for a little explanation. Here D always stand for the derivative as a linear transformation or matrix. For example D/7,«(x«) is the derivative of 77: A * J c R" at the point s*(x*)eA. Similarly Ds*(x*) is the derivative of s*:S-> A at x* e S. The matrix 7: R" -* R" is the identity. Finally in case dim A, = 1, Diag (A) is just the matrix which consists of the diagonal elements of A, making the off diagonal elements zero. For general A, Diag (A) has a natural meaning elucidated at the end of the proof.

PROOF OF THEOREM 3: By differentiating <l>(s,x) = x at (s*, X*)GI, we obtain

(5) DMs*,x*)£ + (DMs*,x*)-I)£=0

for all (s, x) "tangent" to I^QxS at (s*,**). Here (s, s) are infinitesimal variations which keep solutions as solutions. By calculus (see [1]), DJE(s*, x*) (5) = *(x*)andD,E(j*, x*)(x) = Ds*(x*Ki).ThusD,<Ks*,x*)(5) = ZM7,.u->(*<**)) and DM**, **)(*) = DnS'[x-)» Ds*(x*)x. Let y = s(x*), which lies in La where A is a convex set in a linear space LA (and At c LAl). Then (5) becomes:

(6) D77J.(l.,(y) + (7J>77J.u.)£>5*(x*) - I)x = 0,

for all ysLA,xe R". Similarly:

DIIu'iM) + W>i7to.u., • Ds*x*) -1)(£) = 0

fory'eLA„jceRn . Our first order condition in the above corollary may now be interpreted in terms

of (6) to say: For each i 1 , . . . , n, if y'' = 0 for / * i, then i" = 0. Equivalently in terms of (7) x = -(D77J.(jt.,7Ji*U*)-7)- ,D771.u.)(y)

one has, if y = ( y \ . . . , y*")e LA = 77LA,y' = 0,;V 1, then JC' =0 . O.E.D.

The matrix in (7) has a general interpretation, even if s*,x*) is not a Nash solution, y' is an infinitesimal change in f s memory strategies near x* and i = ( x \ . . . , f ) is the effect of this change.

EXAMPLE: Continuous prisoner's dilema. Here

A ( = a T + ( l - a ) E | 0 « a « l (/ = 1, 2)

are the spaces of choices; E is easy, T is tough as in Section 1. The payoff 7 7 : i 4 x A - » 5 c R 2 i s given by the linear extension of Section 1:

771(5i,s2) = Ji-2$2 + 2,

n2(sl,s2) = -2sl + Si + 2,

305

NON-COOPERATIVE GAMES 1633

where Si e A\, s2 € A2. Then

is constant. Suppose s = (s\,s2):S-*Aisti linear memory strategy (the nonlinear case is similar), so

si = ax + by + e, s2 = cx + dy+f,

M: a-COROLLARY OF THEOREM 3: Suppose ((su s2), (x, y)) is an interior Nosh

solution of the memory strategy above. Then a = d = —\.

The proof of the corollary is a simple calculation.

University of California-Berkeley. Manuscript rrceived May, 1979; revision received February, 1980.

REFERENCES

[1] ABRAHAM, R., AND J.ROBBIN: Transversal Mappings and Flows. New York: Benjamin, 1967. [2] AUMANN, R.: "Acceptable Point* in General Cooperative N-Person Games," Annals of

Mathematical Studies, 40 (19S9), 287-324. [3] : "Workshop on Repeated Game*," mimeograph, Inititute for Mathematical Studies in

the Social Sciences, Stanford University, 1978. [4] BRAMS, D., M. DAVIS, AND P. STRAFFIN: "The Geometry of the Arms Race," International

Studies Quarterly, 23 (1979). 567-588. [5] BROWN, G.: "Iterative Solution of Games by Fictitious Play," In Activity Analysis of Production

and Allocation. New York: John Wiley and Sons, 1951, pp. 374-376. [6] COURNOT, A.: Mathematical Principles of the Theory of Wealth, English Edition. New York: A.

M. Kelley, 1960. [7] DESCHAMFS, R: "An Algorithm of Game Theory Applied to the Duopoly Problem," European

Economic Review, 6 (1975), 187-194. [8] FRIEDMAN, J.: Oligopoly and the Theory of Games. Amsterdam: North Holland, 1977. [9] GROFMAN, B.: "The Prisoner's Dilemma Game and the Problem of Rational Choice: Paradox

Reconsidered," in Frontiers of Economics. Blacksburg, Virginia: University Publications, 1975, pp. 101-119.

[10] GROFMAN, B., AND J. POOL: "HOW to Make Cooperation the Optimizing Strategy in a Two Person Game," Journal of Mathematical Sociology, 5 (1977), 173-186.

[11] HIRSCH, M., AND S. SMALE: Differential Equations, Dynamical Systems, and Linear Algebra. New York: Academic Press, 1974.

[12] KURZ, M.: "Altruism as an Outcome of Social Interaction," American Economic Review, 68 (1978), 216-222.

[13] LUCE, R., AND H. RAIFFA: Games and Decisions. New York: John Wiley and Sons, 1957. [14] RADNER, R.: "Satisncing," Journal of Mathematical Economics, 2 (1975), 253-262. [15] : "Can Bounded Rationality Resolve the Prisoner's Dilemma?" mimeographed, 1978. [16] RAPOPORT, A., ED.: Game Theory As a Theory of Conflict Resolution. Dordrecht-

Holland/Boston: Reidel, 1974.

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1634 STEVE SMALE

[17] ROBINSON, J.: "An Iterative Method of Solving a Game," Annals of Mathematics, 54 (1951), 296-301.

[18] SHAPLEY, L.: "Some Topics in Two-Person Games," in Advances m Game Theory, ed. by M. Dresner, L. S. Shapley, and A. W. Tucker. Princeton: University Press, 1964, pp. 1-28.

[19] SMALE, S.: "Dynamics in General Equilibrium Theory," The American Economic Review, 66 (1976), 288-294.

[20] TITCHMARSH, E.: 77ie Theory of Functions. London: Oxford University Press, 1950. [21] VON-NEUMANN, J., AND O. MORGENSTERN: Theory of Games and Economic Behavior, 3rd

Edition. Princeton, N.J.: Princeton University Press, 1953.

397

ChaprrrS

GLOBAL ANALYSIS AND ECONOMICS STEVE SMALE

Uniotnity of California, Brrktley

One main goal of this work is to show how the existence proof for equilibria can be based on Sard's theorem and calculus foundations. At the same time, equations such as "supply equals demand", are used rather than fixed points methods. The existence proofs given here are constructive in some reasonable and practical sense. These equilibria can be found on a machine using numerical analysis methods.

Our motivation for providing a proof of the Arrow- Debreu theorem (Ap­pendix A) is to show that calculus can be used for the foundations of equilibrium theory.

Also in the paper optimization and the fundamental theorems of welfare economics are developed via the calculus. Abstract optimization theorems are proved in Section 3 and applied in Section 4 to pure exchange economies. Debreu's finiteness of equilibria theorem is proved in Section 5. In this section a manifold structure is put on the set of optima and on a certain set of equilibria as well.

The reader can see Smale (1976b) for a general motivation for a calculus approach to equilibrium theory (as well as references to other topics in Global Analysis and Economics). Furthermore some justification is given in this refer-ence for the continued study of classical equilibrium theory in spite of its deep inadequacies for analyzing the problems of our day.

The account here could be used as a basis for a short course and in fact it was written when giving such a course at Berkeley in the winter of 1977. Much of the background for this exposition is to be found in our papers in the Journal of Mathematical Economics.

1. The existence of equilibria

The basic idea of equilibrium theory is to study solutions of the equation; supply equals demand or S(p)~ D(p). For the simple case of one market, where prices are measured in terms of some extra market standard, the familiar diagram below gives some justification for existence of the equilibrium pricey*.

398

332 S. Smalt

D

Figure 1.1

General equilibrium theory treats this problem for several markets. Let us be more precise: Suppose an economy with I commodities is given. Then the space Rt*-(*',...,x')€/l'; x' >Q, each / will play two roles for us: The first is as commodity space; »Jte / l ' + will be interpreted as a commodity bundle. Thus x is the (-tuple (x\...,x') with the first coordinate measuring the units of good number one, etc. But also R'+ with the origin 0 removed will be the space of price systems; thus if p&R'+ - 0 , p — (p\..., p') represents a set of prices of the t goods, pl being the price of one unit of the first good, etc.

We suppose that the economy under study presents us (axiomatically) with demand and supply functions D,S:R'+ -0~+R'+, from the set of price systems to commodity space. Thus Dp) will be the commodity bundle demanded by the economy (or its agents in sum) at prices p. In other words, at prices (/»',..., p'), the vector of goods that would be purchased is Dp). The equilibrium problem is to find (and study) under suitable conditions on D, S a price system p* &R'+ — 0 such that D(p*)—S(p*) (equality as vectors).

Let us write Z(p)~Dp)—S(p) so that the excess demand is a map Z: R\ -0-+R1, and we look for solution p* £/?'+ - 0 of

Z(/>*)-0. (1.1)

The goal of this section is to put conditions on Z which are reasonable from economics and then to show the existence of solutions of (1.1) by a constructive method through the differential calculus. This will be done without passing to the micro-foundations of the excess demand. Then in Section 2 we will give a classical micro-foundational development for the excess demand via aggregation of demand functions of individual economic agents for a pure exchange econ­omy. In Appendix A, we prove the full Arrow- Debreu theorem this way.

Also in this exposition, existence will at First be shown under strong hypothe­ses, so that one can see the methods in .their simplest form. Later the hypotheses will be relaxed.

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Ck 8: Global Analysis and Economies 333

The conditions on the excess demand Z are

Z: /?'+ - 0-»/?r is continuous, (1.2)

Z(\p)-Z(p) for all X>0. (\3)

Thus Z is homogeneous; if the price of each good is raised or lowered by the same factor, the excess demand is not changed. This supposes we are in a complete or self-contained economy so that the prices of the commodities are not based on a commodity lying outside the system,

p-Z(p)-0 I using the dot product, 2 / > ' Z ' ( » - 0 j . (1-4)

This expression states that the value of the excess demand is zero and (1.4) is called Walras Law. One can think of this as asserting that the demand in an economy is consistent with the assets of that economy. It is a budget constraint. The total value demanded is equal to the total value of the supply of the agents. Walras Law is no doubt the most subtle of the conditions we impose on Z here, and a micro-foundational justification will be given subsequently.

Before we state our final condition on the excess demand we give a geometric interpretation of the preceding conditions. Let S^"' —^e/?!j| |;>||2-2(p')2~ 1 be the space of normalized price systems. By homogeneity, it is sufficient to study the restriction Z.S'~leR. By Walras Law Z is tangent to S^"' at each point; p-Z(p)—0 says that the vector Z(p) is perpendicular top. Thus one can interpret Z as a field of tangent vectors on S'+l.

The final condition on the excess demand Z is the boundary condition

Z'(p)>0 if />'-0. (1.5)

Here Z(p)-(Z\p) Z'(/> ))£/*' andp~(p\...,/>'). Condition (1.5) can be interpreted simply as: if the /th good is free then there will be a positive (or at least non-negative) excess demand for it. Goods have a positive value in our model.

Theorem 1.1

If an excess demand Z: R'+ -0-+R' is continuous, homogeneous, and satisfies Walras Law and the boundary condition [i.e., (1.2), (13), (1.4) and (1.5)1 then there is a price system/>•€/*+ - 0 such that Z(^*)-0. This price system^* is given constructively.

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334 S-Smatt

The last sentence will be elucidated in the proof.

The proof of Theorem 1.1 is proved via Theorems \2 and 13. These theorems are general, purely mathematical theorems about solutions of equations systems.

Theorem 1.2 Let/: Dl-*Rl be a continuous map satisfying the boundary condition:

(BD) if x edD' then /(JC) is not of the form fix for any n>0.

Then there is x*€Z)' with/(x*)-0.

Here />'—jce/?'| || JC|| < 1 and 3£>'-;cSZ>'|||;c||- 1.

We use for the proof of this theorem two results that have been central to global analysis and its applications to economics, the inverse mapping theorem (or implicit function theorem) and Sard's theorem. To state these results, one uses the idea of a singular point (a critical point) of a differentiable map, / : U-*R" where U is some open set of a Cartesian space, say /?*. We will say that / is C if its r to derivatives exist and are continuous. For x in t/, the derivative D/(JC) (i.e., matrix of partial derivatives) is a linear map from Rk to R". Then x is called a singular point if this derivative is not surjective ("onto"). Note that if k<n all points are singular. The singular values are simply the images under/of all of the singular points; nndy in R" is a regular value if it is not singular.

Inverse Mapping Theorem If ySR" is a regular value of a C map / : U-*R", U open in Rk, then cither f~l(y) is empty or it is a submanifold V of Uof dimension k-n.

Here V is a submanifold of U of dimension m—k—n if given JC€ V, one can find a differentiable map A: W(x)-*6 with the following properties: (a) h has a differentiable inverse. (b) N(x) is an open neighborhood of x in U. (c) 6 is an open set containing 0 in Rk. (d) A(Af(x)n V) -0nC where C is a coordinate subspace of Rk of dimension m.

Sard Theorem Iff: U-+R", UcRk is sufficiendy differentiable (of class C, r>0 and r>k-n), then the set of singular values has measure zero.

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Ch. 8: Global Anafyju and Economics 335

For a proof see, for example, Abraham and Robbin (1967); general back­ground material can be found here. We say in this case that the set of regular values has full measure. Both of these theorems apply directly to the case of maps / : U-*C where U is a submanifold of dimension k of Cartesian space of some dimension and V is a submanifold of dimension n (perhaps of some other Cartesian space). In that case the derivative Df(x): Tx(U)-+TAx)V) is a linear map on the tangent spaces.

The above summarizes the basic mathematics that one uses in the application of global analysis to economics.

Toward the proof of Theorem 1.2 consider the following problem of finding a zero of a system of equations. Suppose/: D'-*Rl is a C2 map satisfying the very strong boundary condition:

(SB) fx)--x for all xEdD'.

The problem is to find x'eD' with/(jr*)-0. We are following Smale (1976a), influenced by a modification of Varian (1977); for history see the paper by Smale.

To solve this problem define an auxiliary map g: D'-E-tS1'1 by g(x)— f(x)/\\f(x)\\ wnere E-xeD'\f(x)-Q) is the solution set Since g is C2, Sard's theorem yields that the set of regular values of g is of full measure in S'~' (using a natural measure on S'~l). Lely be such a value. Then by the inverse function theorem g~\y) is a 1-dimensional submanifold which must contain —y by the boundary condition. Let V be the component of g~ \y) starting from -y. So V must be a non-singular arc starting from —y and open at the opposite end. Also V does not meet dD' at any point other than -y by the boundary condition and meets —y only at its initial point, since it is non-singular at —y. Now V is a closed subset of D'-E and so all its limit points lie in £. In particular E is not empty and by following along V starting from -y, one must eventually converge to E. This gives a geometrically constructive proof of the existence of x* eD' with /(JC") - 0 .

We remark that to further explicate the constructive nature of this solution, one can show that V is a solution curve of the "Global Newton" ordinary differential equation Df(x)(dx/.dt)— -X/(x) where A» ± 1 is chosen accord­ing to the sign of the determinant of D/(x) and changes with x. If D/(x) is non-singular, then Eulers method of discrete approximation yields

* - * - ! * © / ( * , , _ , ) " ' / ( * « - i ) .

which, with fixed sign, is Newton's method for solving /(*)«» 0. Thus the "Global Newton" indeed is a global version of Newton's method in some reasonable sense. M. Hirsch and I have had some success with the computer

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336 S-Smalt

using the Global Newton as a tool of numerical analysis in solving systems of equations.

Now suppose only that/: D'-*R'is only continuous and still satisfies/(x) - - x for xe9Z>'. Define a new continuous map/0: D[-*Rl by

fo(x)~f(x) for ||*|| <1, /< , (* ) - -* for | |* | |>1.

Take a sequence of e,—>0. For each / we construct a C°° approximation/ of/0, *o \\Mx)-f^.x)\\<e„ all xSD2. One can use "convolution" here. See Lang (1969) for details. Let <p„ be a C° function on R' such that / ? , - 1 and the support of <p, is contained in the disk Dr of radius r.

Then define/0>)« ffQ(y -x)<pr(*)dx- ff0(x)<pr(y-x)dx with r small enough relative to t„ and always r< j . Then /, approximates /0 and / ( * ) - -x for xedZ)2. We can apply the result proved above to obtain JC,€0J with/(*,)-<). Clearly *,€£>'and also x,—»xeZ>'|/(jc)"0 as /'-»oo. This proves Theorem 1.2 in case of the strong boundary condition (SB). Finally, suppose only/: Dl-*R* is continuous and satisfies (BD) as in the Theorem 1.2.

We will define a continuous map / : D[-^Rl such that / ( x ) - -x for xGbD[, as follows:

/ ( x ) - / ( x ) . for | |* | |<1. / (x ) - (2 - | | x | | ) / (x / | | x | | )+ ( | | x t | - l ) ( -* ) for ||*|| >1.

Now by the preceding result there is x*EDl with / ( * • ) - 0 . But ||**|| < 1, for otherwise the boundary condition (BD) would be violated. Thus/(**) 0 and the proof of Theorem 1.2 is finished.

For the main result on the existence of equilibria, we need to modify Theorem 1.2 from disks to simplices. Define

A . - ^ e K ' J S y - l , aA.-^eA.lsome^-O,

and

pc-(l/l,...,l/t)&Aupc being the center of A,.

We will deal with continuous maps <f>:A,-»A0 which satisfy the boundary condition:

(B) $(/>) is not of the form n(p-pc), /»>0 forp£3A,.

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Ck 8: Global Am+ms <md Etmmin 337

If one thinks of +(p) «* a vector based at p in 3Z?,, then <£(;?) does not point radially outward in A, according to condition (B). Theorem 1.3 Let <f>: A, -»A0 be a continuous map satisfying the boundary condition (B). Then there is p* e A, with *(/>•)-0.

For the proof of Theorem 1.3. we will construct a "ray" preserving homeo-morphism into the situation of Theorem 1.2 and apply that theorem. Define A:A,-»A0 by h(p)-p-pc; let \: &0-0-*R + be the map X(p) --(1/fXl/min^,). Then let Z>-Z)'nA0; *: Z)->A(A,) defined by HP)-MP/\\P\\), is a ray preserving homeomorphism.

Consider the composition a: D-»A0,

/> -* (> , ) -> A , - A 0 . We assert that o satisfies the boundary condition (BD) of Theorem 1.2. To

that end, consider ?ed£> and Itt pm,4>(q)+pe^h~,^(q). Now by (B) there is no ji>0 with $p)—p(p—pc) or with ii(p—Pc)"<4.q). Equivalently there is no H>0 with a(<7)-fii//(f), and since i/- is ray preserving that means ct(q)+nq, /i>0. This proves our assertion.

We conclude from Theorem \2 that there is q'eD with a(q*)—0; or if ;>*—V(?*)+^e then $(;>•) «0. This proves Theorem 13.

To obtain Theorem 1.1, define from Z: /?+ -0-»A' of that theorem, a new map ♦ I A . - A O by +p)-Z(p)-(ZZ'(p))p. Note 2«70>)-2Z'</>)-2z'(/>)2/>'"(), so that * is well-defined; + is clearly continuous. Also if/»€dA,, / - O for some i and so *'(;>)-Z'(/>)>0. Thus (B) of Theorem 13 is satisfied for*. Thus by Theorem 1.3thereis^*eA, with«X/>*)-0orZ(j*)«2Z'(/>*)j*. Take the dot product of both sides with Z(p') to obtain, using Walras Law, that \\Z(p*)\\2-0 or that Z( i*)-0 . This proves Theorem 1.1.

There can be natural equilibrium situations where D(p*)+S(p9) as in the following one-market example forp—0.

Fiawtl.2

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338 S.SmaU

Thus for an excess demand Z: R*+ -Q—R1, any p* in R^ - 0 with Z(/>*)<0, i.c, Z'(p*)<0 all /, is sometimes called an equilibrium, e.g. as in Arrow- Hahn (1971). One might also think of such &p* as a. free disposal equilibrium for after destroying excess supplies, one has an equilibrium with Z(/>)-0.

Proposition If Z: R'+ -0-*R* satisfies Walras law, ;>Z( ;>)-<), and Z(p') <0, then for each /, either Z'(;>*)«0 or/>•'-<).

Otherwise for some i, Z'(/>*)<0 and/>•'><); and for all /,pmZi(pm) <0 which contradicts Walras Law.

With weaker hypotheses than those of Theorem 1.1 one can obtain a free disposal equilibrium. Theorem 1.4 (Debreu-Gale-Nikaidd) Let Z: R'+ -Q-*R' be continuous and satisfy this weak form of Walras Law, namely, p-Z(p)<0. Then there is p*&Rl+-0 with Z(/>*)<0. See Debreu (1959).

Note first that Theorem 1.4 implies Theorem 1.1. For let Z satisfy the hypotheses of Theorem 1.1, then by Theorem 1.4 there is p* with Zp*)<0. By the above proposition, for each /, Z'(/»*)-0 or p*'-0. But by the boundary condition of Theorem 1.1, if p " - 0 then Z'(p*)>0, so in fact Z'( />*)-0 and thus Z(p)-0.

For the proof of Theorem 1.4, let B: R-+R be the function /3(/)-0 for f <0, and B(t)-t for />0. Define Z: *'+-0-»rt'+ by Zt(p)-B(Z\p)) for all i,p. Now just as in the proof of Theorem 1.1 above, define $: A,-*A0 by <MP)~ Z(p)-(S,Z'(p))p. This <(> satisfies the hypotheses of Theorem 1.3 and so there is^'eA, with *(;>*)-0 or Z(p*)-2Z'(/>*)p*. Take the inner product of both sides by Z(p) and use the weak Walras to obtain 2Z'(p*)/8(Z*(p,))<0. But tB(t)>0 unless f<0 in which case o8(r)-0. Therefore Z'(p*)<0 all L This proves Theorem 1.4.

Figure 1-3

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Ch. 8: Global Anafytu and Economics 339

Another generalization of Theorem 1.1, and Theorem 1.4 as well, will be proved to account for Z'(p)~*eo as p'-*0, e.g. the phenomenon illustrated in Figure 1.3. This theorem. Theorem 1.S below, is a slight generalization of a theorem in Arrow-Hahn (1971, ch. 2, theorem 3).

Suppose now that the excess demand Z is defined only on a subset ) of /?+ - 0 where "i) contains all of the interior of R'+ and if pG<T), so does Xp for each A > 0. Consider

Z: "ti-tR' is continuous, (12') Z(\p)-Z(p), aU />€<$, X>0, (13') PZ(p)<0, all pe<%, (1.4')

2 Z ' ( p J - o o if />*-*€<$. (1.5')

Theortm 1.5 Let Z: <$e/?' satisfy (1.2'), (1.3'), (1.4') and (1.5'). Then there is a p*e<% with Z(/>•)«).

Let fi: R-*R be as in the previous proof and define a: R-*R by fixing c>0 and letting

a( / ) -0 for f<0, - 1 for t>c, — t/c otherwise.

Define Z: R'+ -0-*R\ by

Z'(/»)-l if pe®,

-(l-a(2Z'(;>)))/9(Z'(,»)) + a(2Z'( / , ) ) otherwise.

Then Z is continuous. Just as in the_proof of Theorems 1.1 and 1.4 above, define <£:A,-»A0 by

<t>(p) — Z(p)—2Z'(p)p. Then <*> satisfies the hypotheses of Theorem 13, and so there is/>*eA, with «fr(/>*)-0 or

Z(;')-2Z'(;V-First suppose that p'e^D. Take the inner product of both sides with Z(pm) to

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340 S. Smalt

obtain Zp*)-Zp*)<0 (using the weak Walras Law). Then

2 (l -a( 2 Z'(^)))z'(/>*)0(Z'(/,-)) + a( 2 Z'(p*)) 2Z\p') < 0.

Since for any t, ta(t)>0, we have as a consequence that

(i-«(2z'(i'*)))2z'(/»*)/8(z'(/»*))<o,

and even

2z'(/)/(z'(/))<o. But /£(/) is strictly positive unless / <0. Therefore Z'(p*) <0 all /._ On the other hand dp* 6 ^ , it follows from the above equation on Z thatp* is

(1, . . . , 1)1/t which is in <$. So in fact^* can't be outside <l). This proves Theorem 1.5.

2. Pore exchange econony: Existence of equilibria

This section has two parts; in the first we make stronger hypotheses and emphasize differentiability, while the second is more general. The two are pretty much independent The existence theorems are special cases of the Arrow-Debreu theorem; see Debreu (1959) and Appendix A.

To start with, consider a single trader with commodity space /»»xe/? ' |x« (x1 x'), x '>0. Thus x in P will represent a commodity bundle associated with this economic agent. It will be supposed that a preference relation on P is represented by a "utility function" u: P-*R so that the trader prefers x to^ in P exactly when u(x)>uy). The sets u~'(c) in P for c in /? are called the indifference surfaces. Strong hypotheses of classical type are postulated:

u: P-*R is C1. (2.1)

Now let g<x) be the oriented unit normal vector to the indifference surface u~l(c) at x, c-w(x). One can express g(x) as gradK(x)/||gradw(x)|| where gradM-(a«/8x1 du/dx*). Then g is a C' map from P lo S'~\ S ' " ' - p £ A ' I H P I I - ! - I* plays a basic role in the analysis of consumer preferences and demand theory.

Our second hypothesis is a strong differentiable version of free disposal, "more is better", or monotonicity,

g(x)ePnS'-l-intS';1 for each x&P. (2.2)

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Ch. S: Global Analytu and Economics 341

The word interior is shortened to int. So (2.2) means that all of the partial derivatives du/dx' are positive.

Our third hypothesis is one of convexity, again in a strong and differentiate form. For xSP. the derivative Df (x) is a linear map from R' to the perpendicu­lar hyperplane s(x)-1- of gx). One may think of g(x)x as either the tangent space 7^X)(.S'~') or as the tangent plane of the indifference surface at x. The restriction of Dg(x) to g(x)-1 is a symmetric linear map of £(x)x into itself,

Dg( x) restricted to g( x)x has « « strictly negative eigenvalues.

We have sometimes called this condition (2.3) "differentiably convex". One can restate (2.3) equivalently as

The second derivative D2u(x) as a symmetric bilinear form restricted to the tangent hyperplane $(x)x of the indif- (2.3') ference surface at x is negative definite.

We can see the equivalence of (23) and (2.3') as follows: Let Du(x): R'-*R be the first derivative of u at x with kernel denoted by KerDu(x). Then since vg(x)-Du(x)(v)/\\gr&du(x)\\, o€KerDu(x) is the same condition as cgradu(x)-0 or vg(x)»0 or yet ©e*(x)x . Let u„Oi eKerPw(x). Then tvg(x)-Dw(xXc,) / | |grad«(x) | | and o,-Dg(xXo2) - D2u(xXt>„ » 2 ) / Hgrad u(x)||. This implies that (2J) and (2.30 are equivalent.

