two-dimensional spaces, volume 2 topology as fluid …two-dimensional spaces, volume 2 topology as...

Two-Dimensional Spaces, Volume 2

TOPOLOGY AS FLUID GEOMETRY James W. Cannon


TOPOLOGY AS FLUID GEOMETRY

10.1090/mbk/109

A M E R I C A N M A T H E M A T I C A L S O C I E T Y

P r o v i d e n c e , R h o d e I s l a n d


TOPOLOGY AS FLUID GEOMETRY James W. Cannon

2010 Mathematics Subject Classification. Primary 57-01, 57M20.

For additional information and updates on this book, visitwww.ams.org/bookpages/mbk-109

Library of Congress Cataloging-in-Publication Data

Names: Cannon, James W., author.Title: Two-dimensional spaces / James W. Cannon.Description: Providence, Rhode Island : American Mathematical Society, [2017] | Includes bibli-

ographical references.Identifiers: LCCN 2017024690 | ISBN 9781470437145 (v. 1) | ISBN 9781470437152 (v. 2) | ISBN

9781470437169 (v. 3)Subjects: LCSH: Geometry. | Geometry, Plane. | Non-Euclidean geometry. | AMS: Geometry –

Instructional exposition (textbooks, tutorial papers, etc.). mscClassification: LCC QA445 .C27 2017 | DDC 516–dc23LC record available at https://lccn.loc.gov/2017024690

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Contents

Preface to the Three Volume Set ix

Preface to Volume 2 xiii

Chapter 1. The Fundamental Theorem of Algebra 11.1. Complex Arithmetic 21.2. First Proof of the Fundamental Theorem 51.3. Second Proof 71.4. Exercises 10

Chapter 2. The Brouwer Fixed Point Theorem 112.1. Statement of the Theorem 112.2. Introducing Extra Structure into a Problem 122.3. Two Elementary Problems 122.4. Three Advanced Problems 182.5. Exercises 27

Chapter 3. Tools 293.1. Polyhedral complexes 293.2. Urysohn’s Lemma and the Tietze Extension Theorem 313.3. Set Convergence 343.4. Exercises 36

Chapter 4. Lebesgue Covering Dimension 374.1. Definition of Covering Dimension 374.2. Euclidean n-Dimensional Space Rn Has Covering Dimension n 384.3. Construction of Partitions of Unity 404.4. Techniques Needed in Higher Dimensions 414.5. Exercises 42

Chapter 5. Fat Curves and Peano Curves 455.1. The Constructions 455.2. The Topological Lemmas 495.3. The Analytical Lemmas 515.4. Characterization of Peano Curves 525.5. Exercises 54

Chapter 6. The Arc, the Simple Closed Curve, and the Cantor Set 576.1. Characterizing the Arc and Simple Closed Curve 576.2. The Cantor Set and Its Characterization 616.3. Interesting Cantor Sets 63

v

vi CONTENTS

6.4. Cantor Sets in the Plane Are Tame 696.5. Exercises 73

Chapter 7. Algebraic Topology 757.1. Facts Assumed from Algebraic Topology 767.2. The Reduced Homology of a Sphere 777.3. The Homology of a Ball Complement 777.4. The Homology of a Sphere Complement 787.5. Proof of the Arc Non-Separation Theorem and the Jordan Curve

Theorem 79

Chapter 8. Characterization of the 2-Sphere 818.1. Statement and Proof of the Characterization Theorem 818.2. Exercises 89

Chapter 9. 2-Manifolds 919.1. Definition and Examples 919.2. Exercises 91

Chapter 10. Arcs in S2 Are Tame 95

10.1. Arcs in S2 Are Tame 9510.2. Disk Isotopies 9710.3. Exercises 100

Chapter 11. R. L. Moore’s Decomposition Theorem 10111.1. Examples and Applications 10111.2. Decomposition Spaces 10211.3. Proof of the Moore Decomposition Theorem 10411.4. Exercises 107

Chapter 12. The Open Mapping Theorem 10912.1. Tools 10912.2. Two Lemmas 11012.3. Proof of the Open Mapping Theorem 11212.4. Exercise 112

Chapter 13. Triangulation of 2-Manifolds 11313.1. Statement of the Triangulation Theorem 11313.2. Tools 11313.3. Proof of the Triangulation Theorem 11513.4. Exercises 116

Chapter 14. Structure and Classification of 2-Manifolds 11714.1. Statement of the Structure Theorem 11714.2. Edge-pairings 11814.3. Proof of the Structure Theorem 11914.4. Statement and Proof of the Classification Theorem 12314.5. Exercises 125