Next we show:

Proposition 2.1 If u: P-*R satisfies (2.3) then M"'[C, oo) is strictly convex for each c.

Proof We show that the minimum of u on any segment can not be in the interior of that segment. More precisely let x,x'SP with u(x)>c, u(x')>c. Let 5 be the segment Xx+(l-A)x'|0<X< 1. Letx*«X*x+(l-A')x' be a minimum for u on S. Then Du(x*Xt>)-0 where o - x ' - x ; since x* is a minimum, D2u(x*Xc,c)>0. This contradicts our hypothesis (23') that D2u(x*)<0 on KerD«i(x*). Therefore u is greater than c on S.

The final condition on u is a boundary condition and has the effect of avoiding problems associated with the boundary of R'+:

The indifference surface u~ '(c) , , A\ is closed in R' for each c.

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342 S.Smaie

This may be interpreted as the condition that the agent desires to keep at least a little of each good. It is used in Debreu (1959).

We derive now the demand function from the utility function of the trader. For this suppose given a price system />eint R'+ (of course int R'+ -/*) and a wealth W€LR+ — W€.R\W>0). This definition of R+ is convenient though maybe not consistent Consider the budget set Bp w—xGP\p-x — w. One thinks of Bf w as the set of goods attainable at prices p with wealth w. The demand f(p, w) is the commodity bundle maximizing satisfaction (or utility) on Bfw. Note that Bp w is bounded and non-empty, and that u restricted to Bp w has compact level surfaces. Therefore u has a maximum x on Bf w which is unique by our convexity hypothesis (2.3) (Proposition 2.1).

Then x—/(/>, w) is the demand of our agent at prices p with wealth w. It can be seen that the demand is a continuous map f: int R'+xR+-*P. Since x — /(p,w) is a maximum for u on Bf m, the derivative DU(JC) restricted to Bf m is zero or g(x)-p/\\p\\. From the definitionp-/(p,w)-w and f(\p,\w)-/(/?,w) for all \ > 0 . Thus:

Proposition 2.2 The individual demand/:int /?'+ xR+-*P is continuous and satisfies

to g(Ap.»))-p/Hp\\, 0>) p-f(p,»>)m'w, (c)f(\p,\w)-f(p,w) if A>0.

Furthermore we will show the following classical fact with a modern version in Debreu (1972).

Proposition 2.3 The demand is C1 (and will have the class of differentiability of g in general).

For the proof, note mat from Proposition 22, we can obtain

9: P-(inlS';l)xR+t <f>(x)-(g(x),xg(x)),

which is an inverse to the restriction of/to (intS^"')x/?+. Since <p is C\ by a version of the inverse function theorem,/will be C1 if the derivative D<p(x), of <p at an arbitrary xSP is non-singular. To show that Dtp(x) is non-singular, it is sufficient to prove, Dqp(.*Xi))™0 implies ij—0. For ijG/?', we may write

D9(x)(v)-(Dg(x)(ri),r,-g(x)+xDg(x)(T,)).

So if D9(xXn)"0. by this expression surely Dg(xXn)"0, so ijeKerD^(jc).

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Ch. 8: Global Analysis and Economics 343

But also i>-s(x)«0. TjegCjc)-*- and we know (3) that Dgx) restricted to g(x)^ is non-singular. In other words g(x)x nK.erDg(x)—0. This proves Proposition 2.3.

Let us elucidate this a bit From what we have just said we may write R' as a direct sum R,^g(x)-L®KtrDg(x) or write ijG/?' uniquely as TJ—TJ,+I)2 with TJr«(Jc)-0. D $ ( J C X I 2 ) - 0 . See Figure 2.1.

Here we are basing vectors at x. We may orient the line KerDg(x) by saying 7iEKerD;(x) is positive if n-g(x)>0. The following interpretation can be given to this line: Since Dg(x) is always non-singular, the curveg~\p) withp—g(x), p fixed in S+~' is non-singular. It is called the income expansion path. At xeP, the tangent line to g~ \p) is exactly KerDgix) (from the definition). This curve may be interpreted as the path of demand increasing with wealth as long as prices are fixed. One may consider wealth as a function w: P-*R defined by w(x)—xg(x). Then w is strictly increasing along each income expansion path, and in fact g~\p) can be differentiably parameterized by w.

Suppose now that the trader's wealth comes from an endowment e in P, and is the function w—p-e of p. Then the last property of the demand is given by:

Proposition 2.4 Let pt be a sequence of price vectors in bit R'+ tending to p* in 9/?'+ as ;-»oo. Then ||/(/>,,/v«)||-n» as i-»oo.

Proof If the conclusion were false, by taking a subsequence and re-indexing we have f(PnPi')-*x*. Since u(f(p„p,e))>u(e) all i, by use of (2.4), x* is in P. Therefore g(x*) is defined and equals p*. But since p*edR'+, we have a contradiction with our monotonicity hypothesis (22). This proves Propontion 2.4.

, KerDg(x) \ / ^g(x) P

Figure 2.1

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344 SSmaJt

A pure exchange economy consists of the following: there are m agents, who are traders, and to each is associated the same commodity space P. Agent number /' for /—1 m has a preference represented by a utility function u,: P-*R satisfying the conditions (2.1)-(2.4). We suppose also that to the ith agent is associated an endowment e,e.P. Thus at a price system, pGRl+ - 0 , the income or wealth of the ;th agent isp-e(.

One may interpret this model as a trading economy where each agent would like to trade his endowed goods for a commodity bundle which would improve or even maximize his/her satisfaction (constrained by the budget). The notion of economy may be posed as follows:

A state consists of an allocation xe(P)m, JC-(JC, xm), x^P togedier with a price system ^es^"'. An allocation is called feasible if 2jr,""2«,. Thus the total resources of the economy impose a limit on allocations; there is no production. The state (JC, p)GP)mX. S+"' will be called a competitive or Walras equilibrium if it satisfies conditions (A) and (B):

(A) 2*,«2*,.

This is the feasibility condition mentioned above. (B) For each /', xt maximizes u, on the budget set B—yGP\py—p-e,).

Note that by the monotonicity condition (2.2) above, (B) does not change if in the definition of the budget set p-y—p-et is replaced by p-y <p-e,.

Note that (B) can be replaced by conditions (B,) and (Bj): (B,) p-x,—p-e, for each /'. (Bj) g,(x,)-p for each i.

With (A), (B,), and (Bj), equilibrium is given explicitly as the solution of a system of equations. We will show: Theorem 2.5 Suppose given a pure exchange economy. More precisely let there be m traders with endowments «,€/», /—I m, and preferences represented by utilities u,: P-*R, each satisfying conditions (2.1)-(2.4). Then there is an equilibrium; i.e., there are xfeP, /'« 1 m, and/>ES$T' satisfying (A) and (B).

We may translate the equilibrium conditions (A) and (B) into a problem of supply and demand. Let S: R'+ — 0-»fl'+ be the constant map, S(/>)»2*,. Let 0:int/?'+-»/*'+ be defined by D(p)-'2,fl(p,pel) where f^p.p-e,) is the demand generated by u, (Proposition 2.2). Define the excess demand Z: int R'+ -*R' by Z(p)~D(p)—Sp)- We note that the equilibrium conditions (A) and (B) are satisfied for (x,p) if and only if Z(p)«0 and x,—fi(p, p-e). So if we

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can find a solution of Z(/>)»0 by Section 1, we will have shown the existence of an economic equilibrium in the setting of a pure exchange economy.

Walras Law for Z [(1.4)] is verified directly; if pEint R'+,

p-Zp)-pD(p)-pS(p)-?lpfl(p,pei)-pJlel-0.

Homogeneity, that Z(A/>)«Z(/>) for A>0 is checked as easily. To apply the existence theorem, Theorem 1.6, we take <% to be int /?'+. It

remains only to verify the boundary condition (2.5'), that if p tends to a point in the boundary of R'+ - 0 , the 2Z'(/>)-»oo. But that is a consequence of Proposi­tion 2.4, using the fact that Z is bounded below. Thus we have shown the existence of p'efy with Z(p*)<0. But by the proposition preceding Theorem 1.4, it must be that Z(p*)—0 since Walras Law is satisfied. This proves Theorem 2.5.

We give another setting for a pure exchange economy where we use only continuous preferences.

For this consider a preference relation on the full /?+ as commodity space (rather than its interior P) represented by a continuous utility function u: R'+-*R. We replace conditions (2.1) to (2.4) simply by:

u: R'+-*R is continuous, (2.1')

and u(\x+(\-\)x')>c if u(x)>c, u(x')>c and 0<X<1. (22')

The latter is a strict convexity condition on the preference relation. Suppose that to each trader, in addition to a preference of the above type, is

associated an endowment«, in P. Thus each agent has a positive amount of each commodity.

Theorem 2.6 Given a utility ut: R'+-*R for agents/-l,... ,m satisfying (2.1'), (220 above and endowments e,GP, /« 1,..., m, there is a ("free disposal") equilibrium (x*,p*). Thus: (a) Z*,* < 2e„ and (b) x* maximizes ut on the budget set x,eR'+\p*-x,<pm-et) at x* for each /.

Proof Before constructing a demand, we cut off commodity space near oo to avoid problems with unboundedness. We are able to get away with this because of the

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346 ZSmaU

feasibility condition. More precisely choose c> ||2^ii and let Dc be the ball of radius c or Z>c —/»e/?'|||/»|| <c). Define an associated false demand function fi^R'* -0)xR+->Xc, XC-DenR'+, by taking/at (/>,*) to be the maximum of u, on Bf w — xSXe\p-x<w). Then since Bfw is compact, convex, and non­empty, by the strict convexity property of «,,/(/>, w) is well-defined. Proposition 2.7 The false demand / : (R'+ -0) x R+-*Xe is continuous, /(A/7, \w) "Up, w) for X>0, and/>•/(/>, w)< w- Also if ||Up, w)\\ <c, then the maximum,/^, w), of u, on B,tW-;ce/t,J/>-.x<H» exists (the true demand!) and f,(p,w)-/(/».w)-

This is straightforward except perhaps for the last. Let *,-/(/>, w) with || JC,|| <C and consider *,€/?,.„ with ut(xt) > u^i,). Let S be the segment between x, and x, in R'+. For any *,'#*, on SnXc, u(x-)>ui(xl) by strict convexity (2.2'), contradicting the choice of xk. This proves Proposition 2.7.

Next define Dp)"'Lff(p, p-e>\ S(/>)«2*„ andZ: R\ -0-*/?' by Z - D - S . Then Z satisfies the weak Walras Law, so by Theorem 1.4, there exists p with

Z(/>)-0. Thus iif,(p,p-e,)"xlt S x . - S * , and ||JC,||<C. Therefore by Proposi­tion 2.7, xl~xt»Up,p'e,) and (x, xm,p) is an equilibrium; Theorem 2.6 is proved.

Suppose u,\ R'+-*R satisfies: No Satiation Condition: u,: R*+-*R has no maximum.

Then we claim that the commodity vector xt~Up,w) at the end of the proof of Theorem 2.6 satisfies pUP> HO-W (rather than inequality). Otherwise choose xf in /?'+ outside Bp „ with ui(xf)>ui(f,(p,w)) by the No Satiation Condition. By strict convexity, as in the proof of Proposition 2.7, we get a contradiction. Thus in this case we have that for the excess demand Z(/>)-2/(/», /*•«,)- S(/»), the usual Walras Law is satisfied at equilibrium and we obtain a more satisfac­tory interpretation of the free disposal equilibrium (see the proposition pre­ceding Theorem I.4.).

The question of relaxing strict convexity in Theorem 2.6, as well as questions of production, we defer to Appendix A.

3. Pirato opdmalhy

Towards the problems of Pareto optimality in equilibrium theory and the "fundamental theorem of welfare economics", we consider abstract optimization problems in this section.

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CK 8: Global Analysis and Economies 347

Our setting is an open set W in R" (W could be a smooth manifold or submanifold in what follows) together with C2 functions ut: W-*R, /-»1 m. One might think of W as the space of states of society and the members of that society have preferences represented by the ut. A point xE.W n called Pareto optimal (or just optimal) if there is no.yE If with u,(y)>u,(x) all / and strict inequality for some /. Such a y could be called Pareto superior to x. If m« 1, an optimum is the same thing as an ordinary maximum. The point xE W is a local optimum if there is a neighborhood N of x and x is an optimum for «,,...,«„ restricted to N. A point x E W is a strict optimum if whenever y E W satisfies ut(y)>u,(x), all /, then y—x (like a strict maximum). Finally a local strict optimum is defined similarly. Note that these definitions apply generally, e.g. to non-open W in R". The goal of this section is to give calculus conditions for local optima. The following theorem is proved in Smale (1975) and Wan (1975); we follow the Smale paper especially, which one can see for more history. Theorem 3.1 Let «,,...,«„: W-*R be C2 functions where Wis an open set in R". If xE Wis a local optimum, then there exist X,,- • •, \m > 0, not all zero and

2 * , D « , ( * ) - 0 . (3.1) Further suppose A, Am, x are as above and

2 A,D2u,(x) is negative definite on the space . . „ oS/r |A,Dtt / (x) (c ) -0 , / - l , . . . ,m .

Then x is a local strict optimum.

Here Du,(x) is the derivative of u, at x as a real valued linear function on R", and D2u,(x) is the second derivative as a quadratic form on R* (one could think of D2u,(x) as the square matrix of second partial derivatives]. 2)A/D2u<(x) is then also a quadratic form.

Note that if one takes *— 1 and «—1, the theorem becomes the basic beginning calculus theorem on maxima. For m— 1, and n arbitrary, the theorem might be in an advanced calculus course. It has been pointed out to me by several people that one can reduce the proof of Theorem 3.1 to this case of m-1 . However the direct proof we will give has some advantages with the geometry and symmetry in the u/s. In the following Im stands for image.

Proof of Theorem 3.1 Let 7»o»-cE/l" , |c-(o„...,t^), ©,>() and Pas its closure. Then the first condition of the theorem may be stated as there is XePos—0 with A-Du(x)—0

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(dot product). Here «»(«i um) maps W into Rm. Let x be a local optimum and suppose ImDw(x)n Pos^Q. Then choose v&R" with Dw(xXu)eA». and a(0 a curve through x in W with a(0)«x and the cr'(0)-u. Clearly for small values of /, M,(a(/))>u,(a(0))»«,(JC) SO that x is no local optimum. Thus we know that lmDu(x)nPos—$.

From this it follows from an exercise in linear algebra that there is some \ePos-0 with A orthogonal to ImOu(x). Thus A D M ( X ) - 0 , and the first part of the theorem is proved.

Suppose that the theorem (second part) is true in case A,>0, all /, and consider the general case. Let the indices be such that A, AA>0, At+1 « . . . -X^-0 . Then conditions (3.1) and (3.2) are the same for optimizing u,,..., um at x and optimizing uu...,uk at x. So (3.1) and (3.2) are satisfied for ux,...,uk also; and since by assumption the theorem is true in this case, x is a strict local optimum for the ux,...,uk. But then it is also a strict local optimum for ui,...,um. From this it is sufficient to prove the theorem in the case all the A, are strictly positive.

We may suppose that x is the origin of R" and u(x)—0 in /?"*, so that the symbol x will remain free to denote any point in W. Then the condition that OG W is a local strict optimum is that there is some neighborhood JV of 0 in W with (u(N)—0)nPos—<>. We will show that under the conditions of Theorem 3.1, indeed there is such an -V.

Denote by A" or KerDu(O) the kernel of Dw(0) as a linear subspace of R* and by K x its orthogonal complement

Lemma 3.2 There exist r,8>0 with the property that when ||x||<r, x —(x,, x2), x,e/T, x 2 et f x and5||x,||>||x2|| then Au(x)<0 if x#0.

Proof Let #-2A,D2u,(0). By (3.2) there is some <r>0 so that H(x, x)< -o | |x | | 2 for xSK. For xER", X«-(X,,X2) , x,eAf, x2GK±, we may write H(x,x) — / / (x„x,)+2//(x„x2)+/ /(x2 ,x2) . Since |ff(x„ x2)| <C||*,| | ||x2|(, |H(x2, x2)| <C,|tx2j|2, we choose ij,S>0 so that if S;|x,U > ||x2jj then Hx,x)< —ri\\x\[2. Write by Taylor's theorem for ||x||<r, u(x)-Du(0)(x) + D2u(0)(x, x) + /?3(x) where | |A-/?3(X)||<TJ/2||X||2 . Taking the dot product with A yields the lemma.

Now writey-ImDw(O) and write u in Rm as u—(ua,ub), uaGJ, ubGJ±.

Lemma 3.3 Given a>0 and fi>0 there is J > 0 so that if | |X | |<J , x«(x, ,x2) , x,eAf, x 2 et f x with ||xj|| >S | |X, | | , then ||ut(x)|| <a||«B(x)||.

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Proof The restriction

Du(0)K.:K±-*lmDu(0)

is a linear isomorphism so there are positive constants c,,r with

| |DU(0X^) | | - | |D«(OXX 2 ) | |>C I | | JC 2 | | all *-<*, .< 2 ) .

>c\\x\\ if llxjl^fillx.ll.

By the Taylor's series

ua(x) + ub(x)-ux) = Du(0)(x) + R(x),

so that given /?>0. we may assume ||/?(JC)|| </3||.xj| for ||x||<some number s. With /?-(/?„. Rb) we have

\\u.(x)\\-\\DuiO)x) + *.ix)\\>(c-fi)\\x\\.

and

II«»(*)II-H*»(*)II<J8||*||,

say with /3 small enough and /3/(c-0)<<*. Then ||u6(x)|| <a||«ft(jr)||, finishing the proof of the lemma.

To finish the proof of Theorem 3.1, choose a of Lemma 3.3 so that if ll«.(*)ll<«ll«,(*)ll to*11 u(x)&Pas-Q. This can be done since lmDu(0)nPos - 0 , all the X,s being strictly positive. Choose a disk around 0 of radius r0, r0<r of Lemma 3.2 and r0<s of Lemma 3.3. Let the S of Lemma 3.3. be given by Lemma 3.2. Now from the two lemmas we have that ux)&Pos if x+0, ||;t|| <r0, proving x to be a local strict optimum and Theorem 3.1.

We pass now to an extension of Theorem 3.1 to the setting of constrained optimization. Thus let C2 functions u, uM be defined on an open set WcR', subject to constraints given by conditions of the form ge(x) > 0, /— 1 k. with g:W-*R of class C2. One may express the problem by defining >f0—jt€ W\gfi(x)>0, /3«1,. . . . A: and seeking conditions for optima of the restrictions M„...,umto W0.

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350 S. Sinai*

Theorem 3.4 Suppose xGlV0 is a local optimum for the functions «,,..., «m on fV0, W0 as above. Then there exist non-negative numbers X,, pff, not all zero such that

£ \,Du,(x)+ 2M/,D^(*)-O, (3.r) / - I

where M/i-0 if * , (x)*0 .

Furthermore suppose xGfV0, Xo>0, p^X) with not all the A,,M/j zero, are given so that (3.10 is true. If the bilinear symmetric form

S ^ D V J O + S M . D ^ X ) (3.2')

is negative definite on the linear space

veR'\v\lgndul(x)-0, all;,and o-M,gradg#(x)-0, all a

then x is a local strict optimum for «„..., MM restricted to W0. For the first part let us suppose that gf(x)—0 (by renumbering if necessary)

precisely for all fi-l,...,k, and define +: W-*Rm+k by * -(«„...,um ,g t , . . . ,gk). Then we claim that ImD^(x)n?ai - f Otherwise let D + ( X X C ) € A H and let «(/) be a curve in W satisfying a(0)-x, a'(0)-o. For small enough c, a(e) is in tVa and a Pareto improvement over cr(0)—x. So x could not be locally optimal So ImD+(x)nfar—+ and there is a vector (A,,...,AM,p,,...,p*)€/Vu-0 normal to ImD+(x), as in Theorem 3.1. This proves die first part of Theorem 3.4.

For the proof of the last part we first note, with $: W-*Rm*k as above, that if x S W0 is a local strict optimum for + on W, then it is also a local strict optimum for «,,..., um on W9. This follows from the definitions. But the hypotheses on x in die second part of Theorem 3.4 imply that x is a local strict optimum of ^ as a consequence of Theorem 3.1. Thus Theorem 3.4 is proved.

We end this section with some final remarks: (1) Note Theorem 3.1 is the special case of Theorem 3.4 when Jfc-0. (2) Suppose the gm satisfies die Non-Degeneracy Condition at x&W0. The set

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Ch. 8: Global Anafyju and Economies 351

Dgp(x) for P with £j(.x)™0 is linearly independent. If this condition is satisfied then in (1) at least one of the A, is not zero.

(3) If in Theorem 3.4 m-1, the first part is related to the Kuhn-Tucker theorem, and if the Non-Degeneracy Condition is met, one has A, — 1.

Theorem 3.4 is in Smale (1974-76, V) and Wan (1975). See also Simon (forthcoming) for further information on this.

4. FuBdaneatal theorem of welfare ecoaomks

We return to a pure exchange economy as in Section 2, with traders preferences represented by C2 utility functions u,: P-*R, P—va.lR'+, i —1,... , m, satisfying the differentiate convexity, monotonicity and strong boundary conditions (22% (2.3), and (2.4). Also as in Section 2, the maps gt: P-*SlZx defined by g,(x)~ grad u,(x)/||grad u,(x)\\ will be used in our approach. While we do not presume that each agent is given an endowment, it will be supposed that the total resources r of the economy are a fixed vector in P.

Thus the set W of attainable allocations or states has the form

W - x e ( / » r | x - ( x 1 xj,x,eP, 2>,-r.

The individual utility u,: P-*R of the /th agent induces a map o,: W-+R, cl(x)-ul(xl). After Section 3 it is natural to ask, what the optimal states in W for the functions c,, / - 1 , . . . , m, are. The answer is in:

Theortm 4.1 The following three conditions on an allocation xSfV (relative to the induced utilities v,:W-*R) arc equivalent: (1) x is a local Pareto optimum. (2) x is a strict Pareto optimum. (3) g,(x,) is a vector in S'+\ independent of /.

Let 8 be the set of x 6 W satisfying one of these conditions. Then 8 is a submanifold of W of dimension m— 1.

In this theorem as in this whole section, we are following Smale (1974-76).

Proof Note (2) implies (1). We will show that (1) implies (3). For this we do not use any conditions on u,: P-*R except that the u, are C1.

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Thus suppose that x^W is a local optimum. We apply the first part of Theorem 3.1 to obtain A, A„>0, not all zero, such that ZA,Du,(x)™0 or 2A,DU,(JC,)—0. We may suppose that A, * 0 by a change of notation. Apply the sum to the vector xG(R')m with Jc—(Jc,,...,*„,), 2 J C , - 0 (a tangent vector to W). If jc-(x„0,...,0, -3r„0,...,0) with —3c, in the /fcth place we have 2\lDut(x,)(xi)-\iDu(xi)(x1)-\kDuk(xk)(xl)-0 for all *,£/?'. Thus \kDuk(xk) is not zero all k and equal to A,Du,(x,). This yields condition (3).

For the equivalence of the three conditions, it remains to prove that if x satisfies (3) then (2), x is a strict optimum. So let x satisfy (3) and leiyGfV with »<O0> «>/(*). all'» or equivalently, ui(y,)>u,(xl), all i.

We use now: Lemma 4.2 Let u: P-*R satisfy differentiable convexity (2.3). UySP,u(y)>u(x) andy+x, then Du(x)(y-x)>Q. Thus also in this case,yg(x)>xg(x). Proof For />0 and /< 1, Proposition 2.1 (strict convexity) implies that u(ty-x) + x) >u(x), and so (d/dt)u(t(y-x)+x)\lmO>0. Therefore by the chain rule Du(x)(y-x)>0. On the other hand by Taylor's series if Du(x)(y-x)-0, u(x + fiy-x))~u(x) + D2u(x)((t(y-x))2)+R3 which yields by differentiable convexity [(23')] u(x + ty-x))<u(x) for small t. This lies in contradiction with the convexity. The lemma is proved.

By the lemma, for each i, y,-gi(xl)>x,-gt(xl) with inequality in case yti*xt. Then let p—g^x,) using (2.8), so Jjp-yi>2p-x, with inequality i f^^x, any /'. But sincere W, S^,-/"—2*, and S p w ^ S p ' * , . Thus.y,—xt, each i,y—x and x is a strict optimum.

For Theorem 4.1 it remains to prove that 8 is an (m— 1) dimensional submanifold. For this we use the inverse function theorem in the form of die transversality theorem of Thom which goes as follows: Let W,Vbe submanifolds of some Cartesian space (or abstract manifolds) and let A be a submanifold of V. Thus given .yeA, there is a diffeomorphism h (differentiable map with a differentiable inverse) of a neighborhood U of Y in V onto a neighborhood N of 0 in Rk, *—dim V, and A(An U) — Nn C where C is a coordinate subspace of Rk. Then a: W-*V is transversal to A if whenever x£ W with a(jr)-.yeA, 7^(K)-ImDa(jc)+7^(A). In other words, the image of the derivative Da(x):Tx(W)-+Ty(V) together with tangent vectors to A at>> spans the tangent space of V at Y. Also one can think of Da(x) mapping subjectively onto the complement of the tangent space of A in Ty(V).

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Then the inverse function theorem implies:

Transversality Theorem

Let a: W-+V\»t transversal to the closed submanifold A of V. Then a~ '(A) is a submanifold of W with either o"'(A) empty or dim W— dim a" '(A)- dim V— dim A (codimension is preserved).

Here, the dimension is shortened to dim. References with details are Abraham and Robbin (1967) and Golubitsky and Guilemin (1973).

For the proof let a(x)«.y€A and apply the usual inverse function theorem to the composition v°h°a: W-*C x with h as above, C x is the orthogonal comple­ment of C above and IT : Rk-*C x is the projection.

Now take the W of the Transversality Theorem as the W in Theorem 4.1 and let V be the Cartesian product of m spheres, V-(S'~l)m and A to be me diagonal in V,

A-7e(s'-Tb-0>I,...,*J,><es'-,,/I-72 ym). Define g.W-*S,~l)m by g(x) having ith coordinate given by f,(x,) where gt-.P-tS1'1 a the normalized gradient of the utility of the ith trader. By definition [first part of Theorem 4.1, condition (3)J, g~\A)<-9. We will show that g is transversal to A as follows:

Let Kx-KerDu(x) where u: W—Rm is the map with the ith coordinate of tt(x) given by u,(x,). Then

KM-[xe(Rt)m\xle#.'2xl-0,xrgl(xl)-0).

Let Lx for xS8 be the set of xSTx(W) with DgixXi^eT^i*) or

L x - [xe(R')m\ 2 Jf/-0, Dg,(x,)(x,) is independent of / .

[Eventually we will see that Lx — Tx(8) is the tangent space to 8 at x.\

Lemma 4.3 LxnKx-Ofot all xeO. Moreover dim Kx—mt-t-m+l. Proof Let p-g,(x,) and y,:px-*px be the restriction of Dg,(xt) to px. Then y( is symmetric with negative eigenvalues [see condition (23)]. Also 2 Y / - 1 is an isomorphism since y,~l is symmetric with negative eigenvalue* and the sum of

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negative definite symmetric linear maps is negative definite (from linear algebra, or look at the corresponding bilinear symmetric forms).

Let x£LxnKx and D&^X * / ) - / • Then y f ' ( / ) - * , since *, S,(*,)-° and 23C 1 »2Y/~ , ( />) — 0 sop—0. Thus also x,—0 each i, proving the first part of the lemma. The dimension of Kx is easily counted.

To finish the proof of Theorem 4.1, let us count more dimensions. It is easy to see that dimW—mt-t, dim(S,~l)—ml-m, dimA-f-1. From these dimen­sions and the lemma, Dg(x) restricted to Kx maps Kx injectively into the complement of 7 (A) in T/((S'~t)m), y-(p p). This proves that g is trans­versal to A and therefore by the transversality theorem, £~'(A) is empty or a submanifold of dimension m— 1. However, it cannot be empty by Theorem 2.5. using any endowments et which sums to r. This finishes the proof of Theorem 4.1.

Remark

By the definitions, Lx—Tx(9) and so dim Lx~m — 1, and so

rx( W ) - Tx( 9)®KX (direct sum).

We give some consequences of Theorem 4.1: Corollary 4.4 Let W be the space of attainable states of a pure exchange economy with fixed total resources r as above. Consider the map u: W-*Rm defined by: u(x) has ith coordinate u,(x,), »'«1,..., m, where u,: P-*R is the utility of agent /'. Let 9 be the submanifold of Pareto optimal points. Then u/9, the restriction of u to 9 is an imbedding of 0,into Rm.

Here an imbedding means that the derivative is injective as a linear map from Tx(9)-*Rm, and the map is injective.

In fact, the corollary is an immediate consequence of the remark that KerDu(jc)nrx(*)-0.

Then since u(9) has codimension 1 in Rm> one may define the Gauss map G: 9-+Sm~* by letting G(x) be the unit normal to u(9) at u(x), oriented so that it lies in R\.. By definition G(x) is perpendicular to the image Du/9(x) or G(x)Du(x)(x)-0 for all xeTx9). Since Tx(9)nK.erDu(x)-0, this is the same as G(x)Du(x)(x)-0 for all x - ( x , xm) with 2 ^ - 0 . Thus if we take X-(A, AM)—\x as in Theorem 4.1 and normalized as well, so that ||XX | |S1, then \X—G(x). In a certain way the Gauss map G is the curvature of the imbedded manifold u(9), so that the \ of Theorem 4.1 may be thought of as a curvature. Note that the previous discussion, in contrast to the rest of this

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Ck 8: Global Anafyju aid Economies 335

article, depends on the utility representations ut, not just the underlying prefer­ence. Remark

In connection with Corollary 4.4, it is worth noting that if x&O, then it can be shown that Du(x): Tx(W)-*Rm is surjectivc. If xG8, then the image (see above) of Du(x): Tx(W)-*Rm has dimension m- 1 and it can be shown that the map u is a fold at x in the sense of singularities of maps. See Smale (1974-76); this aspect of the subject is developed in work of de Melo, Saari, Simon, Titus, and Wan [see Simon (forthcoming) for some references].

Corollary 4.5 Given eGW, there is some x in 8 so that e—xSKx. Furthermore there is a neighborhood N(9) of 8 in W so that for each eSN(8), there is a unique x in Q with e-xEKx. For an endowment vector e in N(8), «—(«,,...,*„,) there is a corresponding unique Walras equilibrium, (x, p), with x€0, p—gt(xt), all /, and the .budget condition p-e^p-x,, all L

For the proof note that for every x&fV the attainability condition of equilibrium is satisfied. If xB8, then the satisfaction condition defining ;>—£,(*,) for some i (hence all /) is also satisfied. Finally the budget conditionp-e^p-x, all i may be restated as&fa.He.-Jc,), all /, or simply as e—xeKx( — Ker Du(x)). Then the first sentence of Corollary 4.5 just re-expresses the existence Theorem 2.5. The uniqueness theorem, second or third sentence of the corollary, follows from the tubular neighborhood theorem of differential topology [see Golubitsky and Guilemin (1973, ch. 2, sect. 7)]. While we are following Smale (1974-76, VI), this is also close to work of Balasko (1975).

Towards the final corollary of Theorem 4.1 we give the concept of welfare equilibrium. We say that a state (x, p)E. WX4S'+ ' is a welfare equilibrium if x, is a (in this case the) maximum of u, on the budget set Bp p.x~ xeP\p-x—p-xt. The subset of welfare equilibria in WxS1'1 will be called A. From this definition it follows that (JC, p), x-(x,...,xm), x/G/>,^€S^"' is in A provided (1E), (2E) hold:

(1E) 2x,-r. (2E) gt(xt)—p, each / - 1 m (from the maximization condition on u().

If one has the further data of individual initial endowments, eteP, /«1 m, summing to r, then a third condition (3E), with (1E) and (2E), defines the equilibria of Section 2 or the Walras equilibria:

(3E) P'€i~P'xi> ' " !• • • • .«• The welfare equilibria are called "equilibria relative to a price system" in

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Debreu (1959). They play a central role in theorems of welfare economics as well as non-tatonmeni dynamics. It is important to distinguish these two kinds of related concepts of equilibria. When there is a danger of confusion, we use the words Walras equilibria with emphasis on the budget condition (3E).

A very sharp, though perhaps not general, version of the fundamental theorem of welfare economics is the following:

Corollary 4.6

All as above, 9, A are (m— l)-dimensional submanifolds, closed as subsets of W, rVxS'+\ respectively, and the map 0: \-*W defined by (x, p)-*x is a diffeomorphism of A onto 9cW.

We recall that a diffeomorphism is a differentiate map with differentiable inverse so that it is bijective (one to one and onto).

The usual form [compare Debreu (1959), Arrow-Hahn (1971)| states that A-»0 is well-defined and surjective, i.e., every optimal allocation is supported by a price system and the allocation part of a welfare equilibrium is optimal.

The proof of Corollary 4.6 goes as follows: Define an imbedding a: W-*Wx. S'~l by <*(*)-(*,«,(' ,)) . Then a(0)-A using Theorem 4.1; a/9 and p/A are inverse to each other with a/9 an imbedding of the submanifold 9. Then A is a submanifold and the corollary follows.

We now indicate how some of this goes without assuming any properties on the utilities u,: P-*R besides differentiability, i.e., C2. Let 9S be the subset of the space W of attainable allocations which consists of local strict optima. Em­phasizing no hypotheses on the «,, we still have:

Proposition 4.7

If x e W is a local optimum for the utility induced functions on W, then

(a) there exists X, > 0 not all 0 with SX.Du^x,)-*) (which implies that £,(*,) is independent of «').

Further let x satisfy (a) and also

(b) SX /D2M j(x,X(^)2) is negative whenever 2 x , - 0 , x ( g , ( x , ) - 0 , all /, and x^Q, some i.

Thenxefl,.

For the proof note that the first part is done (Theorem 4.1). The last part just goes by applying the second part of Theorem 3.1; the situation is similar to the proof of Theorem 4.1.

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The condition (b) is considerably weaker than differentiable convexity at x„ each (. In general one may hope to circumvent convexity hypotheses by using the second-order conditions (as in Theorem 3.1). On the other hand, x may be a strict optimum with no supporting price equilibrium. In that case there is only an "extended price equilibrium'' [see e.g. Smale (1974-76, III)].

We now consider the situation of Theorem 4.1 for commodity space with boundary. Up to now in this section the analysis has been interior. Thus suppose that trader i, for /—l,...,m, has a C2 utility representation u,: R'+-*R of his/her preference (so u, is defined on the full R'+, not just the interior). The conditions of differentiable monotonicity and differentiable convexity of Section 2 will be assumed for the rest of this section. We suppose that each u,: R'+-*R is the restriction of a C2 function defined on some open set of R' containing R'+. Then u, off R'+ will never be used. In this way the derivatives Dul(x),D2ui(x) stiU make sense for x€d/t+ and so the conditions (22) and (2.3) make sense on the boundary as well.

Fix a vector r€int/f'+ of total resources and let Wo-ix^R'+Ffex^r). Then WQ is the space of attainable states of our pure exchange economy. Let W be a neighborhood of % in xe(/J')*|2x,"r on which the functions ©,: W-+ R, can be defined by ©,(*)-«,(*,), i - l , . . . , /n . Let gfiW-tR be given by g*(x)«x*. Then we are in the situation of optimizing several functions subject to constraints, or Theorem 3.4. These g * are constraints as above and bear no relation to the normalized gradients of utility functions. The problem of optima in W0 relative to the v,: W0-*R is equivalent to optimizing the v,: W-+R subject tog*(x)>0.

Theorem 4.8 For i - 1 m, let u,: R'+->R satisfy

gradi^x,) ^ ^ c - i g,(x,)€SSr'. eachx,, (4.1) ||gradu,(x,)||

and

D2u/(x,)ongl(x/)J" is negative definite. (42)

Suppose »o"*G( / ?+)' , , |2x j-r) with v,:W0-*R defined by C,(X)-M,(X,). If xElV0 is a local optimum for the o,: (a) there existspES'+* and \u...,\m>0, not all 0, withi>AfDw,(x,) each i,

where one has equality in the k th coordinate if xf**0. Conversely let p,xly..., xm,Xl,...,\m be as in (a) with/rx^O each i. Then x is a strict optimum.

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358 S.SmaU

For the proof let g/:iV-*R be defined as above so that g/(x)« x/ are constraints for v, on W. Then the derivatives satisfy Dg/xXx)—xj where xe(* ' ) m with x - ( x , x„) and 2 x y - 0 . Also if,-(if/ xj). If x in »P0 is a local optimum for the p,, then Theorem 3.4 applies to yield the existence of A,>0,/i/>0, / - l , . . . , m , y - l t, not all zero with j i / -0 if XjVO and

2 A,Du,(x,)(x,) + 2 M/«/-0. all x, as above.

Take x / - 1, x-'— - 1 , all other components of x zero to obtain

where Dtf,(**y denotes they'th coordinate of DK,(X,). Alternately we see that $—A(Du,(x,)+/i, is independent of < where /*,—

(/i1,,...,//,), /i,>0 and fij-^-O. Note that q+Q, for otherwise all the A, and fi, would be zero [recall Du,(x)^0]. Let p—q/\\q\\ and multiply through q— AjDu^x,)* p, by 1/||?||- By renaming the A,, ji, we have now

J - A , D K , ( X , ) + M „ M/>0, A,>0, M/X/-0.

This yields the first part of Theorem 4.8. For the converse let^eW0, u,(y,)>u,(x,% »- l , . . . ,m, x„y,GR'+. We must

show that y,—x, for each i. By the first lemma in the proof of Theorem 4.1, Du,(x,Xyl-xl)>0 with equality only it y,"x,. By our main condition above p-jc,-A,DM,(x,X*i) ""d *° A,**0 since p-x^O. Then by this same condition P'(yi-x,)>n,-yi otp-y,>P'X„ with equality only ily,~xlt each /. On the other hand S ^ - S ^ - r ; putting this together indeed yields y,—x, each i. This finishes the proof.

Remark Note that if u, satisfies the stronger monotonicity condition, that Du,(x,)G intS^"', theapx^O in Theorem 4.8 can be omitted.

Say that (x,p) is a welfare equilibrium (as before), or (x, p)EAc.W0xS'~i if x, is a maximum of u, on the budget set BftP.x—x&R,+ \p'X<p-x,, each i. Thus for (JC.^)GA, 2)x,-r, since xGWQ.

Proposition 4.9 If (x,^)€A, then there exist numbers A, >0, /— l,...,m, and p,eR', M/ >0 with Xj-ji/ "0and^-Af-Du^x^+ji,. Conversely, given (x,p)e W0 X5+"1, with p-x,+0, all /, and A,,/*, as above withp—\t-Du,(x,)+p,, then (x , ; )6A.

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Ch. &' Global Analysis and Economies 359

Proof Since x, is a mnxii"'"" of u, on Bf px, for each /', there exist \, >0, /i, e/J +, o, > 0 not all zero, with

^ D i ^ K S j + Z M / D f / t o X ^ - a ^ - X f - O . all x ,e* r ,

or o^-A.DuXx,)*/!,, Mi-*<-0.

If the o, were 0, then so would be \„ /i,. Thus we may rescale by dividing by a, to obtainp"\lDul(xl)+nt, / V X , - 0 . This proves the first part For the second let y,GBf with u(yi)>u(xi). Then by Lemma 4.2 in the proof of Theorem 4.1, Du,(xiXyl-xi)>Q, and/> yl>yl\lDul(xt)>p-x„ \,+0, as in an earlier argu­ment Then>'/e£,>/.J,(, contrary to hypothesis. Thus (x,p)eA. This proves the proposition.

For the rest of this section, let us assume for simplicity the strong monotonic-ity hypothesis, that D«i,.(x,)eintS5r1. The projection map W0xS^l-*1Vt, (x, p)-*x, induces a map a: A-*$, from welfare equilibria to Pareto optima. By the proposition above and Theorem 4.8, a is well-defined and it is surjective. While these results have an extensive literature under the topic of "fundamental theorems of welfare economics", the question of uniqueness of a supporting price system seems not so standard. Is a injective?

The answer is affirmative under the further mild hypothesis of "no isolated communities** [Smale (1974-76, V)]. For xe*f0, an isolated community is a non-empty proper subset ££ l , . . . ,m with the property that wherever ieS and x/*0, then x~0 for all k&S.

Theorem 4.10

If x is an optimum in W0 with no isolated communities, then there is a unique supporting price system.

Here we are supposing W^ is the space of attainable states; the utility functions u,: R'+^R are C2 with Dii/(x,)€int.S+~I and D*ut(x,)<0 on KerDwJ(xl).

Lemma 4.11 Suppose x e W0 has no isolated communities and i,qGl,..., m) are two agents. Then there is a sequence »,,..., i„ of agents with /,—i, i„—q, and a sequence of goods j jK such that xft+Q, all * and for any k, either jk+x

mjk or *Vn"'*-

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360 S.S*mli

Proof Otherwise take any agent, say agent number 1 for convenience, and consider all above such sequences (i, /'„), (j\,...,jH) with i'|»l. Let S be the subset of l,...,m of all possible i„ reached in this way. If 5 is proper, then it is an isolated community. This proves the lemma.

To prove Theorem 4.10, first obtain p,X,,n, as in Theorem 4.8, with />» XiDttiix^+n,, \,>0, p,€A+ and n,-x,^0. The problem has to do with the ambiguity of the A,, p4. Suppose by renumbering, that agent 1 has some of the first good so x\^0. Normalize p by taking p ' - l (and not | |^| |-1). Then 1— p l - \ , Du,(x,)' since /i'«0, &nd i i» mus determined. Let q be any other agent; choose a sequence (/„..., i„), i ,« l , i„—q, (j\,...,jm) as in the lemma. We claim that X, is determined for each ik. Suppose inductively that X,ki is determined, and !***«*_ i- Then. jk—jk_lt both agents ik and i't_, have some of goody*. Therefore pJk - A,4_, D u , ^ ^ ^ ^ 1 determines pJk aadpJk - Xik Dult(xu)Jk

determines A,t. Here we used the fact that the corresponding p/'s are 0. Once all the A,'s are determined uniquely, let k be any good. Choose i so that xk+Q. Then pk—X,Du,(xl)k determinespk. This proves Theorem 4.10.

5. Flaiteaeai aad stability of equilibria

The first goal is to give a proof that the pure exchange economy described in the first part of Section 2 has only a finite number of Walras equilibria, at least for almost all endowment allocations. At the same time we show that these equilibria are stable (better "robust") in the sense that they persist under perturbations of the endowment allocation. These results are due to Debreu (1970). Our ap­proach to this result is to define an "equilibrium manifold" without passing to the demand functions. The hypotheses, framework (pure exchange economy), and notation will be the same as in the first part of Section 2.

Thus define the equilibrium "manifold" 2 as follows: The space (P)mx(P)m

consists of («, JC), «-(«„...,«„,), *~C*i»- •»*„) with e„ x^P. Here e will be thought of as an endowment allocation parameterizing an economy. Then 2 will be the subset of (P)mx(P)m of (e, x) satisfying:

2 ei"" 2 xi (* I01*! resource or attainability condition), gt(x,) is independent of /

(the first-order condition; *,(*/) - grad u,(x,)/\\g^d u,(x,)\\),

p • (e, - x,)—0 (budget condition).

(5.1)

(5.2)

(53)

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Ck & Global Anafysu and Economies Ml

Thus if e is fixed, (e, x) e 2 , then (x, p) where p -&(*,), is a Walras equilibrium and conversely [see Section 2, (A), (B,), (Bj)]. Theorem 5.1 2 is a submanifold of (/>)"x(/>)" of dimension mf.

Proof Define a map

+ :(P)mX(P)m^Rlx.Rm-xx.(S,-x)m,

by sending

(e,x)-*'Zel~'Slxl,p-(el-xl),...,p-(em_l-xm_ ,),g,(x,) «•„(*„))•

Then from the definition of 2 we may write 2 -$~ ' (0x0xA) where A -(p,..., p)G(S'~l)m, and we have used the fact that conditions 2 e , « 2 x i and ; ' ( « , - x , ) - f t i - l m-1 , imply /^ . . -xJ -O.

As in Section 4, Theorem S.l would be a consequence of + being transversal to OxOx A, using a simple counting of equations. Following the line of proof of Theorem 4.1; if +(e, x)eA define

^.,-('.Jc)e(ATx(/?,riD*(*,xXe,ic)eoxox7XA), or, equivalently, from differentiating (1), (2) and (3),

/•(«<-*<)+/>*(«/-**)-o-

Here we take/>-£,(x,) a n d / - D g ,(x,Xx",). Now we define a second linear subcpace Ktx of (R*)mx(Rl)m by

Kt x-(e,x)|2e,-0, x,-^-0, i < m - 1 , ^ , - 0 , / < m - l .

Here : R'-tP-1 is the orthogonal projection so that e^v.e^p-e,, each i. This space AT,iJC is motivated only by the proof of Theorem S.l. Clearly dim Kt x—ml, and one also can see that dim /?'x/r ,- ,x(,S ," l)J"-dimA-mf. Thus if LtJlc\ ^ , , 0 , we have that ♦ is transversal to OxOx A, just as the situation was in Section 4.

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362 S. Smalt

Lemma 5.2 L.,xnK,,x-o.

For the lemma let (e, x) belong to the intersection. As in Section 4, y,: p±-*P x

denotes the restriction of Dgt(xt). Then 2*,"0 since 2*,"0 and 2^™2*, . So xtp-0, all »', and y,~'(/)"*/. each i. Also 2 Y , ~ 1 ( / ) " 2 J C / - 0 and / - 0 , therefore jc,-0. Finally one sees that e,—Q proving the lemma and hence the theorem.

We emphasize that we are taking «,: P-+R, /— 1,...,m, as in the first part of Section 2. Theorem 5.3 There is a closed set FC(P)m of measure 0 so that if e€F then there exist a finite (positive) number of Walras equilibria relative to the endowment «-» (<,,..., em). This finite set varies continuously in e as long as e does not meet F. Let ir:(P)mx(P)m-*(P)m be the projection defined by v(e,x)-e. Let ir0: 2-» (P)m be the restriction of v.

Lemma 5.4 The map u : Z-*(P)m is closed. The image of a closed set is closed. Proof Consider a sequence eu\xu)) in (P)mx(P)m,j-l,2,3,..., so that eu> con­verges to eG(P)m. Then by the equilibrium conditions defining 2, and the boundary condition on u,, the xU) have a subsequence converging to some xE(P)m. This is enough to show that w0 is closed.

Let Cell be the closed set of critical points of v0 and F—ir0(C). Then F is closed by Lemma and has measure 0 by Sard's theorem. Theorem now is a consequence of the inverse function theorem applied to the map v0.

A study of comparative statics of equilibria can now be done using these theorems.

While the above approach comes from Smale (1974-76) a closely related way of proving Debreu's theorem is in Balasko (1975).

AppendkA. Exfateace of economic cquUlbrhna with production

We prove the theorem of Arrow-Debreu on the existence of economic equilibrium with production as treated in Debreu (19S9). The reason we include the proof is to show that calculus can indeed be the starting point of equilibrium theory with proofs at least as short and natural as those emphasizing Kakutani's

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Ck. 8: Global Analysis and Economies 363

fixed point theorem. On the other hand, our approach has much in common with that of Debreu; we owe much to his exposition as well as to conversations with him.

Here the treatment is brief. One can see Debreu (19S9) for economic interpre­tations. The proof here is based on Theorem 1.5, and it is somewhat similar to the proofs in Section 2.

An economy consists, first, of a production side. We suppose I commodities including labor. To each of n producers,y »1 , . . . , n, is associated a "technology" YjCR' with the conditions: (T) (a) OE YJ, eachy (possibility of no production).

Let r - 2 YJt

(b) Yn(-Y)—Q) (an irreversibihty condition). (c) y is closed and convex. (d) Y-R\(zY (free disposal).

It can be shown that (d) is a consequence of YD — /?'«. in the presence of (c); see Debreu (1959). Here Yj may be thought of as the set of productions that are available to firm/ We suppose that the firm is driven by profit maximization. Thus if a price system p is operative, the production y e Y, is sought so that the profit p-y is a maximum.

Pass now to the consumer side of the economy. To each of m consumers, z«l, . . . ,m, is associated a "consumption set" XtcRf and a utility function u,: Xj-tR which represents his/her preference. The following is assumed:

(C) (a) X, is a closed convex set (b) X, is bounded below. That is, there exist </„...,</„€/?' with XlcxeR'\x>dl) or X,>d,. (Here x>d( means that each component of x is > the corresponding component of d,.) (c) w, satisfies the convexity condition: if x,x'GX, with u,(x)>ul(x'),

then «,(rx+(l -t)x')>u(x') for each /£(0,1). (d) «, has no maximum (no satiation condition)

Remark One could have used directly a preference relation here, as in Debreu (1959), rather than utility function. No generality is gained as one can see in Debreu's paper.

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364 S. Smalt

Furthermore, to each consumer is associated an endowment e^X, with *, having all coordinates strictly larger than some element of A1,. As in Debreu (19S9), this is an unhappy hypothesis. Finally (private ownership economy) let 9„ be the share of agent / in firm j . Then it is assumed that Q<0,,< 1 and 2,1,0/y-1. If a price system p prevails, then the wealth of agent i is given by

An equilibrium for an economy above is a "state" (x,y,p) with x€.]!"_Xt, y^H'^iY^peS1'1 which satisfies A) Attainability, or 2-*/~2>'y+2«/. B) Each consumer my»itniM« satisfaction or:

x, is a maximum of u, on the budget set

B-[x(=Xi\p-x<pcl+'Z9IJp-yjy

Q Each producer maximizes profit or: y, is a «n"JTni"T| of 11 on YJt where

U^.Y^Risn^y^py.

Arrow-Debreu Theorem For an economy above there is always an equilibrium.

We first give a proof under additional restrictions; then we extend that proof to the General Arrow-Debreu Theorem.

Theorem A.1 Suppose that the economy described above satisfies the further conditions:

(1) Each Yj is closed and strictly convex. (2) Each u, has the strict convexity property of Section 2 or, more precisely, if

U,(JC)>C, u,(x')>c and 0<f< 1, then u(tx+(l-t)x')>c. Then there is an equilibrium.

Toward proving Theorem A.1 we use the following basic lemma for which Bowen gave me this analytic version of my more geometric account:

Lemma A.2 (basic estimate) Let Y be a closed convex subset of R' with Yn(-Y)-0) and Yz>-R'+. Then given beR* and n>0 there is a constant c so that if yx ymGY and 2.ty>6 then \\yj\\ <c eachy.

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For the proof let A"- y e Y \ )| y || - 1 . We prove three assertions: Assertion 1 The origin 0 of R' is not in the convex hull of K.

If a,Jt, + • • • +o r x r -0 with 0<a,< 1, o, + • • • + a , - 1 , x,GK, then — a , J C 1 « a , - 0 + a 2 * 2 + " ' " + «;••*,•£)''.

and o,x, clearly is in y. Thus a,*,SYn(.-Y) reaching a contradiction.

Assertion 2 There is a q—(qv...,q,)GR', each q,>0, such that q-x<0 for every xSK.

As K is compact, so is its convex hull. By Assertion 1 there is a q in R' with q-K<0. If e, is a coordinate basis vector then —e,eK and -^—^(-« /)<0.

Assertion 3 There are constants e>0, /9>0, so that if x&Y then ?• x*8+t-e\\x\\.

Let -e->max9-x|xE^ and /8—max?-jc| ||x|| < 1. The inequality is clear if ||JC|| < 1. For ||*||> 1, xeY, and one has Jc/||jc|feA" since Y a convex and contains 0. Then -e>^-jc/| |x|| orqx< -e||jc||.

We finish the proof of Lemma 1 as follows: Suppose S.ty > b with y £ Y. Then q-b<2q-yJ<n(B+e)-e2\\yJ\\, so 2 | |^ | | <(n(B+e)-q-b)/t.

An analogous lemma for the consumption side is:

Lemma A.3 Given c,6/?', there is a>0 such that if x,€.X„ Xl>dl [as in (Q above] for / '- l , . . . ,m, and 2x ,<c , , then ||jcf||<a, each i.

We omit the very easy proof. Now let 6 -2</ / -2< , and choose c as in Lemma AJ, so that if 2> ,

y>6, then |[^||<c,eachy. Ut ^ - ^ n f l , where 0,-.yeZ>'|||.y||</'. For^e /^-Q, let S^(^)-the maximum of II,: Yj-*R where II,(^)«-^-/. Then ^ is the "false supply function" of finny.

Lemma A.4 Sj:R'+-0-*Yj is well-defined, continuous, Sj(\p)-Sj(p) tor \>0, and if \\Sj(p)\\<c, then Sj(p) is the maximum of II, on / (the true supply).

This is clear from the definitions, recalling that we are in the situation of Theorem A.1, so that Y is strictly convex.

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366 S. SmaU

Remark If Yj is merely assumed closed and convex (not necessarily strictly convex), one still has Sj defined as a correspondence; i.e., Sy. R'+ -0-+S(Yj) is a map with values, convex subsets of Yj. It is homogeneous and when restricted to S'+"' has a compact graph

r-(,,jOes5r,x$|j'e4QO. Furthermore in this case if ySSj(p) has norm ||y]j <c, iheny is a maximum of II, on Yj. Note that U.f(y) is independent of yGSj(p).

Define w,: R'+-0-*R, the "false income" of consumer /', by #,(/»)/>«,+ 3LJ8UP-SJ(P). Then w, is continuous. Let b, c, e, be as above and choose c, GR' such that 2yy+«<c, if ||>/||<<: each j . Choose a by Lemma AJ and let

Define a "false demand" Dt: R'+ —0-*Xt for each i by D,(p)"the maximum of u, on S /—x€^jp-jc< **i(7») (compare Proposition 2.7).