Chapter 15. The Torus 12915.1. Lines and Arcs in the Plane 129

CONTENTS vii

15.2. The Torus as a Euclidean Surface 13115.3. Curve Straightening 13415.4. Construction of the Simple Closed Curve with Slope k/� 13615.5. Exercises 137

Chapter 16. Orientation and Euler Characteristic 13916.1. Orientation 13916.2. Euler Characteristic 14116.3. Exercises 149

Chapter 17. The Riemann-Hurwitz Theorem 15117.1. Setting 15117.2. Elementary Facts from Trigonometry 15117.3. Branched Maps of S2 15417.4. Statement of the Riemann-Hurwitz Theorem 15517.5. Proof of the Riemann-Hurwitz Theorem 15517.6. Rational Maps 15617.7. Exercises 157

Bibliography 159

Preface to the Three Volume Set

Geometry measures space (geo = earth, metry = measurement). Einstein’stheory of relativity measures space-time and might be called geochronometry (geo= space, chrono= time, metry = measurement). The arc of mathematical historythat has led us from the geometry of the plane of Euclid and the Greeks after 2500years to the physics of space-time of Einstein is an attractive mathematical story.Geometrical reasoning has proved instrumental in our understanding of the real andcomplex numbers, algebra and number theory, the development of calculus with itselaboration in analysis and differential equations, our notions of length, area, andvolume, motion, symmetry, topology, and curvature.

These three volumes form a very personal excursion through those parts of themathematics of 1- and 2-dimensional geometry that I have found magical. In allcases, this point of view is the one most meaningful to me. Every section is designedaround results that, as a student, I found interesting in themselves and not justas preparation for something to come later. Where is the magic? Why are thesethings true?

Where is the tension? Every good theorem should have tensionbetween hypothesis and conclusion. — Dennis Sullivan

Where is the Sullivan tension in the statement and proofs of the theorems?What are the key ideas? Why is the given proof natural? Are the theorems almostfalse? Is there a nice picture? I am not interested in quoting results without proof.I am not afraid of a little algebra, or calculus, or linear algebra. I do not careabout complete rigor. I want to understand. If every formula in a book cuts thereadership in half, my audience is a small, elite audience. This book is for thestudent who likes the magic and wants to understand.

A scientist is someone who is always a child, asking ‘Why?why? why?’. — Isidor Isaac Rabi, Nobel Prize in Physics1944

Wir mussen wissen, wir werden wissen. [We must know, wewill know.] — David Hilbert

The three volumes indicate three natural parts into which the material on 2-dimensional spaces may be divided:

Volume 1: The geometry of the plane, with various historical attempts tounderstand lengths and areas: areas by similarity, by cut and paste, by counting,by slicing. Applications to the understanding of the real numbers, algebra, numbertheory, and the development of calculus. Limitations imposed on the measurementof size given by nonmeasurable sets and the wonderful Hausdorff-Banach-Tarskiparadox.

ix

x PREFACE TO THE THREE VOLUME SET

Volume 2: The topology of the plane, with all of the standard theorems of1 and 2-dimensional topology, the Fundamental Theorem of Algebra, the BrouwerFixed-Point Theorem, space-filling curves, curves of positive area, the Jordan CurveTheorem, the topological characterization of the plane, the Schoenflies Theorem,the R. L. Moore Decomposition Theorem, the Open Mapping Theorem, the trian-gulation of 2-manifolds, the classification of 2-manifolds via orientation and Eulercharacteristic, dimension theory.

Volume 3: An introduction to non-Euclidean geometry and curvature. Whatis the analogy between the standard trigonometric functions and the hyperbolictrig functions? Why is non-Euclidean geometry called hyperbolic? What are thegross intuitive differences between Euclidean and hyperbolic geometry?

The approach to curvature is backwards to that of Gauss, with definitionsthat are obviously invariant under bending, with the intent that curvature shouldobviously measure the degree to which a surface cannot be flattened into the plane.Gauss’s Theorema Egregium then comes at the end of the discussion.

Prerequisites: An undergraduate student with a reasonable memory of cal-culus and linear algebra, but with no fear of proofs, should be able to understandalmost all of the first volume. A student with the rudiments of topology—open andclosed sets, continuous functions, compact sets and uniform continuity—should beable to understand almost all of the second volume with the exeption of a littlebit of algebraic topology used to prove results that are intuitively reasonable andcan be assumed if necessary. The final volume should be well within the reach ofsomeone who is comfortable with integration and change of variables. We will makean attempt in many places to review the tools needed.