Lemma A.5 The false demand £,:/?+-0-».£( is well-defined, continuous, D,\p)—Dlp) for A>0, andf •Dt(p)—wt(P)- Aiao tf II A(^) l l < a t n e n A(^) " t n e "M*inmm of a, on the budget set ^ /-jceJf,|^-x<w /(^) and/»-.£,(/>)•*<(/>)•

The proof uses the same arguments as that at the end of Section 2, uses the No Satiation Condition, and the convexity of X,. The continuity uses the fact that e, dominates some element of X, (the basic hypothesis on e,). We leave the detailed proof, which is not difficult, to the reader.

Remark In case u, satisfies the convexity condition (c) of (Q rather than strict convexity of Theorem A. I, then D, is defined as a correspondence with values, convex subsets of Xf. It is homogeneous, and the restriction D,: S'+ X-»X, has a compact graph. Also if xGD, satisfies ||x|| <a, then x is & maximum of u, on x&X,\p-x <w,(p)) aadp'X-w,(p).

^Now^define these aggregate functions from /t'+—0 to /?': 5 — 2 4 + 2 « „ D—"2,D„ and Z—D-5. From Lemmas A.4 and A.5, Z satisfies homogeneity and weak Walras, so Theorem 1.5 applies to produce p* SS'+ ' with Z(p')<0. UiyJ*"Sj*p*\x*"DJ(p*\ so then 2*,* < 2jj*+2«,. Since each xf&X^X,, this implies b<z,yf (definition of b). Thus \\yf\\<c (Lemma AJ), and by

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Lemma A.4, yf is the maximum of II, on Yr By the choices of c, and a, via Lemma A.3, ||x*||<a, each /'. By Lemma A.5, xf is the maximum of u, on <xs*,|j*-x<*,</>»)>. with *,(/>•)-/>* •«,+2Ap*-tf •

We may choose ;6/?'+ so that S x ' - S y f + Se-,-*. Apply />• to this to see (using Lemma A.5 again) p*z-0. Then Zyf-z a in y « 2 l ^ by (7") so we have^e Yj with S.ty-Z.v/-•*• Then/>-2.y,-p-2.y/, which implies that.yy also (as weU isy*) maximizes II, on YJt and (xf,yJtp*) is an equilibrium, proving Theorem A.l. Note in fact y^y* by the strict convexity, but our argument covers the more general case of Theorem A.6.

We next weaken the convexity hypotheses of Theorem A.1 by using the approximation theorem of Appendix B: Theorem A.6 Theorem A.1 remains true if each Yj is closed and convex (rather than strictly convex), and instead of the strict convexity hypothesis on each u,, we only assume (C) as in the Arrow-Debreu Theorem.

Proof Proceed as in the proof of Theorem A.1. As in the remark after Lemma A.4, we can consider Sy. S'+^Yj.j— 1 n, as correspondences.

Suppose *>0 is given. Apply the theorem of Appendix B to obtain continuous functions ^e:5^"'-»/ for eachy-l, . . . , /t , with T^c^IY) . Next note that w,:S'~l-+R, defined by *>,(p)— p-e^^jO^p-Sjip^'u a weft-defined continu­ous function, even with 5 a correspondence. As in the remark after Lemma A.5, we can consider D,: S'+l-*X, defined ju a correspondence. Apply the theorem of Appendix B to obtain functions D„: S'+^X, such that 1$ CBXT^) and l/»-A^^)-*/(P)l<«. B^peS*-1-

Define Z,: S'^^R' by Z . ( / > ) - 2 4 ( * ) - 2 i , . ( / > ) - 2 < „ and Z^-ZJ^p) ~(pZ,(p))p. Thcnj>Z,(p)—Q andpZ,(p)-*0 as e-»0. Apply Theorem 1.5 to obtain/*, such that ZJipj—Q.

Let yjt—SjJip,), xt,mDiJ(p,)- Now take a sequence of tk tending to 0. By taking subsequences we obtain yJti-*yj, xitt-*x,, ptt-*p to obtain an equilibrium. This finishes the proof of Theorem A.6 »& is Theorem A.1.

Now we give the proof of the General Arrow-Debreu Theorem. We need:

Lemma A.7 Let'z'denote the convex hull of a subset Z of Euclidean space. Then

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Proof Since 2 l is convex it contains 2 ^ . We will show A+&CA+B. Let a, >0 with 2 a f » l . Then /T+ycA+y^since 2a,*,+.y-2tf,(*,+.>'). Therefore /?+Be ^ + J . Finally 1+/Tc B+JcA + B, showing indeed that A + BcA + B. By induction the proof of Lemma A.7 is finished.

With the hypotheses and notation of the beginning of Appendix A, let Yf be the closure of the convex hull of /. Recalling /"«2ly, we have:

Lemma A.8

217- Y. Proof

Since YjC Yf, ^YfD^Yj. On the other hand, since the sum of the closure of sets is contained in the closure of the sum, it follows from Lemma A.7 that 2^7 c ^ (recall Y is closed and convex). This proves Lemma A.8.

Apply Theorem A.6 to obtain an equilibrium (x*,y*,p) for the economy above with Yf replacing Yj. Now 2 / / E Y (Lemma A.8) and so 2^,*""2//™>' with^Sr.

Furthermore p-y. •mpyf- This is so sincey* is a maximum of II, on Yf and therefore/ is a maximum of II, on Y. This implies (since/ —2/y) that I I , ^ ) is at least as much as H.y*) and hence equal. The rest follows and the Arrow-Debreu Theorem is proved.

AppeadbtB. A theoraa o« the approxJamtin of BMati-ralaed mappings

We prove the following theorem of Cellina (1969), using extensively an unpub­lished exposition of W. Hildenbrand:

Theorem B.l Let K be a compact set (say in some Euclidean space), T a compact convex set of R', and tp: K-*S(T) a correspondence with values convex subsets of T such that the graph r9»(x,y)SKxT\ye^(x)) is compact. Then given e>0 there is a continuous function/: K-+ Tsuch that TfcB2JiT9).

Here 1 is the graph of / in Kx T and Blt is the open set of all points of Kx T within 2e of T,.

For the proof define <j>': K-*S(T) by <p*(x)-convex hull of UyeBt(x)<p(y)-

435

S.Smai*

Ck «V Global Aitafytu and Economics 369

Lemma B.2 Let e>0 be given. Then there is a 6>0 such that T^cBJtTw).

Proof If the lemma were false, one could take 6->\/n and obtain a sequence (x„,y„) in Kx T, with (x„,ym)<£BjLT9), all n, and^-SX'.^, 2 X > 1 , X'.X^eoXzJ), </(*;, x j < l/n. By taking subsequences, we get x.-wc^-^y', X'.-A,, *J-«- /-x. So .y-SX'/ , X'>0, 2X'-1 and (x,y') is in the closure of T9. Since <p(x) is convex, (x, y) is in the closure of Tw, contradicting (x„ y„)$^BJiT). The lemma is proved.

Next let S be as in the lemma and

Uy-(xeK\yeB(<p\x)) for each yST,

and then choose Uri,...,Uyt a finite covering of K. Let B, be a corresponding partition of unity so ft: JT-»0,1], i - 1 , . . . , Jfe, are continuous functions, A ( x ) - 0 exactly if x&.U, and Z,B,m 1. For example, one could take

& ( * ) - ig < ( X ) "**» a , (x ) - inf <rf(x,x').

y - i

Define/(x)<-2A(x)^. Then / i s clearly a continuous function,/: K-*R*, such that for x£JT,/(x) is a convex combination of those points .y, such that xSUfi ory,GBte\x)).

Since an e-neighborhood of convex sets' is convex, BJ&\x)) » convex and /(x) is in it Therefore (x,/(x))€*,(r^) and by the lemma (x. /Cxflea^r,) proving the approximation theorem.

References

Abraham. R. and J. Robbia (1967), T n w m l napoiats «ad flow*. New York: Beegaoua. Arrow. K. and F. Haba (1971X Oaiaral naeanelilhi aaarjrak. Saa Fraaaaoo, CA: Hold- Day. Balaeko, Y. (1975X "Seeae mate em miipiwaii aad OB eafaJaty of eqaffibriaai m laaaral

eqnilibrinm theory", Jooraal of Mathematical EooaooMca, 2:95-111. CeUiaa, A. (1969), "A theorem OB the appraxaaation of ooaapact motti-vahMd meppiaa*-.

RaadiooBti Acadaaaia National* Lineei, 47:faacj6. Debrev, O. (1959), Theory of valae. New York: WDoy. Defareo, O. (1970), "Eooaoaaki with a fiaha aat of aqnOibrk", Bcoaoeaatrka, 3$ :3S7-39Z Debreo, O. (1972), "Smooth prafenaoaa", Enoao—rtrka, 4fe«B-«16. Ootabitaky, M. aad V. Oaflmiin (1973), Stable anrr"1 *»d their ^ifflnritiii New York:

Spriafer.

436

370 S. Small

Lang. (1969), Real analyst*. Reading. MA: Addison-Wesley. Simon, C (forthcoming), "Scalar aad vector maximization: Calculus techniques with economics

applications'*, in: S. Rmter, eel, Studies ia mathematical economic*, MAA studies in mathematics ssries.

Smale, S. (1974-76X "Global analysis and economics, IIA-VT, Journal of Mathematical Econom­ics, 1:1-14, 107-117, 119-127. 213-221. 3:1-14.

Smak, S. (1975), "Sufficient conditions for an optimum", ia: A. Manning, ed, Dynamical systems— Warwick 1974, Lecture note* ia mathematics series no. 468. New York: Springer.

Smak, S. (1976a), "A convergent process of price adjustment and Global Newton methods", Journal of Mathematical Economics, 3:1-14.

Smak, S. (1976b), "Dynamic* ia general eqnilibrram theory", American Economic Review, 66:288-294.

Variaa. H. (1977), "A rasaark on boundary restriction ia the Global Newton method", Journal of Mathematical Fi aomiTf. 4:127-130.

Wan, H.-Y. (1973), "On local Panto optima". Journal of Mathematical Economic*. 2:35-42.

437

Gerard Debreu Wins the Nobel Prize Steve Smale

I was at the office early Monday, October 17, when my wife Clara called with the news mat our friend Gerard Debreu had won the Nobel Prise in economics. Although I had anticipated this event, it was exciting and a pleasure to hear that it had actually happened.

Gerard and I have been close friends since we met fifteen years ago. I still remember our first encounter. He came to my office to ask about some mathematics needed for his work (he was trying to show that a general economy could have only a finite number of equilibria). I found Debreu friendly; we could com­municate easily about mathematics and economics. I was especially impressed with his ability as a mathe­matician, his clarity and rigor.

His questions were the beginning of my own work in economic theory. In the following years we spent long hours in discussions. (Our day-hikes at Point Reyes have been especially memorable.) We ex­changed questions, his more mathematical and mine about economics. Eventually, Gerard joined the math­ematics department; I joined the economics depart­ment.

Debreu's great contribution is his profound use of mathematics in the central theme of economic theory, consolidating an insight of Adam Smith more than 200 years ago. Debreu has given the foundations of gen­eral equilibrium theory in his classic work "Theory of Value." The award of the Nobel prize to Debreu gives a valuable impetus to basic research in mathematical economics.

In 1776, "The Wealth of Nations," Adam Smith sug­gested an "invisible hand" promoted economic ac­tivity. He wrote:

Every individual endeavors to employ his capital so that its produce may be of greatest value. He generally neither intends to promote the public interest, nor knows how much he is promoting it. He intends only his own secu­rity, only his own gain. And he is in this led by an IN­VISIBLE HAND to promote an end which was no part of his intention. By pursuing his own interest he frequently promotes that of society more effectually than when he really intends to promote it.

However, Smith did not give any mechanism as to how this theory could actually work. The invisible

hand was a mysterious force. To see the problem, one can consider, as in Paul Samuelson's famous text "ECONOMICS", the city of New York:

Without a constant flow of goods in and out of the dry, it would be on the verge of starvation within a week. A variety of the right kinds and amounts of food is required. From the surrounding counties, from 50 states, and from the far comers of the world, goods have been traveling for days and months with New York as their destination.

How is it that that 12 million people are able to sleep easily at night, without living in mortal terror of a break­down in the elaborate economic piuorsaea upon which the city's existence depends? For all this is undertaken tcilhout coercion or antniiuA direction by any conscious body!

This is the operation of the "invisible hand". The price system somehow gives order to a highly decen­tralized economic process. The key to understanding this mechanism lies in the equations of supply and demand. If there were only a single isolated market, i.e., one good, it is not hard to see that the equilibrium price of that good would be such that supply and de­mand would balance.

There are many markets, however, in an actual economy and those markets are highly interrelated (for example gasoline and cars). The theory of a balanced supply and demand in this complex situation is called "General Equilibrium". It is quite subtle even to for­mulate the problem sensibly.

About one hundred years after "The Wealth of Na­tions", Leon Walras made an important initial contri­bution by putting forth a mathematical framework in which the equations of supply and demand were gen­erated by individual agents, producers and con­sumers. Moreover, Walras plausibly asserted that a price equilibrium solution to his equations existed.

It wasn't until the 1990s, however, that Abraham Wald made the Walras theory correct and this was only in a limited economic context.

In the early 1950s Kenneth Arrow (Nobel Prize in 1972) and Gerard Debreu gave a general mathematical economic setting for the ideas of Walras and in this setting proved the existence of an economic equilib­rium. Debreu laid all this forth in his monograph, "Theory of Value".

THI MATHEMATICAL lOTIUJCENCia VOt- ». NO 2 0 ISM Sf i t tW-Vat* H n . YoA 6 1

438

The work on general equilibrium theory does not imply that it should be the model for society. For one thing, in order that the conclusions be valid, it is hy­pothesized that there is no monopoly. But monopoly does arise in a decentralized system. Putting aside mo­nopoly, there is the problem of inequity. Arrow and Debreu showed that at an equilibrium allocation, no one can be made better off without hurting someone else. However, the theory allows grossly unequal shares of the goods of society. Thus, substantial gov­ernment mediation of the decentralized price system is required.

There are serious problems of a different kind—lim­itations of the theory itself. In the Arrow-Debreu

economy the passage of time is not satisfactorily taken into account. This lack of dynamics keeps the theory from explaining why prices adjust to equilibrium or why they stay there. Another, possibly related, weak­ness is that the model makes unrealistic demands on the rationality of its agents. Kven equipped with modern computers, the consumers and pro­ducers could not make the decisions required for the theory.

Although fascinating challenges remain, there is now a framework in which we can try to meet them. That is the basis established by economists of two cen­turies and, in particular. Smith, Wafaas, Wald, Arrow and Debreu.

62 m MATHtMATiou. DvnujcsNcm VOL. «. NO 2. i «

439

the following year where he took the Math6matkrues SpSriales curriculum*

In the summer of 1941 he was admitted to the Ecole Nonnale Supe>ieure where he studied until 1944, (He re/members Henri Cartan was his most influential teacher.) His studies were interrupted in l$44 when he enlisted in-the French Army, but he eventually re­turned to take the Agre^ationdeMathematiquesat the end of 1945,

During the interruption Debreu had become inter­ested in economics and, in particular, the mathematical theory of general economic equilibrium. The two years following his AgrGgation were devoted to his conver­sion from- mathematics to economics while he served as an Attache' de Recherches of the Centre National de la Recherche Scientifique.

At the end of 1948 he received a Rockefeller Fellow-ship that allowed him to spend 1949 and part of 1950 at Harvard, Berkeley, Chicago, Columbia, UppsaU, and Oslo. During this time he was offered, and ac-

. cepted, a position as a Research Associate with, the Cowles Commission for Research in Economics at the University of Chicago, Debreu recalls this period fondly; almost all his time was devoted to research and he began his work on Pareto optima, on the existence of a general economic equilibrium, and on utility theory,

When the Cowles Commission moved from Chicago to Yale in 1955, Debreu moved with them. But in 1962 he accepted a position in Economics at the University of California at Berkeley, where he has stayed, except

-fot frequent leaves, ever since, (Since 1975 he has a Photo by Saxon Donnelly. ^ a p p o i r i t m f l n t m ^ ^ Economics, and Mathematics

at Berkeley.) tjcrarcl D e b r e u Among the many honors Debreu has received before

winning the 1933 Nobel Prize/ he was a Guggenheim Fellow (1968-69), Vice President and President of the

Gerard Debreu was born in 1921 in Calais, France Econometric Society (1969^71), Fellow of the American where he went to school until the age of IS, In 1939, Academy of Arts and Sciences (1970), Member of the at the start of the war, he studied an improvised Math- National Academy of Sciences (1977) and a Distin-cmatiques Speciales Preparatories curriculum in Am- guished Fellow of the American Economic Association bett (Fuy-de-D&me) and went to the Grenoble Lycee (1982).

440

ttohal ssaiysss la siunijtslt theory. The |oal here i> to iUuitrate 'global analysis ia economic*' by putting the main results or rlinicil equilibrium theory into a global calculus context The advantages of this approach are fourfold:

(1) The proofs of tiistnica of equilibrium are simpler. Kakutaai's Axed point theorem is sot used, the main tool being tbe calculus of several variables. (2) Comparative statics is integrated into the model in a natural way, the first derivatives playing a fundamental role. (3) The calculus approach is doscr to the older traditions of the subject (4) In so far as possible the proofs of equilibrium are constructive. Those proofs may be tm permeated by a speedy algorithm, which is Newton's method modified to give global convergence. On the other hand, the existence proofs arc sufficiently powerful to yield the generality of the Arrow-Oebreu theory.

Only two references arc given at the end of this entry, each containing as extensive bibliography with historical notes. The two references themselves grve detailed, expanded accounts of the subject of global analysis ia economic theory.

Let us proceed to an account of due model. The basic equation of equilibrium theory is 'supply equals demand', or in symbols, S(j)-D(f). Since we are in situation of several markets, then are several variables is this equation. Equi­librium prices are obtained by setting the excess demand r - D -S equal to xero and sotviag. Consider this function x on a more abstract level.

Suppose that given an economy of / markets, or of / commodities with corresponding prices written as » , , . . . , p „ the excess demand for the /tb commodity is a real valued function z,~z,(p , / , ) pt>0 and we form the vector r - (r, x>). Thus the exeats demand can be interpreted as a map, which we take to be sumciently c&fcrenuable, from If, to ft, where sV is Cartesian /-space and H - f s s f t l p , > 0 . An economic equilibrium is a set of prices a - ( / , a,) for which excess demand is aero, that is, i(p) « 0.

Economic theory imposes some conditions on the function z which go as follows.

First and foremost is Walras's Law, which is expressed simply by a z ( a ) - 0 (inner product). Written out, this is

i

Iff'to A)-° I - I

and states that the value of tbe excess demand is r*ro. This is a budget constraint which asserts that the excess demand is consistent with tbe total assets of tbe economy. It can be proved from a reasonable microeconomic foundation, as can be seen below.

Second is the homogeneity condition i(Xp)-i(p) for all A > 0. Changing ail prices by the same factor does not affect excess demand. This condition reflects the fact that the econ­omy is self-contained; prices are not based on anything lying outside the model.

The final condition is the boundary condition that r,(p) > 0 Up, - 0. This may be interpreted ac if the ith good is free, then there will be a non-negative excess demand for it

Tbe following result and its gctarebzation* and ramifications lie at tbe heart of economic theory. Exiiunct Tktorem. Suppose thai an excess demand x satisfies Walras's Law, homogeneity, aad the boundary condition. Then there is a price equilibrium.

We will give the proof under the additional mild non-degeneracy condition that the derivative of z is non-singular

global analysis icisiatlr theory

somewhere os tbe boundary of ft*. This proof it bated on Sard's theorem aad tbe inverse function theorem, two basic theorems of global analysis.

Consider a difftrenUaWe map / from a set t/ contained ia R* to R". A vector » • R" is said to be a rtfitr value if at each point xtU w t t a / ( * ) - v . the derivative flffx): R»-»R" is Surjective. A subset of R- is otfuU rmatxrt if its oompssment has measure zero. SanTz Thnrtm. If a map / : C->R* is of sufficient Differentiability class (C\ r >k-n. 0) then the set of regular values of / has full measure.

A subset Kc/R»iscaJleda*-ouiienskmalsiifwts^>tfsffor each point, there it a neighbourhood C/ in K and a change of wordinttat of R* which throws U into a coordinate tubapacc of duucuaion k.

Inctrzt Function Thtortm. If v s ft* is a regular value of a smooth map f: £/ — R\ f/cR», then either / - ' ( v ) it empty or it it t submanifold of dimension k -n.

Let us sketch out the proof of the Existence Theorem. Define the space of normalized prices by

A,-j*«R'./2>-tj A space auxiliary to the commodity space is defined by

A , - | i « R * ^ I r , - o |

From tbe excess demand map z: R, -0-»R', define an associated map «V A , - a , by # ( / ) - Z ( a ) - I x ' ( a ) » . Note that «M/>) is weS-dcfined fi-e. »»(/) t A,) and also smooth. Note moreover that if +(p) - 0, then z(p) - 0. end that a it a price equilibrium. This follows from Walras's Law as follows. If a>fj»)-0. then z(p)-Zz'(p)p and so ; : ( f ) - I r ' ( » ) » - » - 0 . Therefore I r'0>) - 0, since pp r»0. By the previous equation z(p) must be xero.

The boundary coodition on z implies that + satisfies a ssmilar boundary condition. That is, i f » , - 0 then ♦,(»)-z ,(p)>0.

It is now sufficient to show that +(p) -0 for some pcA,. The argument for this proceeds by defining yet another map * b y

l*(*)l where £ -$"'(()) and S1'1 is the set of unit vectors in A,.

By definition the set £ is the set of price equilibria, which is to be shown not empty.

Let p, be a price vector on the boundary of A, where the derivative D+(p,) is non-degenerate (our special hypothesis implies the existence of this a,). One applies Sard's theorem to obtain a regular value v of 4 in S1'1 near 4(Pt), where i~'(y) it non-empty.

From the inverse function theorem it follows that i ~'(y) it t smooth curve in A, (a 1-dimensional submanifold). From the boundary condition and a short argument which we omit, it follows that this curve cannot leave A,.

Since the curve i~'(y) is a closed set m A , - £ and has no end points (the inverse function theorem implies that) it mutt tend to £. In particular, £ is not empty and therefore the existence theorem is proved.

The above proof is 'geometrically" constructive in that a curve y -4~'(y) it constructed which leads to a price equi­librium. This picture can be made analytic by showing that y is a solution of the ordinary differential equation 'Global Newton', o>/d/ - U>t(p)-'Wj>\ where I is +1 or - 1 deter­mined by tbe sign of the determinant of D+(p). At a con-

533

441

sequence the Eulef method of approximating the solution of an ordinal? differential equation can be used to obtain a discrete algorithm for locating a price equilibrium. By an appropriate choice of steps, ±1 . this discrete algorithm near that equi­librium is Newton's Method; thus the appellation Global Newton' for the differential equation.

One would like to understand the process of convergence to equilibrium in terms of decentralized mechanics of price adjust­ment. Unfortunately the situation in this respect is unclear.

Next we give a brief picture of bow gjohal analysis relates to a pure exchange economy. This will allow a microeconomic derivation of the excess demand function discussed above, so that the existence theorem just proved wiD impiy an existence theorem for a price equilibrium of a pure exchange economy. Continuing in this framework one can prove Debreu's theorem on generic finiteness of price equilibria, by putting the structure of a diflerentiable manifold on the big set of price equilibria. The equilibrium manifold is a natural setting for comparative statics.

A trader's preferences will be supposed to be represented by a smooth utility function u: /> —R, where p ~xsU,x,>0 a commodity space. The indifference surfaces are those H "' (c) c P. We make strong versions of classical hypotheses on this function.

Monotonictty. The gradient, grad u(x), has positive coordi­nates.

Convexity. The second derivative D'u(x) is negative diftaite on the tangent space at x of the corresponding indifference surface.

Boundary condition The indifference surfaces are closed sets in If (not just P).

From the utility function, one defines for the individual trader a demand function. / : R", x R. -»P of prices p e Ut and wealth w > 0. For this, consider the budget set *A» - x t P\p x — w). Then f(p, w) a the maximum of /

One can prove:

Proposition. The demand function / satisfies (a) gradu(/(^. w))- Xp, for some X > 0

fl>) p/(j>.w)-» (c) f(Xp,Xw)- f(p,w) s n y x > 0 (d) / is smooth.

A pure exchange economy will be a set of m traders, each with preferences as discussed above, associated to utility func­tions M,, i — 1,. . . m defined on the same commodity space P. Abo associated to the lUi trader is an endowment vector e,eP. At prices p. this trader's wealth is the value of his endowment pt,-wr A state is an allocation (x,. ., x„), x,eP tnd a price system peR,

Feasibility is the condition:

A kind of satisfaction condition of a state is (S) For each i, x, maximizes u, on the budget set

B-yt P\py -p '.) An economic equilibrium of a pure exchange economy (e *., u, u„) is a state [(*, * , ) . p] satisfying (F) and (S).

7Vof»m. There exists a price equilibrium of every pure exchange economy.

5J4

The proof goes by applying the previous existence theorem above. Define the excess demand Z - D - S as follows:

SU>)-l.e,. D(p) -£/,<j>. pe,X where / is the above denned demand of the i th trader. One then shows Walras's Law: pZp)-p DM-p SW-ZfMp.p e.)-pY.e,-0

using (b) of the proposition above. Use (c) of the proposition to confirm the homogeneity of r.

The use of the boundary condition is more technical. But under the rather strong hypotheses, this gives a fairly complete existence proof for a price eqiiilibium of a pure exchange economy.

This existence proof extends to prove the Arrow-Debreu theorem in the generality of the latter's Theory of Value.

STEVJ SHALE

See also CATASTaorw raoar; ODKOM. squiuuruu; KATtaounCAi. SCONOMCS: aacuuut Kxyxomes.

srauoOBArHY Mat-CoteO. A. 19*5 TV Theory of General Economic Eeuiltbrnm. e

Dtfferenliable Approve*. Cambridge: Cambnda* Uniwrocy Presi. Smalt, S. I9tl. Global analysis and eeooonna. In Hendoook of

Mathematical Economics, Kefenr /, cd. K. I Arrow uta M. D. Intriligstor. Amsterdam: NonaHolland.

SCIENTISTS AND THE ARMS RACE

Steve Smale

I would like to express my appreciation to the organizers

of this meeting. In particular, allow me to mention Professors

Brieskorn, Starlinger, Durr, and Kreck. This gathering is impressive

and important. On the one hand, it will have a direct political effect

on decelerating the arms race. On the other hand, discussions on the

program can help us better understand this threat of war; moreover,

they can aid in generating a framework for dealing with this threat.

Eighteen years ago I almost left my profession as mathema­

tician to join ranks with the movement against the Vietnam War.

In the summer of 1965 I organized militant demonstrations against

trains carrying U.S. troops to departure points for Vietnam. I wrote

leaflets. I argued with police chiefs about routes of marches.

Together with Jerry Rubin and other friends, I organized a "teach-in"

at Berkeley which lasted 36 hours and brought tens of thousands of

people. We wrote:

"Vietnam...lays bare the reality: a white nation bombing

a colored people...a rich highly developed nation laying waste

the resources of an underdeveloped land. Vietnam lays bare

the violence beneath the smiles and informality of American

life. It gives us every reason to look at ourselves and say:

'We are accomplices in murder.'"

At that time WP established an anti-war politics of confronta­

tion. Our goal was reached many years later when dominant politicians

Talk given at Congress of Scientists for Peace, Mainz, W. Germany,

July 2, 1983

445

said that the Vietnam War had to be ended because the war was tearing

the country apart.

The problem of the arms race today is not the same as the

problem of the Vietnam War, and the nature of the opposition has

changed substantially.

As far as scientists go, it is the well-established ones who

have led the opposition to President Reagan's military policies.

This contrasts with their policies at the time of the Vietnam War.

Many of these same scientists were then either quiet or else working

on military strategies, and it was the students who were central to

the dissent.

As an illustration, let me recall that subgroup of the Insti-2 tute for Defense Analysis called Jason. In the late 1960's, Jason

consisted of about 40 elite scientists working on Vietnam military

strategy for the U.S. Government. According to the Pentagon Papers,

Jason scientists met with Defense Secretary McNamara in 1966, and

presented him with a well-thought out proposal to put a military

barrier along the "demilitarized zone" in Vietnam. This proposal

gave highly detailed advice as to the kind of mines, sensors, bombs,

to implemen.t the barrier. Here are the names of some Jason scientists of that time:

Richard Garwin, Murray Gell-Mann, Sydney Dreli, Wolfgang Par.ofsJcy,

Steven Weinberg, Freeman Dyson, Marvin Goldberger, Donald G]aser,

Herbert York.

Many of these same men are now putting serious effort into

the politics of disarmament. Consider, for example, Richard Garwin.

446

Not only was he a very active member of Jason, but he worked on the

H-bomb as well. Edward Teller, in 1981, characterized Garwin's

contribution to the first H-bomb as follows:

"In the end, the shot was fired almost precisely

according to Garwin's design, and it worked as expected."

That was 30 years ago. Now, Garwin and Carl Sagan have organized a

"Petition for a Ban on Space Weaponry" (with signers who include

Drell, Panofsky, and York). Moreover, Garwin writes: "...my judgment

now is that this nation has lost its way." He has recently lobbied

against the MX missle.

One of the first to become inactive in Jason, bubble chamber

inventor Don Glaser appeared a year ago at a press conference to announce

the Nuclear Freeze initiative in California. Freeman Dyson recently

gave a talk at Oxford against the arms race. In April, 1983,

I listened to a panel organized by the U.S. National Academy of

Science wherein Garwin, Goldberger, and Panofsky spoke in support of

various aspects of limiting or ending the arms race. In fact, at

its annual 1982 meeting, the National Academy of Science passed

overwhelmingly a resolution asking for a substantial increase in

efforts toward "an equitable and verifiable agreement between the

United States and the Soviet Union to limit strategic nuclear arms

and to reduce significantly the numbers of r.uclear weapons and delivery

systems."

Thus, much of the science establishment has come to a posi­

tion of at least mild opposition to administration armament policy.

The sane is true of organized religion and other sectors of society.

447

448

Ex Presidential Security Advisor McGeorge Bundy, ex Secretary of

Defense Robert McNamaca, Ex C.I.A. director William Colby and other

leading government figures of the Vietnam War years are now speaking

against Reagan's military decisions. In truth, a major factor in the

disarray of the American left is that the left-wing opposition has

been pre-empted by those in the center who have much more power.

Handing out leaflets protesting U.S. intervention in El Salvador

is pretty trivial when CBS sends the same message more dramatically

to millions across the country on color TV.

How is one to explain this phenomenon? Why has so much of

mainstream America come into the opposition? Certainly there are

several reasons. Lessons have been learned from Vietnam. The power

of the government to intimidate critics has been weakened over the

last decades. But I wish to discuss another factor.

The U.S/Soviet arms race today is a threat to the survival of

every individual in the United States (in contrast to the threat posed

by the limited, far-off Vietnam War). When my life and my family's

and friends' lives are at stake, I will try to do something about

it. The same is true for the people I have discussed above. Moreover,

individuals enjoying power can see more easily the effect of their

words; hence they are particularly motivated to speak out when danger

of world war looms. During Vietnaa, much protest came out of a shame

for what our country was doing, and that we citizens were guilty of

complicity. Worry about extension to world war existed, but was

dulled by a thaw in relations between the Soviet Union and the United

States. In 1972, at the same time that Nixon was escalating the

bombing of North Vietnam, he was establishing the most cordial

U.S./Soviet relations since World War II.

In contrast, we now have Reagan, who is a danger to the peace

of the world. This man seems to be obsessed with anti-communism and

is trying to force a U.S. military buildup no matter what the consequence.

He is resistant to developing any kind of accord with the U.S.S.R. Reagan

is unleashing the C.I.A. in an attempt to overthrow the Nicaraguan

regime. He threatens Cuba and will intervene in El Salvador to the

extent that Congress will permit. At the same time that Reagan is

renewing U.S. grain shipments to Russia, he is attempting to coerce other

Western nations into trade embargos against the Soviets. Reagan is

now pressuring NATO countries into accepting new nuclear missiles

able to hit Soviet territory. The installation of these weapons

with their proximity to Moscow is provocative and destabilizing for

the armaments balance of the two great powers.

Why is Reagan doing these things? How can one explain the

rise of American militarism between Nixon's detente and the present?

It is not just Reagan who is responsible, since at the end of his

presidency Carter was sharply increasing the military budget. Without

intending to justify this rise of American militarism, I would

cite as the primary cause, the militaristic actions of the U.S.S.R.

in the previous decade. I have in mind the large increase of Soviet

449

450

armaments (e.g., the S-S 20's), and the invasion of Afghanistan, as

well as less direct events in Poland, Angola, Cambodia and Laos.

These Soviet moves in the presence of a post Vietnam military

de-emphasis in the United States proved that the right-wing anti-

Soviet politicians who had anticipated such events were correct.

It helped pave their way to the power that they hold in Washington

today.

The Soviet and the U.S. leadership is each telling the truth

when it accuses the other of unjustifiable military aggressiveness; and

each is using the aggressiveness of the other to defend its own.

There is an important dynamic in this arms race, a strong

interaction between U.S. and Soviet moves in initiating new weapons

and setting armament budgets. And now both sides are playing tough.

It is sometimes said that there is no logic in the arms race

since both the U.S. and U.S.S.R. would be substantially better off

without it. "The arms race is illogical," I believe this view is

a mistaken one and conceals a fundamental tragic force which creates

a political equilibrium destructive to both participants. In terms

of the mathematical theory of games, this phenomena is described by

the "Prisoner's Dilemma". The argument is subtle, but proceeds in

this manner, oversimplified for clarity.

The United States must decide whether or r.ot to possess the

H-bomb. It reasons that if the Soviets don't obtain it, then in the era

of atomic bombs, the U.S. is better off with it. On the other hand, if

the Soviet Union does obtain the H-bomb, then the U.s. surely needs

it. In either case, the U.S. gains by building the H-bomb. Now the

same considerations appLy to the Soviet decision. Therefore, both

sides develop the K-bo^b if they are raticr.al, ar.d both become worse off.

This scenario is especially convincing in a static world where

there is no sequence of decisions. Beyond this logic there are

other effects such as lobbying from the armaments industry and military

establishments within each country.

The situation is more complex in that there is a dynamic

process in which peaceful or belligerent moves on one side encourage

similar actions on the other side. Of course, decision making within

each country is a complex process. A belligerent action bv one side

gives arguments and power to the militarists on the other side. Moreover,

soze events affect relations over a long time period. For exan-.ple, the

Russians have not forgotten the presence of U.S. troops on their soil

in 1920.

Using a little mathematics, and, taking the passage of time 4 strongly into account, I made an analysis which suggests a resolution

of this Prisoner's Dilemma. A concise translation of the mathematical

language into words is: "Play on the easy side, but don't let oneself

be exploited." Such a strategy can be defended on theoretical grounds

as leading to a peaceful, equilibrium with rational people. Perhaps this

conclusion has implications for big power strategies in the arms race.

The relationship between science and war is old and deep. It

is said that Archimedes; designed weapons which kept the Romans out of

Syracuse for three years. The end of the story is told in mural in

the Berkeley mathematics department named "The Death of Archimedes" -

a scene of a conquering Roman soldier stabbing the scientist.

451

452

Modern American science owes a gr^at dea) to the military.

The system of federal support to science was established during World

War II, especially through the efforts of Vannevar Bush and the Office

of Scientific Research and Development. Moreover, the military has

given crucial support to the computer, from the first postwar ENlAC's

to the most powerful CRAY'S of recent years. In turn, the computer

has profoundly affected much of contemporary science.

In the other direction, the military debt to science is even

more clear cut, as the most casual glance at modern weaponry shows. Now

that these weapons are threatening the survival of civilization, the past

responsibility of scientists for this precarious situation implies a moral

responsibility to deal with it today. We cannot evade this responsibilityl

Bertolt Brecht revised his great play "Galileo" after the U.S.

dropped the atom bomb on Hiroshima. Charles Laughton translated it

into English. I hope you will forgive my presumption in reading

to a German audience a bit of this translation from the German.

"...as a scientist I had an almost unique opportunity...At

that...time, had one man put up a fight, it could

have had wide repercussions. I have come to believe that I

was never in real danger; for some years I was as strong

as the authorities, and I surrendered my knowledge to the

powers that be, to use it, no, not use it, abuse it, as

suits their ends."

And then Galileo adds:

"I have betrayed my profession."

453

1. "A Statement from Leaders of the Vietnam Day Committee" in We Accuse, Diablo Press, 1965, Berkeley California.

2. Science Against the People, the Story of Jason, Sespa, Berkeley, 1972.

3. Anatol Rapoport, Strategy and Conscience, Harper and Row, N.Y., 1964.

4. Steve Smale, The Prisoner's Dilemma and Dynamical Systems Associated to Non-Cooperative Games. Econometrica, Vol. 48 (1980) pp. 1617-1634.

5. Daniel S. Greenberg, The Politics of Pure Science, New American Library, N.Y., 1967.

6. Bertolt Brecht, Galileo, Grove Press, 1966.

454

On The Steps Of Moscow University* Steve Smale

From the front page of the New York Times, Saturday, August 27,1966:

August 15, 1966, was a typical hot day in Athens and I was alone at the airport. My wife and children had just left by car for the northern beaches. My plane to Moscow was to depart shortly.

I had arrived in Paris from Berkeley at the end of May. It was a great pleasure to see Laurent Schwartz again. Though tall and imposing, Schwartz was soft in voice and demeanor. He was a man of many achievements. As a noted collector of butterflies, he had visited jungles throughout the world; Schwartz was an outstanding mathematician; he was also a leader of the French left. With Jean-Paul Sartre, he had been in the forefront of intellectuals opposing the French war in Algeria. He was accused of disloyalty, and his apartment bombed. More recently, Schwartz and I had been in frequent correspondence about in­ternationalizing the protest against the Vietnam War. He had asked me to speak at "Six Hours for Vietnam" which he and other French radicals had organized.

The Salle de la Mutuality—I remembered that great old hall on the left bank for the political rallies I had

* The following article U a portion of a full length autobiography the author is currently writing.

attended there fifteen years earlier. Now there were several thousand exhuberant young people in the au­dience, and I was at the microphone. My French was poor and since my talk could be translated I decided to speak in English. I still wasn't at ease giving non-mathematical talks; even though I had scribbled out my brief talk on scratch paper, I was nervous.

There was a creative tension in that atmosphere that inspired me to communicate my feelings about the United States in Vietnam. As I was interrupted with applause my emotional state barely permitted me to give the closing lines: ". . . As an American, I feel very ashamed of my country now, and I appreciate very much your organizing and attending meetings like this. Thank you." Then 1 was led across the stage to M. Vanh Bo, the North Vietnamese representative in Paris; we embraced and the applause reached its peak.

The next day I eagerly read the news story in "Le Monde"; and Joseph Barry, "datehned Paris", in the Village Voict June 2, 1966, gave an account of the meeting:

Steve Smale, 1966. Photo by Caroline Abraham.

TM MAmaMATICAL M n U J C a N C n VOL «. NO 1 C 19*4 SfrfnjB-VMta, Ntw Yo* 2 1

Moscow Silences a

Critical American By R A Y M O N D H , A N D E R S O N

Sppqiii s T l w N i w YocfcTbnM MOSCOW, Aug. 26— A University of

California mathemat ics professor was taken foe a fast and unscheduled- automo­bile ride through t h e streets of Moscow, questioned and then released today after he tuid criticised both the Soviet Union a n d the United States at an informal news con-fererice--

Speqking on the steps- of the University/ of Moscow, the professor. Dr. Stephen. Smater. i . -^ . I

455

. . . The dirty war was one's own country's and sev­eral thousand came to the Mutualite to condemn it. "Six Hours for Vietnam" it was titled. Familiar names, such as Sartre, Ricoeur, Schwartz, Vidal N'a-quet, were appended to the call. And it was orga­nized by the keepers of the French conscience: French teachers and students.

There was some comfort in the presence of Amer­ican professors—from Princeton, Boston, and Berkeley. Indeed Professor Smale of the University of California got almost as big a hand as Monsieur Vanh Bo of North Vietnam. Such is the sentimen­tality of the left. Even the chant that followed Smale's speech—"U.S. go home! U.S. go home!"— was obviously a left-handed tribute. . .

After that I very quickly returned to the world of mathematics. I remember spending the next evening at dinner with Ren* and Suzanne Thorn. Ren* Thorn was a deeply original mathematician who was to be­come known to the public for his "catastrophe theory." A few days later Rene and I drove together to Geneva to a mathematics conference, and I to a rendezvous with my wife, Clara, and young children, Laura and Nat.

Both Schwartz and Thorn had won the Fieids medal. Two—sometimes four—of these medals are awarded at each International Congress of Mathematics (held every four years). It is the most esteemed prize in mathematics, and I had been greatly disappointed in not getting the Fields Medal at the previous Interna­tional Congress in Stockholm in 1962. That disappoint­ment led me to deemphasize the importance of the Fields Medal; I was (and I still am) conscious of the considerations of mathematical politics in its choice. Thus I showed little concern about getting the medal at the Congress that summer in Moscow. Neverthe­less, when Rene Thorn told me during the ride to Ge­neva that I was to be a Fields medalist at the Moscow Congress, it was a great thrill. Thorn had been on the awarding committee and was giving me informal ad­vance notice. Georges de Rham notified me officially in Geneva a few days later.

The time in Geneva was beautiful: it included in­tense mathematical activity, seeing old friends, and Alpine hikes with my family.

Medal or no medal, I had been planning to go to the Moscow conference where I was to give an hour-long address. Clara and I had ordered a Volkswagen camper to be delivered in Europe, and we drove it from Geneva to Greece via Yugoslavia, camping with our kids on the route. The plan was for Clara and the kids to stay in Greece while 1 was in Moscow. They would meet me at the airport on my return. We would go by auto-ferry to Istanbul, where we had reservations at the Hilton Hotel (a change from camping!). The return was to feature an overnight

2 2 THE MATHMAIKAI. INTIUJGINCE* VOL ». NO 2. MM

roadside stop in the region of Dracula in Transylvania. It was our first visit to Greece and we enjoyed the

beaches, visited monasteries at Meteora and traveled to the isle of Mykonos. I was especially moved by the home of the orade at Delphi, 1 knew it well from sto­ries my father had read to me when 1 was a child.

Now at the Athens airport, 1 reviewed those beau­tiful memories; tomorrow 1 would be receiving the Fields medal in front of thousands of mathematicians gathered in Moscow from all over the world.

At first I was only slightly annoyed when the cus­toms official stopped me, objecting to something about my passport. I knew my passport was ok; what was this little hassle about? Slowly I began to understand. When we had come across the Greek border, customs had marked in my passport that we were bringing in a car. (The government was concerned that we would sell the car without paying taxes.) Now the Greek of­ficials were not letting me leave the country witiSout that car. The customs officials were adamant. Unfor­tunately, the car was out of reach for the next ten days. The plane left the ground—it was the only one to Moscow until the next day. My heart sank. There was no possibility of getting to Moscow in time to get my medal.

The American embassy was closed, but after some frantic efforts, I managed to find a helpful consular official, Mr. Paul Sadler. He became convinced of my story, waived the protocol of the embassy, and wrote to the Director of Customs:

Dear Sir As explained in our telephone conversation with

Mr. Psilopoulos. . . The Embassy is aware of the customs regulations

which require an automobile imported by a tourist to be sealed in bond if the tourist leaves the country without exporting the automobile. Unfortunately, Mr. Smale's wife has departed Athens for Thessa-loniki with the car and Mr. Smale is unable to deliver the car to the customs.

Since it is of such urgency that Mr. Smale be in Moscow no later than the afternoon of August 16, the Embassy would greatly appreciate your approval to permit Mr. Smale to leave Greece while his car remains in the Country. The Embassy will guarantee to the Director of Customs any customs duties which may be due on the automobile in the event Mr. Smale does not return to Greece and export the car by the end of August, 1966. . .

I was able to take the plane the next day. At a stop in Budapest, Paul Erdos, a Hungarian mathematician I knew, boarded the plane. Erdos gave me the startling news that the House Unamerican Activities Com­mittee had issued a subpoena for me to testify in Congressional hearings. I guessed immediately it was

456

because of my activity on the Vietnam Day Committee According to press report! the committee haa sub-in Berkeley; I had been co-chairman with ferry Rubin. poenaed several persona. No Berkeley student [my We had organized Vietnam Day and made an attempt emphasis] was Included . . . to stop troop trains. Moat of all, there were the days of protest—the "International Days of Protest." Not only was I under fire from HUAC, but my Uni-

I arrived late in Moscow and rushed from the airport versify, it seemed, wouldn't even acknowledge me. to the Kremlin where I was to receive the Fields Medal On the other hand, my friends and the many other at the opening ceremonies of the International Con- mathematicians came to my defense. Serge, who was grass. Without a registration badge the guards at the in Berkeley at the time, and my department chairman, gate refused me admission into the palace. Final- Leon Henldn, acted quickly to dear my name. Sup-ly, through the efforts of a Soviet mathematician portive petitions were circulated in Berkeley and who knew me, I obtained entrance and found a rear Moscow. An article in the San Fnmruco Qtronicle (Aug. seat. Rene Thorn waa speaking about me and my 6) which followed the Examiner was terrific. (See box work: page 24)

Next, I received a letter from Beverly Axebod, which . . . si les oeuvres de Smale ne poaaedent peut-etre further lifted my spirit. (See box page 25) I took special pas la perfection formeDe du travail definitif, c'est pleasure at her addressing me as "Provocateur Ex-queSmaJeestunpkxmierquipreridsesrisquesavec tnwrdmaire"; and that preceded my most provocative un courage tranquule;. . . act by a week!

The HUAC hearings in Washington were the most After Thorn's talk I found a letter from my friend, tumultuous ever. Jerry Rubin appeared in costume as

Serge Lang, waiting for me. He wrote diat the Sim an American Revolutionary soldier; members of the Francisco Extmmer (August 5) had written a slanderous Maoist Progressive Labor party proudly proclaimed article about me and added . . . "I hope you sue them themselves communists. In the New York Times, Au-for everything they are worth . . . I promise you 2,000 gust 18, (1966) Beverly Axelrod was quoted aa saying dollars as help in a court fight. . ." The article started that she felt in physical danger staying after lawyer as follows: Arthur Kinoy was dragged out of the hearings by court

police. . . . . . . Another Times article ttat dsy was captioned:

UC Prof DodffCS: AWJiBDtE^SW6C€m«MXSHOU^UNn: \ P CCTASEMAtKEMftTiaAWUNABEETCI-

OTifcJ"peita,,. s k i p s TESIOJ^ AT HEARINGS - TLJ.S. for M O S C O W A e ^ , , j w o u l d ^v* welcomed the chance to

testify in Washington. Now, politics was playing a sec-Stevhen Smale ondary role to mathematics, and so it was just as well Suvoorter of to h"ve n u ,* e d ** »uhpoena. I had the best of both y D C - F S M worlds, the "prestige" of having been called, but not

the problem of going. m, -i.n-T¥iii.i were quite exciting. I renewed friendships with the

Dt Stephen. Smale-, UraVerrity o* Qli- Russians, Anosov, Arnold and Sinai, whom I had met forma-mxifesso* and backer of the Vietnam . w . * - * . .•- # # . . u •_ i^^cSrum<t«andoIdF[«Sf«edfM0ve- « Moscow m 1961. The four of us were working in merit, is either on his- way orLs In Moscow, the fast developing branch of mathematics called dy-Hw-EMnraneriearned; today. nanucal systems and had much to discus* in a short

In leaving the country, h e hasdodged a ., subpoena* directing hn» te» appear before- time. the House Committee on Un-American Ac- I became caught up in the social life of a group of

non-establishment Muscovites, visiting their homes One of a number of subpoenaed Berkeley . ^_ . , it_ . _ ^ .—_ » ,

anfrwattcdvisr. Dr. SmaWoois ieaee-or «nd attending their parties. These Russians were absence from UC and [eased, h is home SCOmful o f CXHTimUTUSm, R*ICaStiC a b o u t the ir gOVCm-

11 ment, and, in the privacy of their own company, free , J and uninhibited. This was quite a change from the

serious, cautious, apolitical manner of Soviet mathe­maticians to which I had become accustomed. Conaid-

And it went on to quote a University of California ering tfiat dissent was being punished at that time spokesman: by prison (recall Daniel and Sinavsky), it was surpris-

THI MATtaaAIEAL MnUJCSNCtt VOL I N O . i l M 23

457

O S«n FranciKo Chronicle. 1966. Reprinted by permtMon.

ing to find how open they were in my presence. I loved the Russians in that crowd. Nevertheless,

we argued heatedly about Vietnam. Some of those dis­sidents were so anti-communist that they wished for an American victory in Vietnam. Chinese communism was akin to Stalinism and they hoped that the Amer­icans could stop both Soviet and Chinese expansion.

There was political agitation among the 5,000 math­ematicians in Moscow on two fronts. One aim was to obtain condemnation of the United States intervention in Vietnam, and 1 was quite active in that movement. The other aim was to pressure the International Con­gress to condemn HUAC's attempt to subpoena me. A Times story, datelined Moscow, Aug. 21, led off with:

The Nea York Times story caught the attention of the State Department which became concerned enough to initiate an investigation. The National Academy of Sci­ences, Office of the Foreign Secretary, cooperated by contacting the official delegates at the Congress in Moscow:

. . . we would appreciate any comments you may be able to make to us about the political subjects raised in the context of the Congress which we in turn could pass on to the Department of State . . .

Indeed, the writing and circulation of a petition was in progress:

. . . We the undersigned mathematicians from all parts of the world express our support for the Viet­namese people and their right to self-determination, and our solidarity with those American colleagues

2 4 T M MATHBKATICA1. tNTBiJCINCB VOL. (, NO. 2. HM

457

UC Prof's Subpoena May Boomerang on Probers

fly Jack Smith Subpoenas were served on a ematics at UC from Columbia Uni-number of those involved Thurs- versify, Dr. Smale is driving with

The attempt to subpoena a bri]- day, but Professor Smale, a one- his wife and two children from Ge-liant University of California math- time Vietnam Day Committee lead- neva to Moscow for the fnterna-ematician for anti-war activities er, was out of the country. tional Congress on Mathematics,. may boomerang on the House Un- scheduled August 16 through 26. American Activities Committee. REPORT During the congress, Dr. Smale The Chronicle was told yesterday. will be given the Field Medal, the

"American [sic] is going to be the highest honor in mathematics and laughing stock of the world for Refuting a somewhat exagger- comparable to the Nobel Prize. treating such a way in that way," ated published report that the 36- Smale was one of the organizing his colleague Dr. Serge Lang de- yeflr-old professor had "dodged" members of Berkeley's Vietnam dared. the subpoena, colleagues noted Day Committee,- which began as

The highly respected mathema- that he had applied for a leave of the sponsor of a 36-hour "teach-in" tician, Professor Stephen Smale, absence at least six months ago. He in May, 1%5. Its anti-war activities was named along with seven others . left here in June and spent two continued afterward, and Smale in the Bay Area to appear before the . months at the University of Geneva was one of the leaders of demon-Congressional committee on Au- in Switzerland. strations seeking to halt troop gust 16 because of anti-Vietnam At present, according to Dr, trains last fall and took part in war activities. Lang, a visiting professor of math- campus rallies.

O S«n FranciKo Chronicle, 1966. Reprinted by penntuion.

ing to find how open they were in my presence. The New York Times story caught the attention of the I loved the Russians in that crowd. Nevertheless, State Department which became concerned enough to

we argued heatedly about Vietnam. Some of those dis- initiate an investigation. The National Academy of Sci-sidents were so anti-communist that they wished for ences, Office of the Foreign Secretary, cooperated by an American victory in Vietnam. Chinese communism contacting the official delegates at the Congress in was akin to Stalinism and they hoped that the Amer- Moscow: icans could stop both Soviet and Chinese expansion.

There was political agitation among the 5,000 math- . . . we would appreciate any comments you may emaadans in Moscow on two fronts. One aim was to be able to make to us about the political subjects obtain condemnation of the United States intervention raised in the context of the Congress which we in in Vietnam, and 1 was quite active in that movement. turn could pass on to the Department of State . . . The other aim was to pressure the International Con­gress to condemn HUAC's attempt to subpoena me. Indeed, the writing and circulation of a petition was A Times story, datelined Moscow, Aug. 21, led off in progress: with:

. . . We the undersigned mathematicians from all SQENTISTS URGED TO CONDEMN U.S. o f ^ woM e x ^ ^ ^ V i e t .

name* people and their right to JelWetermination, TARGETS AT WORLD PARLEY J ^ o u r ' S k l i l l t y ^ ^ j ^ , ^ colleague,

2 4 T » MATHSMATIGU. INrBiJCINCB VOL. t. NO Z. KM

458

who oppose the war which is dishonoring their country.

This activity brought me together with Laurent Schwartz and Chandler Davis once more. Davis had served a prison term stemming from his HUAC ap­pearance back in Michigan. Unable to find a regular job in the United States, he had become professor of mathematics at the University of Toronto. I had been friends with Chandler as a student at Ann Arbor and had been present at hi* HUAC hearings.

The three of us were invited by four Vietnamese to a banquet on Tuesday, August 23, in appreciation for our opposition to the war. Without any assistance from the Russians, these Hanoi mathematicians put a great amount of energy into preparing a fine Viet­namese feast under difficult conditions; the banquet took place in a dormitory. I gained a convincing pic­ture of the struggle that these particular Vietnamese

were going through, continuing to do their mathe­matics white American bombs fell on their capital dry. And for the first time I could see directly the meaning of the American anti-war movement for the Viet­namese. Even though our language and culture were different, much warmth flowed between us.