Comments on exercises: Most exercises are interlaced with the text in thoseplaces where the development suggests them. They are an essential part of thetext, and the reader should at least make note of their content. Exercise sectionswhich appear at the end of most chapters refer back to these exercises, sometimeswith hints, occasionally with solutions, and sometimes add additional exercises.Readers should try as many exercises as attract them, first without looking at hintsor solutions.

Comments on difficulty: Typically, sections and chapters become more diffi-cult toward the end. Don’t be afraid to quit a chapter when it becomes too difficult.Digest as much as interests you and move on to the next chapter or section.

Comments on the bibliography: The book was written with very littledirect reference to sources, and many of the proofs may therefore differ from thestandard ones. But there are many wonderful books and wonderful teachers thatwe can learn from. I have therefore collected an annotated bibliography that youmay want to explore. I particularly recommend [1, G. H. Hardy, A Mathematician’sApology ], [2, G. Polya, How to Solve It ], and [3, T. W. Korner, The Pleasure ofCounting ], just for fun, light reading. For a bit of hero worship, I also recommendthe biographical references [21, E. T. Bell, Men of Mathematics ], [22, C. Henrion,Women of Mathematics ], and [23, W. Dunham, Journey Through Genius ]. AndI have to thank my particular heroes: my brother Larry, who taught me aboutuncountable sets, space-filling curves, and mathematical induction; Georg Polya,who invited me into his home and showed me his mathematical notebooks; my ad-visor C. E. Burgess, who introduced me to the wonders of Texas-style mathematics;R. H. Bing, whose Sling, Dogbone Space, Hooked Rug, Baseball Move, epslums and

PREFACE TO THE THREE VOLUME SET xi

deltas, and Crumpled Cubes added color and wonder to the study of topology; andW. P. Thurston, who often made me feel like Gary Larson’s character of little brain(“Stop, professor, my brain is full.”) They were all kind and encouraging to me.And then there are those whom I only know from their writing: especially Euclid,Archimedes, Gauss, Hilbert, and Poincare.

Finally, I must thank Bill Floyd and Walter Parry for more than three decadesof mathematical fun. When we would get together, we would work hard everymorning, then talk mathematics for the rest of the day as we hiked the cities, coun-trysides, mountains, and woods of Utah, Virginia, Michigan, Minnesota, England,France, and any other place we could manage to get together. And special thanksto Bill for cleaning up and improving almost all of those figures in these bookswhich he had not himself originally drawn.

Preface to Volume 2

The first of three volumes in this set was devoted to the measurement of lengthsand areas, and to some of the consequences that study had in number theory,algebra, and analysis. Euclid was able to solve quadratic equations by geometricconstruction. But when mathematicians tried to extend those results to equationsof higher degree and to differential equations, a number of fascinating difficultiesarose, all involving limits and continuity, best modelled by topology.

In this second volume we assume that the reader has had a first course intopology and is comfortable with open and closed sets, connected sets, compactsets, limits, and continuity. Two good references are W. S. Massey [25] and J. R.Munkres [24].

The following discoveries led to the topics of this second volume.(1) The solution of cubic and quartic equations required serious consideration of

complex numbers, thought at first to be mysterious. But the mystery disappearedwhen it was seen that complex numbers simply model the Euclidean plane. Abeland Galois proved that equations of degrees 5 and higher could not be solved inthe relatively simple manner by formula as had sufficed in equations of degrees 1through 4. But Gauss, without giving explicit solutions, managed to prove theFundamental Theorem of Algebra that ensured that complex numbers sufficed fortheir solution. Gauss gave proofs involving the geometry and topology of the plane.

(2) Newton showed that the study of motion could be greatly simplified if, in-stead of examining standard equations, one examined differential equations. Prov-ing the existence of solutions to rather general differential equations led to problemsin topology. One of the standard proof techniques involves Brouwer’s Fixed PointTheorem. This volume proves that theorem in dimension 2 and outlines the proofin general dimensions.

(3) Descartes demonstrated that mechanical devices other than straight edgeand compass can construct curves of very high degree. Once curves of very generalform are accepted as interesting, further delicate questions of length and area arise:finite curves of infinite length, finite curves of positive area, space filling curves,disks whose interiors have smaller areas than their closures, 0-dimensional setsthrough which no light rays can penetrate, continuous functions that are nowheredifferentiable, sets of fractional dimension. This volume gives examples of many ofthese phenomena.