At that banquet, I was asked to give an interview to a reporter named Hoang Thinh from Hanoi. I didn't know what to say and struggled with the problem for the next day. I felt a great debt and obligation to the Vietnamese—after all, it was my country that was causing them so much pain. It was my tax money that was supporting the U.S. Air Force, paying for the na­palm and duster bombs. On the other hand, I was a mathematician, with compelling geometrical ideas to be translated into theorems. There was a limit to my ability to survive as a scientist and weather further political storms. I was conscious of the problems that could develop for me from a widely publicized anti-

THIkUTHaMATCALMnUJ(aNCIXVOL.i.NO 2.UM 25

B E V E r t Y A J C E L f c O D A ' T r o t N E Y - A T L A W

U H wmnm x uuHMh " MA*wt n»st August 14, 1966

Dr. Steven Smale Provocat enr_ Ex t r aord ina i r e Mbacowy tl.S.S.Hi,

Dear Steve H

THIS lawsuit i s not ju s t another legal exercise ; i t offers the possibil i tyCl£ r ea l ly dealing a death blow t o HOAC, Mastj i f not a l l , tbe othera who have been, subpoenaed7 w i l t be par t ies to this - ac t ion, and we intend to- ge t "^prestige" names t o be added after— wardsv Y.cmrS: would be- the most important name, for obvious reasons . We- hope- you w i l l agree and authorise. the- us & of your name as p la in t i f f „

Also, as I explained to teon Hertfcin, we would l i k e to be able t o use the names of as many of your colleagues as are w i l t i ng to pa r t i c ipa t e . The theory i s t h a t everyone suffers an injury from the a c t i v i t i e s of the Committee, and pa r t i cu la r ly those who, because of t h e i r profess ion,-feel » kinship with you.

I "nr working on th i s in associat ion with a number of (Tew York lawyers, including Kuntsler & Kinoy,. and nat ional A.C.L.U. , which is footing the b i l l .

Please read the enclosed complaint, and, as soon as1 possible,, at any hour, phone me, or Arthur Kinoy, or_ William Kimseler, at . the- Congressional Hotel, Wash. D.C. phone 20 2 JJL 6-^6611^ Cc^-t i ,S~'

We cin~take telephone authorization for use of names., to- be followed, up by a l e t t e r s imilar to the one enclosed.

Hope your 3tay in Moscow is a great experience- I envy you.

Bes t regards ,

469

Steve Smalt in Moscow, 1967.

U.S. interview given to a Hanoi reporter in Moscow. In particular, I knew that what I said might come out quite differently in the North Vietnamese newspaper, and even more so when translated back into the U.S. press.

This was the background for my rather unusual course of action. On the one hand, I would give the interview; on the other hand, I would ask the Amer­ican reporters in Moscow to be present so that my statements could be reported more directly. I would give a press conference on Friday morning, August 26.

The mechanics for holding such a press conference in Moscow were not simple. I surely wasn't going to become dependent on the Russians by asking for a room. Since Moscow University was the location for the Congress, it was natural to hold the interview on the steps of this university. Out of respect for my hosts, I invited the Soviet press. There was obviously a provocative aspect to what I was doing, so for some protection, I asked a number of friends to be present, including Chandler Davis, Laurent Schwartz, Leon Henkin, and a close friend and colleague, Moe Hirsch. I was already scheduled to leave for Greece early the following morning.

The interview with Hoang Thinh was now set up under my conditions. Then late Thursday I received word that Thinh wouldn't be present! His questions were given me in writing and I was to return my an­swers in writing.! began to draft a statement.

As I wrote down my words attacking the United States from Moscow, I felt that I had to censure the Soviet Union as well. This would increase my jeop­ardy, but, having just received the Fields Medal and being the center of much additional attention because of HUAC, I was as secure as anybody. If / couldn't make a sharp anti-war statement in Moscow and crit­icize the Soviets, who could?

The Congress was coming to a dose and my ren­dezvous with the journalists was close at hand.

Amidst my friends and all the mathematicians I felt a certain loneliness. No one was giving me encourage­ment for what I was doing now. Chandler showed some sympathy, and Thursday night the two of us walked together locked in discussion and debate.

That morning, Friday, August 26, as the hour ar­rived, the main question on my mind was: how would the Soviets react to my press conference? Everything so far seemed to be going smoothly as the participants assembled on those broad steps with the enormous main building of the University behind. A woman from a Soviet news agency asked if she could interview me personally, alone. I said yes, but afterwards. Then I read the following statement:

This meeting was prompted by an invitation to an interview by the North Vietnamese Press. After much thought, I accepted, never having refused an interview before. At the same time, I invited repre­sentatives from Tass and the American Press, as well as a few friends.

I would like to say a few words first. Afterwards I will answer questions.

I believe the American Military Intervention in Vietnam is horrible and becomes more horrible every day. I have great sympathy for the victims of this intervention, the Vietnamese people. However, in Moscow today, one cannot help but remember that it was only 10 years ago that Russian troops were brutally intervening in Hungary and that many courageous Hungarians died fighting for their in­dependence. Never could I see justification for Mili­tary Intervention, 10 years ago in Hungary or now in the much more dangerous and brutal American Intervention in Vietnam.

There is a real danger of a new McCarthyism in America, as evidenced in the actions of the House Un-American Activities Committee. These actions are a serious threat to the right of protest, both in

2 6 Ttfl MATMD4AHCAL K n U I C a N C B VOL. <. NO. 2.1«M

460

Steve Settle speaks at the Chicago National Convention, 1968.

the hearings and in the legislation they are pro­posing. Again saying this in the Soviet Union, I feel I must add that what I have seen here in the dis­content of the intellectuals on the Sinavsky-Daniel trial and their lack of means of expressing this dis­content, shows indeed a sad state of affairs. Even the most basic means of protest arc lacking here. In all countries it is important to defend and expand the freedoms of speech and the press.

After I had read my prepared remarks, a woman, whom I didn't recognize, approached me to say that I was wanted immediately by Karmonov, the organ­izing secretary of the Congress. Keeping cool, I re­plied, "O.K., but wait till I finish here." There were a few more questions from the American correspond­ents, which I answered. Now there were die two re­maining pieces of business, which seemed to be con­verging. The woman from the Congress Committee reappeared to lead me to Karmonov, and the Soviet reporter was content to follow along. Sensing a story, the American press followed, as did Moe Hirsch and some other friends.

Karmanov was very friendly; no mention of politics or the press conference came up. It was an unhurried conversation about my impressions of Moscow, incon­sistent with the urgency with which I had been called. Karmonov was quite solicitious, and gave me a big picture book about the treasures of die Kremlin (in German!). He next wished to accommodate me in seeing the museums and other touristic aspects of Moscow. A car and guide were put at my disposal. At that point I had no interest whatsoever in sight­seeing. However, I had promised the woman from the Soviet press a personal interview and reluctantly acceded to go with her by car, not at all clear who was going, where we were going or why. I felt pressured and a little scared. But all the while I was treated not just politely, but like a dignitary. It was hard to resist.

As several of us left KarmonoVs office, die Amer­ican reporters were waiting, and asked me what was happening. I really didn't know; in any case, I had no chance to answer as I was rushed out into a waiting car. The newsmen were pushed aside by Russians on each side of me. Moe yelled out, "Are you all right, Steve?" I only had time for a hurried "I think so," before the car was driven off at high speed \vith the American press in hot pursuit. I was to read later in the newspapers how most, but not all, of the American reporters were eluded. The car stopped at the head­quarters of the Soviet news agency, Novosti, and I was taken inside. The same kind of red carpet treatment continued. The top administrators showed me around, but there was no interview. My notes were copied, but otherwise my visit to Novosti seemed to have no pur­pose. We were just passing the time of day, and, fi­nally, I insisted that I be returned to the Congress activities.

I felt an enormous release of tension as I rejoined my colleagues at the closing reception and ceremonies of the International meeting. There had been a lot of concern for me, and friends told me: "Stay in crowds." With trepidation I returned alone after midnight to my room at the Hotel Ukraine. The phone was ringing. I. Petrovskii, the President of die Congress, wanted to see me the first thing next morning. That was impos­sible since I was leaving by air at 7:00 A.M. The phone rang again. This time it was the American Embassy. Was I all right? Did I need any assistance? I replied that I probably didn't need any help.

Indeed, I was on that 7:00 A.M. plane. Clara, Nat and Laura had been enjoying the sea and sun during this time and were unaware of the mix-up with cus­toms and the events in Moscow. It certainly was a welcome sight to see them at the Athens airport. Department of Mathematics University ofCalifornia Berkley, California 94720

nauAT«n«*nc*i.»n»ujCD*.Bivc«.».NO. n w 27

461

1 Some Autobiographical Notes* STEVE SMALE

Part I. Origins and Youth THE MAELSTROM, October 1, 1924, Vol. I, Number 1

The Fortnightly Publication of the Maelstrom Society of Albion College

Lawrence Albert Smale,1 Editor

The Maelstrom is the official organ of the Maelstrom Society. Unfortunately the present membership is limited to one; but it is hoped before the next issue of the Maelstrom appears, such a youthful host of poets, philosophers, artists, disciples of free-love, Bohemians, budding anarchists, communists, nihilists, atheists, agnostics, pacifists, citizens-of-the-world, and any others of the anti-Philistine persuasion attending Albion College, if such there be, will have clamored for admittance, as to constitute a group that will pale the heterogeneity of Mendoza's band of outlaws in Barnard Shaw's celebrated play. Before petitions for membership will be considered however, it will be necessary for conscientious members of the following organiza­tions to fully apostatize.

All Greek-letter fraternities K.u Klux Klan All Other 100-Percenters Kiwanis Rotary American Legion Republican Party Democratic Party National League of Women Voters W.C.T.U.

* For the proceedings of the "Smalefest," Berkeley, August 1990. f Lawrence Albert Smale died in March 1991.

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4 S. Smale

The Maelstrom at various times will attempt to expose the follies that these or­ganizations are guilty of. The editor laments that he has not the pen of a Nietzche for invective cogent enough to hurl against their absurd doctrines, mawkish traditions, and stupid customs.

October 3, 1924 Flint Daily Herald page 1 Headline

FLINT STUDENT EDITOR OF ALBION COLLEGE PAPER ATTACKS PRESIDENT AND IS PROMPTLY EXPELLED

Within one half hour after the first sale an Albion college campus this forenoon of the "Maelstrom", a self-confessed free-lance sheet, the lone editor, Lawrence Albert Smale, of Flint, a freshman, had been expelled from the college.

The "Maelstrom's" 12 pages were devoted to attacks on President, religion, frater­nities of the college, college spirit and the usually accepted conventions of life.

October 6,1924 Detroit Free Press page 1 Headline

OUSTED ABELION COLLEGE FREE THOUGHT EDITOR JAILED

Like the fabled Frankenstein, which destroyed its man-creator, the "Maelstrom", exotic publication which shocked, horrified and interested Albion College, arose to­day from its ashes of oblivion to cause the arrest of its editor and sponsor, Lawrence Smale. Smale is charged with selling and distributing obscene literature, in a warrant issued in Marshall.

Pop had a dual life. He worked in a ceramic laboratory at AC Spark Plug, a General Motors plant in Flint, Michigan. It was a modest white-collar job, and he didn't enjoy it. He was also a Marxist. His party, the Proletarian Party, had diverged from the Communist Party in the twenties, considering the latter to be too reformist. Perhaps even more than a Marxist, he was an atheist. He had been expelled from college for publishing his own magazine in which he had asked if God could make a stone so large that he couldn't lift it. Pop also didn't like partiotism. To him, the Boy Scouts of America repre­sented God and Country, the worst aspects of American life.

I remember that one of the few early traumas in my life occurred when I decided to join the Boy Scouts. Pop was gentle and easy-going, but never would allow that to happen. Even though he bought me presents to compen­sate, I cried for days.

My father certainly has been a big influence in my life. I was 20 years old when I first set foot in a church. That was Notre Dame in Paris. I have kept to this day a healthy skepticism about our country's institutions.

On the other hand, I tended to be more concerned with current problems. The Proletarian Party believed in emphasizing theory and analysis; and Pop followed it in considering some of my activities as reforming the capitalist system. That would only postpone the day of socialism.

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1. Some Autobiographical Notes 5

My father, mother, younger sister Judy, and I lived in the country 10 miles from Flint. At first we had neither electricity nor plumbing. Judy and I walked a mile to a one-room schoolhouse which we attended until high school. I still marvel at that little school. One teacher, a woman with a year or two of college, taught nine grades of students; each grade had classes of reading, math, history, and so on. In addition, she was librarian, did the janitorial work, the cooking of school lunches, and everything else. Yet, we did obtain a good education.

I never completely adjusted to high school. Coming from a rural area tended to make me feel socially awkward. Many of my interests were outside of school. I became intensely involved in chess, participating in three national chess tournaments. I also studied organic chemistry and built a laboratory of sorts in the loft of a former chicken house.

In high school, my first political conflict took place. The story of the Scopes "Monkey" trial had made a strong impression on me, and I was upset that our high school biology teacher, Joe Jewett, wouldn't teach evolution. He simply skipped that chapter in the text (at least it was better than my grade school, where such books were screened from the library). I circulated a petition among my fellow students to get evolution into the course. But only one classmate signed!

Some years later, my parents sent me a clipping from the "Flint Journar, Nov. 15, 1959, headed:

MATH SHARK IMPRESSED TEACHERS

It quoted Jewett: ... Joseph L. Jewett, who had Smale in one of his biology classes, described him as a "very attentive boy, very much interested in the subject and always asking questions".

Jewett said it wasn't a 'one-way' proposition for Smale because he 'contributed a lot' to the class.

"It wasn't a case of all learning from him," Jewett said. "He gave out some ideas also."

Jewett described Smale as quiet and courteous. He was respected by his classmates, Jewett said.

A little different perspective is given by the characterization of me in the high school yearbook, on my graduation in 1948:

"I agree with no man's opinions—I have some of my own"

Going to college at Ann Arbor opened up a new world. I found many students who were similar to me, made friends (male friends, mostly; it was still several years more before I could relate to women comfortably).

I took part in campus life, and, for example, helped organize a chess club. More important for our story, I gradually became involved in the left-wing political life at the University of Michigan.

At that time the liberal-left political party was the Progressive Party; it ran

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6 S. Smale

Henry Wallace for president in 1948 and Vincent Hallinan in 1952. There was a campus branch called the Young Progressives, or YP. Even though it wasn't very visible, I think that it is fair to say that the Communist Party was the main force on the left. Its youth group, the Labor Youth League, or LYL, held secret meetings, secret membership, but had certain "open" spokesmen.

I went to a communist lead "Youth Festival" in East Berlin in the summer of 1951 and then returned to organize the "Society for Peaceful Alternatives'' on campus. I was active in the YP, became a campus leader of the LYL, and, for a short time, was even a member of the Communist Party.

There were two related phenomena going on at the same time in this political movement. On the one hand, there was the vocal activism on such issues as civil rights, the Korean War, nuclear weapons, McCarthyism, and the Rosenberg case (reformism, my father would say, although he did con­tribute to the Rosenbergs). The left's position and activities on these issues weren't compromised (compared to the liberals') by McCarthyist pressures. That is why I (and the others) was drawn in; I have no regrets for my partici­pation in these programs.

On the other hand, we were involved in studying and teaching a Soviet version of Marxist studies, and eventually recruiting for the LYL and the CP. This aspect of the movement, especially, had much in common with funda­mentalist religion. Stalin, before his death, was God, and Soviet policy was gospel. We studied the works of Marx, Lenin, and, of course, Stalin and his followers. A main document we read was the History of the Communist Party of the U.S.S.R., written in Moscow. As Balza Baxter, the LYL state chairman put it, as quoted by the Michigan Daily, around the end of 1952:

The LYL is a youth organization in which Communists participate as equals with non-Communists, and LYL uses a guide to the explanation of things what we call the scientific study of Marxism Leninism.

I might say that the League has a fraternal relationship with the Communist Party. We do study the Stalin versions of Marx and Lenin. After all, the Soviet Communist Party has had much more experience with Communism than anyone else.

There was also a system of rewards and discipline. The rewards included admission to membership in the LYL and the CP, or even leadership in those organizations. Social ostracism and excpulsion were the ultimate discipline. Discipline was threatened for such things as personal political independence and fraternizing with Trotskyites.

I remember once on a trip to New York City visiting briefly the headquar­ters of the Socialist Workers Party (followers of Trotsky) with a friend. On my return, I mentioned this visit casually to fellow members of toe LYL. They did not take the matter lightly and eventually brought it to the State Chairman, Balza Baxter. He was a relatively easy-going fellow, who I liked a lot. Baxter smoothed it over. "Just youthful intellectual curiosity,'' he said.

Although there were some strains and tome signs of my independence, I had accepted the basic doctrines of the Communist Party and defended them.

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1. Some Autobiographical Notes 7

Today, when I ask myself how I could have acted so foolishly, it is not difficult to find an answer.

There are strong tendencies for just about everyone to want something basic to believe in without questioning. Most often this takes the form of traditional religion. But even atheists and revolutionaries must rest their beliefs on some foundation. They are not immune to the problem of reason­ing in circles.

But still, to accept the Communist Party? Consider my frame of reference at that time. I was sufficiently skeptical of

the country's institutions to the point that I couldn't accept the negative newspaper reports about the Soviet Union. I so believed in the goal of a Utopian society that brutal means to achieve it could be justified. I was un­sure of myself on social grounds, and the developing social network of leftists around me gave me security. Then, these were the times of McCarthyism, the Rosenberg executions, the Korean War hysteria; the CP was the main group giving unqualified resistance to these forces.

In the spring of 1952, there was an extended campus conflict at the U. of M. over the free speech question, anticipating in a minor way the FSM struggle at Berkeley 12 years later. There were big differences. McCarthyism was strong, HUAC was in its heyday, and old left tactics dominated the protest. It is useful to go into greater depth on this story, as it illuminates the tenor of those times; it also illustrates my early political style.

On March 4, 1952 the front page headline story of the Michigan Daily read:

TWO SPEAKERS HALTED BY "IT

Committee Fears Talks 'Subversive'

Two men associated with allegedly subversive organizations were temporarily banned from speaking on campus yesterday in an apparently unprecedented move by the University Lecture Committee.

The speakers, proposed by two student organizations, were denied permission to appear "until sufficient evidence is produced" to satisfy the committee that the speeches would not be subversive.

Permission was withheld from: 1) Abner Greene... 2) Arthur McPhaul, excecutive secretary of the Michigan Chapter of the Civil Rights Congress, also branded subversive.

Some students weren't accepting the speakers' ban and a mild form of defiance took place. On May 7, 1952, the Michigan Daily ran the headline story:

MCPHAUL ADDRESSES 'PRIVATE MEETING' AT UNION

Blasting the House Un-American Activities Committee as "an arm of the million­aire forces of Wall Street", Arthur McPhaul, banned Monday from speaking on

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8 S. Smale

campus by the University, put in a colorful appearance at a private Union dinner last night under mysterious sponsorship...

Next in the Michigan Daily:

PROBE BEGUN OF MCPHAUL APPEARANCE

A University investigation was launched yesterday into mysterious circumstances surrounding the appearance of banned speaker Arthur McPhaul at a private Union dinner Thursday night.

and then:

CHARGE STUDENTS VIOLATED 'IP RULE IN MCPHAUL TALK

Charges of violation of a University student conduct by-law have been made before the Joint Judiciary Council against all of the students known to have attended the McPhaul dinner March 6 at the Union, it was learned yesterday...

The By-law allegedly violated states: No permission for the use of University prop­erty for meetings or lectures shall be granted to any student organization not recog­nized by University authorities, nor shall such permission be granted to any individ­ual student.

But as we defendents were to say in our full page ad:

We could not possibly have violated the regulation... As it is now written, this regulation is a restriction upon the discretion of University

authorities to grant the use of University property. Since no student is in any position to grant permission for the use of University property, he cannot possibly violate this regulation.

My reaction to some of these events is contained in the letter I wrote to my parents at that time, reproduced here without change:

Dear Mom and Pop, Just a word to let you know more details of the disciplinary aciton against us. There

is some question of expulsion as indicated in personal feelings of some people in the newspapers. I, however, feel that the "officials" date not take such a step, although they would tike to. The Joint Studies Jucic met Sat. morn., Sat. afternoon, Sun. morning and Sun. afternoon (today). So far it has been impossible (no time yet) to get the defense of the accused much publicity. Nevertheless the support of the student body is remarkable. For example, we wanted to put 1/2 page ad in the daily with a long statement of our case... Cost $80. Well Fred and a friend of his and mine went around to the J.C. Students here and collected $80. Maybe more later. In just a few hours we have collected $70. It is really wonderful to find the students standing by us. All the details of the case you can find in the clippings.

As yet I am the only student that refused to tell whether I was at the dinner. I had a most pleasant time today. We went petitioning to put the Progressive Party

on the ballot in Willow Run. 9000 homes worse than tenements without a single tree. A Negro girl from Detroit and I went throuigh the Negro district there. There was a wonderful response. The residents never had a good word for the Democrats or Republicans. 80% signed that were registered. I found much more courtesy than in the corresponding white district

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1. Some Autobiographical Notes 9

Sat. night we had a large party—3 cars from Detroit. Well all for now,

love-Steve

The investigation, as well as the number of news stories and editorials kept espanding. Then, Anally, May 4, headline story of the Michigan Daily:

FIVE STUDENTS ON PROBATION

The smoke-screen lifted over the controversial McPhaul dinner investigation yes­terday as the Joint Judiciary Council and the Sub-committee on Discipline cleared fifteen students of breaking a Regents' by-law, but put five of them on probation for their conduct before the council.

Eight confusing weeks of deliberations involving four University bodies and hun­dreds of pages of testimony reached a climax as students received official notification of their acquital or punishment from University officials yesterday morning.

Five of them were put on probation for failure to give the Judiciary the coopera­tion students should reasonably be expected to give a student disciplinary body*. All will be forced to drop out of extra-curricular activities where regular eligibility is required. This consists of elective offices or other positions where the student repre­sents his group or the University, according to an Office of Student Affairs ruling. The penalty will last until Jan. 31,1953. Those disciplined are: VALERIE M.COWEN... STEVEN SMALE, Grad., 21 years old who will be required to drop out as secretary-treasurer of the Chess Club and treasurer of Society for Peaceful Alternatives. Shaffer, Sharpe, Smale and Luce all graduated with honors and Luce is a Phi Beta Kappa.

We appealed without success. In June, 1933, the department chairman, T. Hildebrandt, called me into his

office. He told me that unless my mathematics grades improved, I wouldn't be allowed to continue in graduate school. This had some effect on me as I already was moving away from the political arena. A couple of years earlier I had failed a physics couse, and then, as a senior, changed my major from physics to mathematics. It was mainly out of momentum that I continued in mathematics into graduate school at the University of Michigan. But now I was 23 years old and beginning to wonder where I was headed.

In the fall of 1953, the combination of concern about my future, Hildebrandt's warning, and a great mathematics teacher, Raoul Bott, con­verted me into a very serious student of mathematics. It turned out that this mathematics phase was to last until the FSM in the fall of 1964.

I was lucky to meet Clara Davis in the fall of 1954, and we were married at the beginning of 1955. This relationship was important for me in many ways. In particular, it was helpful in my moving away from a rigid political ideology and solidifying my work in mathematics.

I received a Teaching Assistantship and started teaching in the mathemat­ics department in the fall of 1954. After five class meetings, Hildebrandt called me in again and told me that I was fired because of my previous leftist activities. He put the responsibility on the University administration. On the

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10 S. Smaie

other hand, Hildebrandt had found a research contract that would continue my pay. I didn't protest publicly since the money was coming anyway, and I was more interested in doing mathematics. However, the next fall, the pay stopped too. Fortunately, Clara obtained a job as a librarian in Dearborn, and her wages supported us during the last year of my graduate work.

It was not surprising that my political past continued to haunt me as I sought a job. For example, I recall a friendly professor (Ray Wilder) telling me that I should discontinue letters of reference from Chairman Hildebrandt who was warning potential employers that I was a leftist.

In the fall of 19S6, we moved to Chicago where I had accepted my first teaching job. It was not in a regular mathematics department, but at the "College" of the University of Chicago. I just remember a couple of political notes from that time. One was that Soviet military intervention in Hungary helped reinforce my evolving alienation from Communism. Also, one morn­ing I left my Chicago apartment to find two FBI men waiting for me. They wished to discuss the political events at Ann Arbor, but they accepted my quick refusal to do so.

Nat was born in 1957, Laura in 1959. The mathematics went well. After two years in Chicago, I held successive positions in the Institute for Ad­vanced Study (Princeton, N.J.), Instituto de Matematica, Pura ed Aplicada (Rio de Janeiro), the University of California (Berkeley), and Columbia Uni­versity (New York).

Clara, Nat, Laura, and I were living in New York in October of 1962 when we heard the first news that the Soviets were putting nuclear missiles in Cuba. I reacted sharply to the growing threat that atomic war could start any day. I became intensely angry at Kennedy, being aware that the United States had already missiles located on the Soviet border in Turkey. When I became convinced that Soviet missiles were en route to Cuba, I became angry at Kruschev as well. It didn't make sense to die in a nuclear war due to the insane militarism of the two countries.

So, Clara and I with Nat and Laura packed a few of our belongings and started driving to Mexico!

There were a few faculty at Columbia who knew what we were doing and they were helpful. I still have a debt of gratitude to Ralph Abraham and Serge Lang who saw that my class kept running smoothly. My father and mother were visiting us in New York at the time and were also sympathetic and cooperative. The long drive to Mexico helped to ease my tension; also, the tension of the superpower confrontation eased too, as we passed into Mexico.

We were in touch by phone with my friends at Columbia, and as the missile crisis was fading, I learned that my departure from Columbia was by no means irrevocable. Thus, I flew back to New York, taking up my class again. Clara drove our car back with the kids, arriving a few days later. Few people were aware of what had happened.

Were we crazy or were we acting quite intelligently? Even today it is hard for me to analyze our decision to leave the country as atomic war seemed

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1. Some Autobiographical Notes 11

imminent. Clara, at the time, put the question on our minds this way: At what time should a Jew have decided to leave Germany?

We moved to Berkeley from New York in the summer of 1964. The events of the Free Speech Movement, or FSM, erupted soon after the start of classes in the fall. The huge scale, audacity, drama and sincerity of that movement captivated me. The New Left" politics were refreshing.

Friends of ours, Mike Shub, Kathy, and David Frank were students at Columbia University who moved out to Berkeley at the same time we did; they drove our car for us when we came by plane. All three, together with Mike's wife Beth Pessen, were arrested in the famous FSM sit-in. I had visited them at Sproul Hall just before their arrest, and afterwards Clara and I helped get them out of jail. Eventually, Mike was to write a thesis with me and he remains one of my closest friends.

I didn't become part of the FSM itself, but supported the movement through activity in faculty groups. I was among that very small minority of faculty which supported the FSM, both goals and tactics, without qualifica­tions and with enthusiasm.

After the big arrest and further struggle, the faculty senate voted to support the demands of the FSM: The FSM victory became clear. The whole fall experience was one of great exhilaration for me.

During that activity I came to know some of the FSM leaders, including Mario Savio and Steve Weissman. I admired both Savio and Weissman. They were brilliant strategists and articulate speakers. Savio's moving speeches on the steps of Sproul Hall were crucial to the FSM success. Weissman played a central role in obtaining faculty support. I could never have guessed that within a year Weissman and I would be on opposite sides for the crucial decision.

Part II. Vietnam Day May 23, 1965 The New York Times

33-HOUR TEACH-IN ATTRACTS 10,000

MANY CAMP OUT FOR NIGHT AT BERKELEY VIETNAM DEBATE

At 12:55 this morning a bleary-eyed, bearded young man whose gray sweatshirt bore the inscription, "Let's Make Love, Not War", stretched out on the grass outside the Student Union and went to sleep in the darkness.

Nearaby, a girl with straight black hair and bare feet wrapped herself in an Army blanket to ward off the night-time chill and plaintively asked her esort, "Don't they ever run out of things to say?"

Orators addressing the 33-hour teach-in demonstration on the University of California's Berkeley campus this weekend never did run out of things to say, or of students to say them to.

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12 S. Smale

The demonstration, billed as a protest against Government policies in Vietnam, attracted a peak crowd of nearly 10,000 to hear such speakers as Senator Ernest Gruening, Democrat of Alaska, and Dr. Benjamin Spock, the specialist in child care.

Even as the speeches droned on past midnight into the chill morning hours, as many as 4,000 students camped out on the field in sleeping bags and blankets to listen. Scores of stray dogs raced yelping in and around the sprawled bodies hunting for scraps of food in the darkness.

... Show-business trappings were in evidence at Berkeley this weekend. A score of folk singers strummed out tunes about civil rights and the draft. A satiric group called The Committee performed a short play about an automated sit-in.

One group of militants collected pints of blood for the Dominican rebels while another collected signatures from students promising that they would refuse to an­swer a draft call.

Tables were set up around the field selling everything from "The Young Socialist" to "The Soul Book"; one leaflet urged students to 'join the revolutionary organization of your choice1, arguing that any revolution would be better than the status quo.

If the Berkeley teach-in had its denigrators on the faculty, it also found its champi­ons. In opening the teach-in, David Krech, professor of psychology, said the demon­stration represented 'the finest hour of this great university'.

Among those who accepted invitations were Dick Gregory, the comedian; Kenneth Rexroth, the poet; Norman Mailer, the novelist; Norman Thomas, and Ruben Brache of the Dominican rebels.

A majority of the audience clearly agreed with the critics of the Administration. Senator Gruening drew a standing ovation when he demanded that the President seek an immediate negotiated settlement in Vietnam on any terms available.

"The white man cannot settle the internal problems of Asia", he said.

Johnson extended the war against Vietnam in February 1965 by bombing the North. I followed the stories from Vietnam with growing apprehension, only too conscious of the danger of atomic war. For more than a decade, life had gone especially well. I felt a great satisfaction with my work, my family, my job. My main insecurity came from the knowledge that all this would likely be destroyed if the U.S. and U.S.S.R. went to war.

This knowledge affected my actions. Three years earlier during the Cuban missile crisis, I abandoned (tempo­

rarily, it turned out) an enviable position on the Columbia University faculty to take my family to Mexico. Now I reacted differently. I became increasingly committed to political action to try to stop Johnson.

Why was I so presumptuous as to think that anything I could do would have a significant effect? There were several reasons. First, I had an activist background. In my student days, for example, I had organized a "Society for Peaceful Alternatives." Secondly, I had just seen and been part of the un­folding of the Free Speech Movement. The power generated by this force was already enormous. Finally, I had the time, security, and prestige that came from a tenured professorship.

My involvement in the Vietnam protest came slowly at first. A teach-in on the Berkeley campus was organized by some faculty to support the first

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1. Some Autobiographical Notes 13

teach-in at Ann Arbor in March of 1965. I said a few words there. Then I represented the local faculty AFT (The American Federation of Teachers) union as chairman of "The Political Affairs Committee" in the organization of an anti-war march in San Francisco in April 1965.

Although the march was called to support an SDS (Students for Demo­cratic Society) march in Washington, D.C., it was pretty much dominated by the old left. Every poster or slogan had to be approved in advance by the organizing committee. The march turned out to be moderately large, but lacked drama and media coverage. The more radical groups had split off to have their own march and it was quite small.

It must have been at one of those organizational meetings that Jerry Rubin had noticed me.

The war kept escalating. So, when Barebara Gullahom and her boyfriend, Jerry Rubin, came by my office toward the end of April 1965 to ask if I would help organize a town meeting on Vietnam, I was ready. That was before Jerry became famous, and I didn't recognize either him or Barbara. But I liked their spirit and approach. I supported them in bringing together one or two dozen anti-war activists of varied backgrounds. Our first meetings were in Jerry's tiny apartment on Telegraph Avenue near the campus. Wc decided early on the form of the protest. It was to be a combination "teach-in," community meeting, and educational protest, to be held at the University of May 21. It would be called Vietnam Day.

Leaflets were distributed to get help and funds. Speakers were invited and some accepted, the organizational meetings became larger, and the major media picked up the story. Support for our effort grew.

I obtained the early endorsement of the faculty AFT. This was important to legitimize our activity and to obtain campus space.

I tried, with some success, to enlist the help of people closest to me. Clara was involved in "Women for Peace" then and we began to work

together on aspects of Vietnam Day. She designed some of the leaflets and I would take them to the printer.

Moe Hirsch and I had been close friends for almost 10 years. Wc worked in the same field of mathematics, attended conferences together, and were now colleagues at Berkeley. Moreover, Moe and I had a lot in common when it came to politics. We had participated in the FSM together. Now he joined the organization of the big teach-in with enthusiasm. Moe was a good speaker and writer, and he spoke for us on Sproul Hall steps and helped draft some of the key documents we would need, both before and after Vietnam Day.

Another mathematics teacher in my department, John Lewis, joined our efforts. Mike Shub was a graduate student in mathematics. Recall that I had known him from Columbia and that he had been arrested in the FSM. Mike worked with us especially on questions of liaison with the campus commu­nity. Among many others, I remember especiany an older Socialist Workers Party Member, Paul Montauk, who worked as a chef in San Francisco.

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14 S.Smale

Paul was especially constructive and his many connections were helpful to us.

And, of course, there was Jerry. I liked him a lot, and we became good friends. Jerry was there all the time with his great enthusiasm, sharp insights about dealing with the media, and, especially, his "do it" philosophy.

We didn't spend much time on analysis and theory. He knew what we wanted and had some good instincts about how to obtain it. Therefore, our mode was one of continually doing things, all kinds of things, which would make Vietnam Day into a bigger and sharper anti-war protest. Jerry and I did, however, talk for many hours about how to deal with various problems that came up.

At some point the de facto organization that was preparing for the teach-in was formalized. It was called the Vietnam Day Committee or VDC, and Jerry and I were chosen co-chairmen.

It was a lot of fun to work in the Vietnam Day Committee. We didn't think in terms of duty. It was more like an exiting creative challenge: How to make something that would be the greatest teach-in, the biggest Vietnam war pro­test. How to make Johnson cringe.

For example, we would sit around in Jerry's apartment making sugges­tions for speakers. It was like a competition to see who could propose the biggest, most provocative names. There was practically no limit to our ambi­tions; Bertrand Russell, Fidel Castro, the ex-President of the Dominican Republic, Norman Mailer, U.S. Senators, Jean-Paul Sartre, and so on. Then we would call them up then and there, or send telegrams. And sometimes it worked and the invitations were accepted.

But there was another important aspect to sending out these invitations. It helped gain the attention of the media, as the following example shows.

At that time, the U.S. invasion of the Dominican Republic was a big news story. It was only natural that the Berkeley Gazette would put at the top of its front page, May 10:

EX-COMINICAN CHIEF BOSCH INVITED HERE

Juan Bosch, deposed president of the Dominican Republic, has been invited by the sponsors of the May 21 University of California 'teach-in' on Viet Nam to give his view of the island crisis.

A cablegram, signed by Stephen Smale, professor of mathematics, and Jerry Rubin, co-chairmen of the Viet Nam Day being planned for 30 continuous hours May 21 and 22, was sent to Bosch who now lives in Puerto Rico.

Then on May 15 the Gazette carried the story:

REBEL LEADER DUE

Juan Bosch, former political leader of the Dominican Republic, has informed the organizers of the Viet Nam teach-in scheduled for May 21 on the University of Cali­fornia campus, that he is very interested in speaking at the event...

472

1. Some Autobiographical Notes IS

The appearance of Rubin Brache has been confirmed. France and the Soviet Union have been attempting to seat Brache as the United Nations representative for the rebel government of the Dominican Republic.

This bold style helped build the big teach-in. Another important example was our invitation to Dean Rusk, the Secre­

tary of State: The San Francisco Chronicle wrote on May 2:

UC FACULTY CHALLENGE TO RUSK Three University of California faculty groups challenged Secretary of. State Dean

Rusk yesterday to debate this Nation's "disastrous policy in Vietnam" at a "teach-in" demonstration on campus May 21.

The challenge was issued in a telegram sent to Rusk by mathematics professors Morris W. Hirsch and Stephen Smale ...

Then a little later in the Daily Cal:

Letters and telegrams continue to pour in from people who are accepting invita­tions to speak at a massive Community Meeting on Vietnam scheduled May 21 and 22.

The State Department has replied to a telegram sent by Professors Stephen Smale and Morris Hirsch of mathematics. In a phone call to Hirsch a spokesman for the State Department said department officials were thinking of sending a high level official such as Averell Harriman or William Bundy.

Finally, on the front page of the Berkeley Gazette:

STATE DIPT. OUT OF TEACH-IN GATHERING HERE

The Viet Nam Day Teach-In' on the University of California campus Friday will be carried on without the two representatives of the State Department, leaders of the 'teach-in' announced today.

The U.S. State Department has withdrawn its offer to send two representatives to the event, one of whom was to have been William Bundy.

This marks the second State Department withdrawal from debate on U.S. policies in Viet Nam in the last four days. McGeorge Bundy withdrew from the national radio debate at the last minute on Saturday, just before the event...

The committee also confirmed the appearance of comedian Dick Gregory at the 'teach-in'...

Of course, we were busy writing press releases, on the phone to the news­papers, and holding press conferences during this time.

There were other audacious steps we took that kept us on the front pages as Vietnam Day approached. An example from the front page of the Daily Cal, May 20, speaks for itself:

REQUEST CLASSES BE STOPPED FOR VIETNAM DAY TOMORROW

The much-heralded Vietnam Day—a solid 30 hours of intense discussion of the war in Vietnam—gets off the ground at noon tomorrow.

473

16 S.Smale

Its organizers are requesting that all classes be cancelled beginning at noon tomor­row so that all students may attend.

Acknowledging that 'this is a very unusual request', a telegram sent last night to the Regents and to Acting Chancellor Martin Meyerson declared, "We have become bold enough to make this request because of the extraordinary circumstances that developed to make Vietnam Day an event of international prominence and impor­tance for the University of California."

Besides support from foreign individuals and peace groups, explained Jerry Rubin of the Vietnam Day Committee, the 'extraordinary circumstances' included a sympa­thy strike scheduled for tomorrow by a national Japanese student union.

The telegram, signed by Rubin and Stephen Smale, professor of mathematics, said 'Professors are already cancelling classes and office hours and many students have already accepted the fact that they will not attend classes during this event'.

The University local of the teaching assistants' union, said the telegram, has asked all TA's not to hold their classes.

"The University encourages maximum discussion of all pressing social issues,*' re­plied Chancellor's Special Assistant Neil Smelser when informed of the telegram, "but such discussion should not interfere with regular activities."

"Class schedules and office hours will proceed as normal..." The list of speakers, performers, debaters, comedians and singers is staggering—

and so is the marathon schedule.

It was crucial for the success of our venture to have a good campus loca­tion. Luckily, the specter of the FSM was still haunting the university admin­istration. A year earlier, a Vietnam Day would have been impossible. Now we were not willing to compromise significantly on this issue and the Chan­cellor's office was accommodating. The story in the Berkeley Gazette May 14 describes the cooperation well:

UC CURBS RULES FOR TEACH-IN

University of California rules on campus political activity are being relaxed for the May 21-22 Viet Nam Day teach-in, according to sponsors of the event and Prof. Neil Smelser of the Chancellor's office.

The University will make available the lower Student Union Plaza and the adja­cent baseball field and will construct a passageway to the field to accommodate the crowds expected at the event. The speakers* platform also will be raised to make it more visible from all sides.

The 72-hour notification rule applying to off-campus speakers will be reduced to notification on the day of the event...

Co-chairmen for the event, Prof. Stephen Smale and Jerry Rubin, stated "The University has been very cooperative in making available the areas needed for the teach-in."

Vietnam Day was obtaining support now from a wide spectrum. It was almost unprecedented to see so many different liberal and radical groups who had so often fought each other come together to help organize, endorse, and appear on the program together.

We also had some international help, for example, from the eminent mathe-

474

!. Some Autobiographical Notes 17

maticians Laurent Schwartz, Paris, S. Iyanaga, Tokyo, and Bertrand Russell, England.

We also had opposition. As the teach-in drew near, two defenders of the Vietnam War, whom we

had invited, made a public attack. They were Robert Scalopino and Eugene Burdick, both professors at the University in Berkeley.

The San Francisco Chronicle, May 20, wrote:

BURDICK BLAST AT TEACH-IN 'CIRCUS'

This wasn't so bad; we had worked to create a carnival aspect to Vietnam Day. We wanted it to be fun to go to, as well as a big protest. The Berkeley Gazette on the eve of the event (May 20) carried a big red front page headline:

UC'S SCALAPINO RIPS TEACH-IN MEETING ON CAMPUS TOMORROW

A further element of tension as the teach-in hour arrived was indicated in the San Francisco Examiner, May 21:

Berkeley Police Capt. William N. Beall Jr., head of the patrol division, said he will have all his men on standby alert for possible duty.

This will be the first time such an alert has been ordered since the Sproul Hall sit-ins, Dec. 2-3.

Vietnam Day, May 21, 1965, arrived and we waited anxiously; Jerry, me, Barbara, Moe, and the others who had contributed so much effort. The success of the big teach-in became an end in itself and would help justify our own existence. This success hinged on the number of those who would come. And whether they would enjoy it enough to stay. It was supposed to last 33 hours. Of course, real success couldn't be measured just by the number in the audiences, but that number was the most tangible sign.

I had been quoted in the San Francisco Examiner that morning (May 21, p. 18):

Professor Stephen Smale, one of the Vietnam Day Committee members, said hope­fully yesterday he expects 'about 10,000 people to attend at the peak hours' of the thing.

Well, a few hours later, we saw it happen. The first report confirmed it. The Berkeley Gazette afternoon, May 21 in a red front-page headline:

10,000 ASSEMBLE FOR INAUGURAL SESSION OF TEACH-IN* ON CAMPUS

The Vietnam "teach-in" started on the University of California campus at noon today as some 10,000 filled the lower Student Union plaza.

The carnival air prevailed and the scene was gaudy with banners. Any number of food concessions were open. Earlier charges that the program had been turned into an outright protest of U.S. policy instead of an academic discussion appeared to have no effect on the spirit of the crowd.

475

18 S. Smale

We also had succeeded in making Vietnam Day an enjoyable event. One of the stories on Vietnam Day in the May 22 San Francisco Examiner headlined:

VIETNAM DAY AT V.C.I ALL FUN OF A CARNIVAL

Indeed, the crowds had filled up the available space and they stayed. The newspapers gave widely varying estimates of peak crowds, but 10,000-12,000 seemed to be typical numbers. The Oakland Tribune was the newspaper most biased against us. There were three different headings about Vietnam Day in their May 23rd paper

WOMAN HIT IN EYE NEAR TEACH-IN (front page)

TEACH-IN LIMPS TO FINALE (front page of city news section)

and

VIET POLICY PROTESTS FAR SHORT OF GOALS (same)

Yet, their news story stated:

... Finally, about 1 p.m. an estimated 9,000 persons were mustered to hear and cheer [Norman] Thomas' blustery harangue against United States involvement in Vietnam

The peak crowds were one thing, and the total attendance something else, since it was to last a day and a half. As the San Francisco Chronicle wrote May 22:

... There was a constant ebb and flow of spectators, but their total never waned below the 5,000 mark despite gusty winds and mostly sun-less sky...

One could only guess at the total number of participants, and Jerry and I did that in a subsequent VDC letter. We said 50,000 came.

During the first evening of the event, there was a fund raising dinner where Benjamin Spock, Senator Gruening, Norman Thomas, I.F. Stone, and others were present. This generated a substantial amount of money. Other funds were solicited directly from the crowds. I believe, altogether, we obtained quite enough to pay for all the expenses, including travel expenses for most of the speakers. Clara was at a VDC member apartment to help count the cash and recalls how it was piled high on the floor. We had a system of monitors which, besides collecting money, used a system of walkie-talkies for meeting troubles that might have arisen. However, Vietnam Day turned out to be rather peaceful.

The media coverage was heavy. Besides the national press, Bay Area news­papers carried several main front-page stories, and there was some rather sympathetic reporting. A local radio station, K.PFA, ran most of the program with a live broadcast

Vietnam Day lasted the full programmed 33 hours and actually went be­yond that for two or three hours more. Activity never stopped, even during

476

1. Some Autobiographical Notes 19

the early morning hours, although the crowds dwindled. I myself did go home to get a couple of hours sleep.

The main message of the speakers was "Get out of Vietnam" and this was greeted enthusiastically.

The May 23rd San Francisco Examiner wrote on its front page:

Cal's teach-in a-go-go hit full stride yesterday with a full roster of a-go-go-out-of-Vietnam advocates...

Sitting on the grass, munching hot dogs and swizzling soft drinks, they listened raptly and cheered lustily such speakers as Socialist Norman Thomas, Assemblyman John Burton, comedian Dick Gregory, novelist Norman Mailer and dozens of others.

The marathon protest continued until 11:45 p.m. last night, or 3 hours and IS minutes past its scheduled end, caused in part by the reappearance of several of the guest speakers, Gregory, for instance, made three separate addresses to the students.

I still remember vividly the powerful oration of Isaac Deutscher. After his socialist-oriented attack on the United States in the chilly morning hour of 1:30 A.M., 10,000 or so gave him a prolonged standing ovation. He is reported to have said afterward: "This is the most exciting speaking engagement I have had since I spoke to the Polish workers thirty years ago. It is extra­ordinary—simply extraordinary" (Teach-ins: USA, eds. L. Menashe and R. Radosh, Praeger, New York, 1967, p. 33).

And I also remember well Norman Mailer's description of Lyndon Johnson. Ralph Gleason in a review of Vietnam Day in the San Francisco Chronicle said:

... a veteran journalist remarked, as Mailer stood on the platform, the applause thundering in the air, that his was the most impressive political speech in a generation.

Here is more from Gleason:

The revolution of creativity marked the two-day affair at the University of California, too. It throbbed day and night with the vitality of improvisation. At times over 8000 people packed the lawns and the plaza in a atmosphere of morality that, despite the criticisms of what Paul Krassner called the 'teach-in drop-outs' was as peaceful as a religious revival.

... Norman Mailer spoke of the reality of death and life and how the society's very existence depends upon a moral revolution, a return from formality to reality, from symbol to the thing itself.

Before Vietnam Day, I was quite undecided as to whether I would con­tinue my involvement, especially at such a level. In fact, I had made plans to go to Hawaii at the end of June for a vacation with my family. With the success of Vietnam Day, I recommitted myself with such fervor that I would become close to abandoning my career as a mathematician. In fact, I kept up an ever-increasing pace of organizing demonstrations until October 16. But that is getting a little ahead.

What could out-do Vietnam Day? A clue came from the teach-in talk of Straughton Lynd. He said:

477

20 S. Smale

... is the creation of civil disobedience so massive and so persistent that the Tuesday Lunch Club that is running this country—Johnson, McNamara, Bundy, and Rusk— will forthwith resign ...

Jerry and I agreed to call a new teach-in coupled with massive civil disobe­dience. We chose the dates October 15-16, to maximize the momentum, taking into account that classes started in September. Our friends in the VDC concurred.

Here is a summary that Jerry and I wrote at the time (from old notes of mine):

Vietnam Day, Berkeley by J. Rubin A S. Smale

On May 21-22 a student faculty group sponsored a community protest meeting on the Berkeley campus which lasted for 35 hours, bringing about 50,000 different people to the event over the period and with broadcasts reaching an estimated half a million people in the Bay Area.

The speeches centered around U.S. participation in Vietnam with most of the talks attacking various aspects of U.S. policy, with liberal and radical points of view both well represented.

Initially the sponsor of the protest meeting ws the Berkeley Faculty AFT. This was later joined by the Faculty Peace Committee and the employed graduate student AFT. However, the people from these groups and others who were actually doing the preparation formed an organization called the Vietnam Day Committee which came to be an important organization in its own right, and by the time Vietnam Day had arrived, was the de facto sponsoring organization. The Vietnam Day Committee still flourishes as an active peace group in Berkeley.

The working principle of the Vietnam Day Committee was to put forth an interest­ing and exiting program with wide publicity. Emphasis was on peaceful alternatives to Johnson's policy, but strong attempts, with partial success, were made to put de­fenders of Administration policy on the platform at Vietnam Day. The meeting was called an educational protest.

It was an important policy of the Vietnam Day Committee to make no concessions to respectability, or to weaken the protest. For example, we resisted much pressure to de-emphasize the student and campus element of the program by removing the event from the campus. (Berkeley students are not "respectable"!). Also, we opposed pressure to remove the most radical speakers. One of the major successes of the project was the fact that speaking on the same platform were such diverse men as Norman Thomas, Senator Gruening, Dr. Benjamin Spock, Isaac Deutscher, Dick Gregory, Norman Mailer, the Editor of the National Guardian, and speakers from the Dubois Club, YSA, Progressive Labor, and pacifist groups.

Publicity, besides dramatic use of the press, included the distribution of thousands of posters and hundreds of thousands of fliers.

Following similar principles, the Vietnam Day Committee in future activity is im­plementing its educational protest with direct action.

In addition to a picket of Johnson when he comes to San Francisco for the 20th Anniversary of the U.N. June 25-26, the Vietnam Day Committee is plAnning an­other protest meeting on October IS with massive civil disobedience on October 16.

478

1. Some Autobiographical Notes 21

All peace and political groups, nationally and even internationally are being asked to support these days of protest...

[Smale ends his chapter here. He is writing his autobiography in chapters like these, several of which are reproduced in this volume. Future chapters will continue the story. (Editor's note)']

479

480

STEVE SMALE

Mathematical Problems for the Next Century1

v I. Arnold, on behalf of the International Mathematical Union, has written to a

number of mathematicians with a suggestion that they describe some great

) problems for the next century. This report is my response.

Arnold's invitation is inspired in part by Hubert's list of 1900 (see, e.g., [Browder, 1976]) and I have used that list to help design this essay.

I have listed 18 problems, chosen with these criteria:

1. Simple statement Also preferably mathematically pre­cise.

2. Personal acquaintance with the problem. I have not found it easy.

3. A belief that the question, its solution, partial results, or even attempts at its solution are likely to have great im­portance for mathematics and Its development in the next century.

Some of these problems are well known. In fact, included are what I believe to be the three greatest open problems of mathematics: the Riemann Hypothesis, the Poincarf Conjecture, and 'Does P = NPT Besides the Riemann Hypothesis, one below is on Hubert's list (Hubert's 16th Problem). There is a certain overlap with my earner paper "Dynamics retrospective, great problems, attempts that failed* [Smale, 1991].

Let us begin.

Of the zeros of the Riemann zeta function, defined by an­alytic continuation from

Re(s) > 1,

are those which are in the critical strip 0 s Re(s) s 1 all on the line Re(s) = ±? This was Problem #8 on Hubert's list There are many fine books on the zeta function and the Riemann hypothesis which are easy to locate. I leave the matter at this.

pTODMfn mi TrlM P"Mrtowv ConJocVAtrai Suppose that a compact connected S-dimensional mani­fold has the property that every circle in it can be deformed to a point. Then must it be homeomorphic to the S-sphere?

The n-sphere is the space

S" = | ieR"M | l l = D, IWF = X :

A compact n-dimensional manifold can be thought of as a closed bounded n-dimensional surface (differentiable and non-singular) in some Euclidean space.

'Lacwa given on the occasion of Arnold's eorn Ortidoy at Ine Raids biatrrute. Toronto. June 1997

O 1MB SPnNQBt-WlJ lG NEW YOFK V G U M : 20, NLAOGR2. IS9B 7

481

The TwUmensional Poincare conjecture asserts that a compact n-dimenslonal manifold M having the property that every map/: S* -»M, k < n (or equivalently, k s n/2) can be deformed to a point, must be homeomorphic to S".

Henri Poincare studied these problems in his pioneer­ing papers in topology Poincarf in 1900 (see |Polncar£, 1963], pp. 338-370) announced a proof of the general "-di­mensional case. Subsequently (in 1904) he found a counter­example to his first version of the statement ([Poincare, 1953], pp. 436-498). In the second paper he limits himself to n » 3, and states the 3-dimensional case as the problem above (not actually as a "conjecture').

My own relationship with this problem is described in the story (Smale, 1990]. There I wrote,

I first heard of the Pioncarg conjecture in 1965 in Ann Arbor at the time I was writing a thesis on a problem of topology. Just a short time later, I felt that I had found a proof (3 dimensions). Hans Samelson was in his of­fice, and very excitedly I sketched my ideas to him After leaving the office, I realized that my *proof" hadn't used any hypothesis on the 3-manifold.

In 1960, "on the beaches of Rio," I gave an affirmative answer to the n-dimensional Poincare. conjecture for n > 4. In 1983, Mike Freedman gave an affirmative answer for n - 4. (Note: for n > 4,1 proved the stronger result that M was the smooth union of two balls, M = D" U Lr\ that re­sult is unproved for n = 4, today.)

For background on these matters, besides the above ref­erences, see [Smale, 1963].

Many other mathematicians after Poincare have claimed proofs of the 3-dimensional case. See [Taubes, 1987] for an account of some of these attempts.

A reason that Polncarf's conjecture is fundamental in the history of mathematics is that it helped give focus to a manifold as an object of study. In this way, Poincare In­fluenced much of 20th-century mathematics with its at­tention to geometric objects, including eventually algebraic varieties, Riemannian manifolds, etc.

I hold the conviction that there is a comparable phe­nomenon today in the notion of a "polynomial-time algo­rithm." Algorithms are becoming worthy of analysis in their own right, not merely as a means to salve other problems. Thus I am suggesting that as the study of the set of solu­tions of an equation (e.g., a manifold) played such an im­portant role in 20th-century mathematics, the study of find­ing the solutions (e.g., an algorithm) may play an equally important role in the next century.

Pll»lll» 3l D M * P ' HPT I sometimes consider this problem as a gift to mathemat­ics from computer science. It may be useful to put it into a form which looks more like traditional mathematics.

Towards this end, first consider the Hilbert Nullstellensatz over the complex numbers. Thus let/i /* be complex polynomials in n variables; we are asked to decide if they

have a common zero f € CV The Nullstellensatz asserts that this is not the case if and only if there are complex polynomials gi,..., gk in n variables satisfying

Z 9ift = 1 (1) i

as an identity of polynomials. The effective Nullstellensatz as established by BrownaweD.

(1987) and others, states that in the above one may assume that the degrees of the g, satisfy

deg g( s max(3, D)n, D = max deg /,.

With this degree bound the decidability problem becomes one of linear algebra. Given the coefficients of the/j, one can check if (1) has a solution in the coefficients of the ft. Thus one has an algorithm to decide the Nullstellensatz. The number of arithmetic steps required grows exponen­tially in the number of coefficients of the/, (the input size).

Conjecture (over C). There is no polynomial-time al­gorithm/or deciding the Hilbert Nullstellensatz over C.

A polynomial-tune algorithm is one in which the num­ber of arttuuetic steps is bounded by a polynomial In the number of coefficients of the/*.

To make mathematical sense of this conjecture, one has need of a formal definition of algorithm. In this context, the traditional definition of Turing machine makes no sense. In [Blum-Shub-Smale, 1969] a satisfactory definition is proposed, and the associated theory is exposed in [Blum-Cucker-Shub-Smale (or BCSS), 1997],

Very briefly, a machine over C has as inputs a finite string of complex numbers, and the same for states and outputs. Computations on states include arithmetic opera­tions and shifts on the string. Finally, a branch operation on "xi = 0?" is provided.

The size of an input is the number of elements in the in­put string. The time of a computation is the number of ma­chine operations used in the passage from input to output Thus, a polynomial-rime algorithm over C is well-defined.

Note that all that has been said about the machines and the conjecture use only the structure of C as a field, and hence both the machines and the conjecture are meaning­ful over any field. In particular, if the field is Z2 of two el­ements, we have the Turing machines. Consider the decision problem: Input k polynomials in n variables uHth coefficients in Z* Is there a common zero I £ (Z2)V

Conjectwe. There is no polynomial-time algorithm over It deciding this problem.

This is a simple reformulation of the classic conjecture P*NP.

In the above I have bypassed the basic ideas and theo­rems related to NP-completeness. For the classic case of

8 T X u*T>cMkncAL HTHuaecot

482

Cook and Karp, see [Garey-Johnson, 1979], and for the the- Problem 6: Finiteness of the Number of Relative ory over an arbitrary field, see BCSS. Equilibria in Celestial Mechanics

Given positive real numbers m\,... ,mn as the masses Problem 4: Integer Zeroes of a Polynomial in the n-body problem of celestial mechanics, is the Let me start by defining a diophantine invariant rmotivated number of relative equilibria finite? by complexity theory. A program for a polynomial/ £Z[f] of one variable with integer coefficients is the object (1, t, The problem is in Wintner's book (1941) on celestial me-uh... uk~), where uk = /, and for all (, ue = ui" Uj, i,j < (, chanics. A relative equilibrium is a solution to Newton's and ° is + or - or X. Here «0 = t, M_I = 1. Then T ( / ) is equations which is induced by a plane rotation. the minimum of k over all such programs. For the 3-body problem there are five relative equilib­

ria: three found by Lagrange, two by Euler. For 4 bodies Is the number of distinct integer zeroes offpolynom- the finiteness is unknown. inally bounded by T(/)? In other words, is In [Smale, 1970], I interpreted the relative equilibria as

wt critical points of a function induced by the potential of the ZaUi-TU) J U> planar re-body problem. More precisely, the relative equi-

where Zaf) is the number of distinct integer zeros of libria correspond to the critical points of f and c is a universal constant? y ,„ _ A-W^WI —» R

Mike Shub and I discovered this problem in our com- where Sk = \x e (R2)n | I mtXi = 0, '^Zmilkll2 = 1), A = plexity studies. We proved that an affirmative answer im- [x e Sk \ xt = x some i *j\. The rotation group SO(2) acts plied the intractability of the nullstellensatz as a decision on Sk - A, and Vis induced on the quotient from the po-problem over C and thus P * NP over C. See [Shub-Smale, tential function 1995] and also BCSS. „ m

Since the degree of/ is less than or equal to 2T+1, T = V(x) = 2_ i| _ n T ( / ) , there are no more than 2 r+1 zeros altogether. ,<J " * j"

For Chebyshev polynomials, the number of distinct real Note that V: Sk -» R is invariant under the rotation group zeros grows exponentially with T. SO(2) and that the quotient space Sfc/SO(2) is homeomor-

Many of the classic diophantine problems are in two or phic to complex projective space of dimension n - 2. more variables. This problem asks for an estimate in just Mike Shub (1970) has shown that the set of critical one variable, and nevertheless seems not so easy. points is compact, and Palmore (1976) that even for n =

Here is a related problem. A program for an integer m 4, Vmay have degenerate critical points. is the object (1, m i me), where me = m, m0 = 1, m, = Relative equilibria play an important role in celestial me-mi ° TO,-, i,j < q, and ° = +, - or X. Then let r(m) be the chanics, for example, as in the bifurcation of the angular minimum of (, over all such programs. Thus r(m) repre- momentum map. Moreover there are "the Trojans" in the sents the shortest way to build up an integer m starting solar system, which correspond to the Lagrange relative from 1 using plus, minus, and times. equilibria.

Problem: Is there a constant c such that •</<;!) s (log kf Kuz'mina (1977) has found explicit upper bounds in the for all integers fc? One might expect this to be false, so that generic case. k\ is "hard to compute"; see [Shub-Smale, 1995]. Further background may be found in [Abraham-

Marsden, 1978]. Problem 5: Height Bounds for Diophantine Curves

Can the feasibility of a diophantine equation f(x,y) = Prob,~ 7: Distribution of Point, on the 2-Sphere 0 (input/e Z[u, v\) be decided in time 2*c where c is het V>^> - ^i*i<i*N log p ^ j , where x - (x, xN), a universal constant? the x, are distinct points on the 2 sphere S2 C R3, and ||n - x3\

is the distance in R3. Denote min^ Vftix) by VN. Here s = s(f) is the size off, defined by Find (xh ..., xN~) such that

v-. i i v- VpJx) - VN £ c log N, c a universal constant. (2) s(f) = Y (logkl + 1), Kx,y) = V aax°iy°2, ^ >

'^d ^d To "find" means to give an algorithm which on input N a = (ai,a_), and |a| = ai + a_, at > 0. outputs distinct X\ xN on the 2-sphere satisfying (2).

Moreover, / is feasible if there exist integers x, y with To be precise one could take a real number algorithm in f(x,y) = 0. The Turing model of computation is supposed. the sense of BCSS (adjoining a square root computation)

This problem is posed essentially in [Cucker, Koiran, with halting time polynomial in N. and Smale, 1997]. The size s(f) is a version of the "height" This problem emerged from complexity theory, jointly of/. Conjectured height bounds for example as in [Lang, with Mike Shub [Shub and Smale, 1993]. It is motivated by 1991] could prove helpful in settling this problem. See also finding a good starting polynomial for a homotopy algo-[Manders-Adleman, 1978]. rithm for realizing the Fundamental Theorem of Algebra.

VOLUME 20, NUMBER 2. 1998 9

483

An (xt xN) = x such that Vj/i) = Vy Is called an JV-tuple of elliptic Fekete points (see [Tsuji, 19G9|).

The function Vf, as a function of N satisfies

VN= - fog f - ^ 2 - \N log N + OOT It is natural also to consider the functions

V W ^ I f c T ^ . V ^ m i n V ^ ) ,

* as before and 0 < s < 2. The original V«(r), V* corre­spond in a natural way to s = 0, and for * = 1, V/Xx, 1) is the Coulomb potential, and Vw(l) corresponds to an equi­librium position of N electrons constrained to lie on the two-sphere.

I had asked Ed Saff for some help in dealing with the main problem above. Subsequently, he and his colleagues produced a number of fine papers dealing with the subject and its ramifications. See [Ktujlaars and Saff, 1997] and |Saff and Kidjlaars, 1997] for background and further references. In (Rakhmanov, Saff, and Zhou, 1994], one can And nu­merical evidence (N = 12,000) to support these problems.

Another way of looking at our main problem here is to optimize the function

HW.X) = (exp v^x ) ) - ' = H I k " *jll-

However, as was written in |Shub and Smale, 1993),"... this may not be so easy since there are saddle points of index N (on a great circle in S2, evenly spared N points, x\,.. , XN). Also the various symmetries that WN possesses will confuse the picture."

PvoajMsn 8s MVocMrtlon ©i DytNMnlcs into BooMMnio T I M O I V The following problem is not one of pure mathematics, but lies on the interface of economics and mathematics. It has been solved only in quite limited situations.

Extend the mathematical model of general equilibrium theory to include price adjustments.

There is a (static) theory of equilibrium prices In eco­nomics starting with Walras and firmly grounded in the work of Arrow and Debreu (see [Debreu, 1969|). For the trivial case of one market this amounts to die equation "supply equals demand," and a natural dynamics is easily found [Samuelson, 1971]. For several markets, the situa­tion is complex.

There is a function called the excess demand, Z(p) = D(p) - S(p), from the space of prices to the space of com­modities. Both demand D and supply 5 are defined by ag­gregation over the Individual agents. Economics Justifies conditions on individual behavior which lead to axioms on Z. These axioms for the excess demand map Z : R* -» R' are:

1. Z(Ap) = Z(p), allp = (p, pt),Pi a 0, A e R, A > 0. 2. Sf., ptZi(p) = 0, Walras's law (the total value is zero). 3. Z((p) > 0 if pt = 0 (positive demand for a free good).

By (1), (2), (3), Z may be interpreted as a vector field on the intersection of the (€ - l>sphere with the posi­tive orthant, pointing inward on the boundary. The exis­tence of an equilibrium price vector p* follows from Hopf s theorem, so that Zp*) = 0, and "supply equals de­mand."

Problem 8 asks for a dynamical mode), whose stales are price vectors (perhaps enlarged to include other economic variables). This theory should be compatible with the ex­isting equilibrium theory. A most desirable feature is to have the time development of prices determined by the in­dividual actions of economic agents.

I worked on this problem for several years, feeling that it was the main problem of economic theory [Smale, 1976]. See also (Smale, 1981] for background.

PTDMWII wi I M un#av Pro0TMMiiwf Snrooiwii Is there a polynomial-time algorithm over the real numbers which decides the feasibility of the linear sys­tem of inequalities Axzb?

The algorithm requested by this problem is that given by a real number machine in the sense of BCSS (see also Problem 3). The system Ax aft has as Input an m x n real matrix A and vector J £ R * and the problem asks, Is there somexG WwithZf.jOflXj — Morallt = 1,. . . ,m?Time is measured by the number of arithmetic operations. This problem is in BCSS.

This is a decision version of the optimization problem of linear programming: Given A, 6 as above and e E R", de­cide if

max e x subject to Ax 2 6

exists, and if so, output such an x. The famous simplex method of Dantzig provides an al­

gorithm for both problems (over R), but Klee and Minty showed that it was exponentially slow in the worst case. On the other hand, Borgwardt and I, with subsequent im­portant support from Haimovich, showed that it was poly­nomial time on the average. For all of these things, see (SchrUver, 1986).

In terms of the Turing model of computation, using ra­tional numbers Q, and cost measured by "bits," there is a parallel development Starting with ideas of Yudln and Nemirovsky, Khachian found a potynomial time algorithm (the ellipsoid method) for the linear programming prob­lem, Subsequently Karmarkar with his "interior point method* found a practical algorithm for this problem, which he showed ran in polynomial time in the Turing model. For all these things one can see IGrOtschel, Lovasz, and Schryver, 1993] as well as |Scnnjvw, 1986].

Closer to the main problem above over R is a similar problem asking for a "strongly polynomial algorithm" over Q for solving these linear programming problems. This no­tion of algorithm demands that the number of arithmetic operations as well as the number of bit operations be polynomial in the size of the input (m X (n -t- 1)). Partial

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results are due to MegkWo and especially Tardos (see [Grotschel-Lovasz-Schrijver, 1993|).

For the problem over R there are also references [Barvinok-Vershik, 1963) and [Traub-Wogsiiakowski, 1962].

PTODMSR 101 in# dMsnaj LvfiHVM Let p be a non-wandering point of a diffeomorphism S : M-*tiqfa compact manifold. Can S be arbitrar-Uy well approximated with derivatives Of order r (C-approximation) for each r, by T:M-*M so that pis a periodic point of T?

A non-wandering point p e M is one with the property that for each neighborhood V otp there is a * 6 Z such thatS'[ /n(/* <t>. Here S* is the l iterate bfS; pis a pe­riodic point of period m if Tm(p) = p.

This is the discrete form of the famous "closing lemma," which in the C'-case has been solved affirmatively by Charles Pugh (1967).

There is an easy ( -approximation with the desired property. Peixoto observed that this argument failed for C-approximations, correcting a mistake of Rene Thorn (which Rene has told me was his biggest mistake).

Pugh and Robinson (1983) have proved the closing lemma with C'-approximations for the llarailtonlan ver­sion. Peixoto gave an affirmative answer with C-approxi-mations (any r) for the circle, as well as the continuous-time version for orientable 2-dimensional manifolds. Recently the closing lemma has been given additional im­portance by the work of Hayashi (1997); see also [Wen-Xia, 1997).

Problsm 11: la Pus Phii—lsiial Dynamics

Can a complex polynomial T be approximated by one of the same degree with the property that every criti­cal point tends to a periodic sink under iteration?

This is unsolved even for polynomials of degree 2. Here a polynomial map 7 C -» C is considered a discrete dy­namical system by iteration. So if z e C, its orbit in time, z = zo, z\, *2, . . . is defined by Z\ = Tfa-i) *nd t may be interpreted as time (discrete). A fixed-point w of T (7\u>) = to) is a sink if the derivative T'(w) of T at w has absolute value less than 1. A periodic sink of T of period p is a sink for T". A critical point of 7" is just a point where die derivative of T is zero.

While the problem is now made precise, it is useful to see it in the framework of hyperbolic dynamics from the 1960's.

A fixed-point x of a diffeomorphism 7*: U -»M is hy­perbolic if the derivative m\x) of T at x (as a linear auto­morphism of the tangent space) has no eigenvalue of ab­solute value 1. If x is a periodic point of period p, then x is hyperbolic if it is a hyperbolic fixed-point of V. The no­tion of hyperbolic extends naturally to II, the closure of the set of non-wandering points (see Problem 10).

A dynamical system r e VW.M) is called hyperbolic (or

satisfies Axiom A) if the periodic points are dense in fl and n is hyperbolic [Smale, 1967] or (Sraale, 1980]. We assume also a no-cycle condition. The work of many people, es­pecially Ricardo MaM, has identified hyperbolic dynamics with a strong notion of the stability of the dynamics called structural stability. There is even the beginning of a struc­ture theory for this class of dynamics.

While hyperbolic systems constitute a large set of dy­namics, an even larger set, including applied chaotic dy­namics, lies beyond The concept of hyperbolidty extends bom the invertible dynamics to the case of our problem above, polynomial maps from C to C. Classical complex variable theory permits recasting the problem to an equiv­alent one:

Can a polynomial map T.C — Cbe approximated by one which is hyperbolic?

The theory of complex one-dimensional dynamics was begun by Fatou and Julia towards the beginning of this cen­tury. In the 1960's I asked my thesis student John Oucken-heimer to look at this literature and try to solve the above problem (among other things). His thesis (see [Chem and Smale, 1970]) contains the affirmative answer, but with a gap in the proof. Now the problem stands open as the fun­damental problem of one-dimensional dynamics.

John's paper is one of many on the problems of this es­say with a wrong proof, starting with Polncare.

Complex 1-dimensional dynamics has become a flour­ishing subject and includes important contributions of Douady and Hubbard, Sullivan, Yoccoz, McMullen, among many others. See [McMullen, 1994].

There is a parallel field of real 1-dimensional dynamics of a smooth map T : / - » / , / = (0, 1].

Problem: Can a smooth map T: [0, 1] -» [0, 1) be ap­proximated in C, all r > I, by one which is hyperbolic?

About the time of Guckenheimer's thesis, I asked Ziggy Nitecki to study this problem. My earlier negligence was compounded in not catching the mistake in Nitecid's the­sis (see [Chem and Smale, 1970]), which purported to give an affirmative proof.

Subsequently, Jakobson (1971) answered the problem for ( '-approximations, but the general case remains open. See [de Melo-van Strien, 1993] for background.

PraoMin 18 GsmfrBlnMrv of OwrSonM pMoflM Can a diffeomorphism of a compact manifold M onto itself be (^approximated, ailr& l,by one T: M-*U which commutes with only its iterates?

Thus the centralizer of T in the group of dWeomor-phisms, Diff(Af) should be (T* | k 6 Z).

I had started thinking about the centralizer in [Smale, 1963], but it was after Nancy Kopell's thesis with me (see [Chem and Smale, 1970)), answering the question affirma­tively in case dim M - 1, that 1 proposed this problem

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[Smale, 1967]. Today it remains unsolved even for 2-di-mensional manifolds M.

One may also ask if the set of difTeomorphisms of M with trivial centralizer is dense and open in DtfffAf) with the C-topology.

The main work on these problems has been done by Palis and Yoccoz (1989), with almost complete answers in the case of hyperbolic dynamics (see Problem 11) for any manifold.

I wrote in |Smale, 1991], "1 find this problem interesting in that it gives some focus in the dark realm, beyond hyper-bottdty, where even the problems are hard to pose clearly."

PYSSJMCRI i3t HMbsrrs 19flfi PVOMSMH Consider the differential equation in R2

% = IKx,y), % = ^ ,5 , ) , ( 3 )

where P and Q are polynomials. Is there a bound K on the number of limit cycles ofthe form Ks i f , where d is the maximum of the degrees of P and Q, and q is a universal constant?

This is a modem version of the second half of Hubert's 16th Problem. Except for the Riemann hypothesis, it seems to be the most elusive of Hubert's problems.

In fact, since a paper of Petrovskii and Landis (1967) purporting to give a positive solution, the progress seems to be backwards. Earlier, Dulac (1923) claimed that the sys­tem (3) always has a finite number of limit cycles. After a gap in Petrovsldi and Landis was found (see [Petrovskii and Landis, 1969]), Dyashenko (1966) found an error in Dulac's paper. Moreover Shi Songling (1962) found a counter­example to the specific bounds of Petrovsku-Landis for the case d = 2. Subsequently, two long works have appeared, independently, giving proofs of Dulac's assertion [Ecalle, 1992] and [Ilyashenko, 1991]. These two papers have yet to be thoroughly digested by the mathematical community.

Thus one has the finiteness, but no bounds. We will con­sider a special class where the finiteness is simple, but the bounds remain unproved.

The following corresponds to Lienard's equation (see, e.g., [Hirsch and Smale, 1974])

<te - „ dy ... — = y - Ax), —- = -x, (4) dt y J ' dt where/(r) is a real polynomial with leading term i2* Tl and satisfying/(0) = 0.

If f(x) - x 2 - x, then (4) is van der Pol's equation with one limit cycle. More generally, it can be easily shown that all the solutions of (4) circle around the unique equilibrium at (0,0) in a clockwise direction. By following these curves, one defines a "Poincare section," T: R* -» R* where R* is the positive y-suds. The limit cycles of (4) are precisely the fixed-points of T. In various talks I raised the question of es­timating the number of these fixed-points (via some new land of fixed-point theorem?), in response, Lao, de Mek>, and Pugh (1977) found examples with k different limit cycles and

conjectured this number * for the upper bound Still no up­per bound of the form (deg/)' has been found. Because T is analytic, (4) has a finite number of limit cycles for each/.

For more background see |Browder, 1976], (Ilyashenko and Yakovenko, 1996], |L)oyd and Lynch. 1968), and [Smale, 1991].

PrsMssa 14c LsratM Atvawtvc Is the dynamics of the ordinary differential equations of Lorenz that of the geometric Ixfrenx attractor of Williams, Guckenheimer, and Yorke?

The Lorenz equations are:

x = - lQr + lOy y = 28x - y - xx i = -§* + xy.

Lorenz (1963) analysed these equations by computer to find that most solutions tended to a certain attracting set, and in so doing, he produced an important early example of "chaos." However, mathematical proofs were lacking. This numerical work inspired the rigorous mathematical development of a geometrically defined ordinary differen­tial equation which seems to have the same behavior (Yorke, [Williams, 1979], 'Guckenheimer and Williams, 1979]). This geometric attractor has been analysed in de­tail and one can fairly say that it is understood.

Problem 14 asks if the dynamics of the original equa­tions is the same as that of the geometric model. The most complete positive answer would be to describe a homeo-morphism of R3 to R3 which would take solutions of the Lorenz equations to solutions (a member of the family) of the geometric attractor—or rather, of a member of the two-parameter family of geometric attractors.

An answer to this problem would be a step in estab­lishing foundations for the field of applied dynamical sys­tems. Up to the present time, in the equations of engi­neering and physics, chaos has only been established in a weaker sense, that of proving the existence of horseshoes (e.g., Melrdkov, Marsden, and Holmes; see (Guckenheimer and Holmes, 1990]).

Geometric, structurally stable, chaotic attractors in dy­namics are in my paper [Smale, 19671 But these did not arise from any physical system.

Some partial results on problem 14 are due to Rychlik and Robinson, see [Robinson, 1969|.

rTQSJMfH i 9 I P U M S ^ S X S H M B4|(fSJ0QnS, Do the Navier-Stokes equations on a S-dimensumal do­main fl in R3 have a unique smooth solution for all time?

This is perhaps the most celebrated problem in partial differential equations. Let us be a little more precise. The Navier-Stokes equations may be written

^ + (u-V)u - vAu + grad p = 0. div u • 0,

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486

where a C-map u: R+ x n -♦ R3 andp : ft -» R are to be found satisfying these equations, u prescribed at t = 0 and on the boundary dil. Here R+ = 10,°°), v-7 is the operator X? Ui—, and v is a positive constant See, eg., [Chorin-Marsden, 1903] for details.

Many mathematicians have contributed toward the un­derstanding of this problem. An affirmative answer has been given in dimension 2 and in dimension 3 for t in a small interval |0, T\. See [Temam, 1979) for background.

The solution of this problem might well be a funda­mental step toward the very big problem of understanding turbulence. For example, it could help realize the ideas of (Ruelle and Takens 1971], which put the notion of a chaotic attractor into a model of turbulence. See also [Chorin, Marsden, and Smale, 1977].

In [Smale, 1991] I asked if the solutions of the 2-dimen­sional Navier-Stokes equation with a forcing term on a torus must converge to an equilibrium as time tends to in­finity. Babik and Vtshik (1983) had given some evidence to the contrary. Subsequently Liu (1992) provided examples to show convergence to a more complicated attractor.

PTOMMI I IOS TtM vsjooMsjn v#nf#cvutfSi Suppose f : C" -» C* is a polynomial map with the property that the derivative at each point is non-sin-gvlar. Then must/be one-to-one?

Here/(z) = (/i(z)....,/»(?)), and each/, is a polyno­mial in n variables. The derivative of / at z, Df[z) : C™ -» O, may be thought of as the matrix of partial derivatives, and the non-singularity condition as Det Qftz) * 0.

If/ is indeed injective, then it is surjective and has an inverse which is a polynomial map. For an elementary proof of this see [Rudin, 1996].

The problem goes back to the 1930's- See the excellent surveys [Bass, Connell, and Wright, 19S2] and [van den Essen, 1997] for the importance, background, and related results.

It, iifcl, I, 4 v . * - ^ ^ M F F A I g w n a l l i i i i m w n ii Miwinsj i"viyBQinw s ^ i i N m

Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average, in polynomial time with a uniform algorithm? The final theorem in my five-paper series with Mike

Shub |Shub and Smale, 1994] Is exactly this result without the word "uniform."

I review the definitions. Consider/: C »C\/(i) = J\(z\ ,/nW). * = (*i. • • • F *») where each/, is a polynomial in n variables of degree a\. It is reasonable to make the/, into homogeneous polynomials by adding a new variable Zo, work in the corresponding protective space, and then translate the algorithm and results back to the initial affine problem.

"Approximately" can be defined intrinsically using Newton's method, and is necessary because of Abel, Galois, el al. Time is measured by the number of arithmetic operations and comparisons, s , using real machines (as in Problem 3), if one wants to be formal.

A probability measure must be put on the space of all such/ for each d = du ..., O . and the time of an algo­rithm is averaged over the space of/. Is there such an al­gorithm where the average time is bounded by a polyno­mial in the number of coefficients of/ (the input size)?

In [Shub and Smale, 1994] it is proved that this can be done, but the algorithm is different for each d and even for each desired probability of success. A uniform algorithm is one that is independent of d (d is part of the input).

Certainly finding zeros of polynomials and polynomial systems is one of the oldest and most central problems of mathematics. This problem asks if under some conditions, specified in the problem, it can be solved systematically by computers. If there is no poh/nomial-time way of doing it, then no computer will ever succeed.

There is a more recent development putting the prob­lem of zeros of polynomials into a universal role. The Hilbert Nullstellensatz (as a decision problem) is NP-conv plete over any field (see Problem 3).

A similar, more difficult, problem may be raised for the real numbers.

ProMam 18i Limit* of I n f M l f n c s What are the limits of intelligence, both artificial and human?

Penrose (1991) attempts to show some limitations of ar­tificial intelligence. His argumentation brings in the inter­esting question whether the Mandelbrot set is deddable (dealt with in [Blum and Smale, 1993]) and implications of the Godel incompleteness theorem.

However, a broader study is called for, one which in­volves deeper models of the brain, and of the computer, in a search of what artificial and human intelligence have in common, and how they differ. I would look In a di­rection where learning, problem-solving, and game the­ory play a substantial role, together with the mathemat­ics of real numbers, approximations, probability, and geometry.

I hope to expand on these thoughts on another occa­sion.

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VOUME 30. NJMB0 H M 18

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9TEVKSMALE Mathematics Deflarttnertt

Crty University of Hong Kong Tat Ches Avenue

Kbwtoon. Hong Kong

China

e-rnail. masrnefe&rnath.cityu.edu.hk

Steve Smele. Ph.D. of the Untofsity of Michigan [1957) and long-time professor at the University of California Berkeley. . | won a Raids Medal and other honors for his mathematics. He is quite distinguished also as a collector and photographer of beautiful minerals, and We Maftamafcsf Intetffgencer was grateful for his letting us publish a sample of his collection on our cover, ml. 14 (1932), no. 3.

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