(4) The study of solutions to equations became more unified when all variableswere considered to be complex variables. Riemann modelled complex curves bysurfaces, which are 2-dimensional manifolds and are called Riemann surfaces. Theanalysis of 2-dimensional manifolds led naturally to notions, such as triangulation,genus, and Euler characteristic. These notions are explained in this volume.

xiii

xiv PREFACE TO VOLUME 2

All of these considerations required the study of limits and continuity, and theabstract notion that models limits and continuity in their most general settings isthe notion of topology. Henri Poincare wrote:

As for me, all of the diverse paths which I have successivelyfollowed have led me to topology. I have needed the gifts of thisscience to pursue my studies of the curves defined by differentialequations and for the generalization to differential equations ofhigher order, and, in particular, to those of the three bodyproblem. I have needed topology for the study of nonuniformfunctions of two variables. I have needed it for the study ofthe periods of multiple integrals and for the application of thatstudy to the expansion of perturbed functions. Finally, I haveglimpsed in topology a means to attack an important problemin the theory of groups, the search for discrete or finite groupscontained in a given continuous group.

Bibliography

Plain Fun

(top recommendations for easy, but rewarding, pleasure).

[1] Hardy, G. H., A Mathematician’s Apology, Cambridge University Press, 2004 (eighth print-ing).

[2] Polya, G., How to Solve It, Princeton Univerity Press, 2004.[3] Korner, T. W., The Pleasures of Counting, Cambridge University Press, 1996.

- More Fun

[4] Davis, P. J. and Hersh, R., The Mathematical Experience, Houghton Mifflin Company, 1981.[5] Rademacher, H., Higher Mathematics from an Elementary Point of View, Birkhauser, 1983.[6] Hilbert, D., and Cohn-Vossen, S., Geometry and the Imagination, (translated by P. Nemeyi),

Chelsea Publishing Company, New York, 1952. [College level exposition of rich ideas fromlow-dimensional geometry, with many figures.]

[7] Dorrie, H., 100 Great Problems of Elementary Mathematics: Their History and Solution,Dover Publications, Inc., 1965, pp. 108-112. [We learned our first proof of the fundamentaltheorem of algebra here.]

[8] Courant, R. and Robbins, H., What is Mathematics?, Oxford University Press, 1941.

Classics

(a chance to see the thinking of the very best, in chronological order).

[9] Euclid, The Thirteen Books of Euclid’s Elements, Vol. 1-3, 2nd Ed., (edited by T. L. Heath)Cambridge University Press, Cambridge, 1926. [Reprinted by Dover, New York, 1956.]

[10] Archimedes, The Works of Archimedes, edited by T. L. Heath, Dover Publications, In.,Mineola, New York, 2002. See also the exposition in Polya, G., Mathematics and PlausibleReasoning, Vol. 1. Induction and Analogy in Mathematics, Chapter IX. Physical Mathemat-ics, pp. 155-158, Princeton University Press, 1954. [How Archimedes discovered the integralcalculus.]

[11] Wallis, J., in A Source Book in Mathematics, 1200-1800, edited by D. J. Struik., HarvardUniversity Press, 1969, pp. 244-253. [Wallis’s product formula for π.]

[12] Gauss, K. F., General Investigations of Curved Surfaces of 1827 and 1825, Princeton Uni-versity Library, 1902. [Available online. Difficult reading.]

[13] Fourier, J., The Analytical Theory of Heat, translated by Alexander Freeman, CambridgeUniversity Press, 1878. [Available online, 508 pages. The introduction explains Fourier’sthoughts in approaching the problem of the mathematical treatment of heat. Chapter 3 ex-plains his discovery of Fourier series.]

[14] Riemann, B., Collected Papers, edited by Roger Baker, Kendrick Press, Heber City, Utah,2004. [English translation of Riemann’s wonderful papers.]

159

160 BIBLIOGRAPHY

[15] Poincare, H., Science and Method, Dover Publications, Inc., 2003. [Discusses the role of thesubconscious in mathematical discovery.] Also, The Value of Science, translated by G. B.Halstead, Dover Publications, Inc., 1958.

[16] Klein, F., Vorlesungen uber Nicht-Euklidische Geometrie, Verlag von Julius Springer, Berlin,1928. [In German. An algebraic development of non-Euclidean geometry with respect to theKlein and projective models. Beautiful figures. Elegant exposition.]

[17] Hilbert, D., Gesammelte Abhandlungen (Collected Works), 3 volumes, Springer-Verlag,

1970. [In German. The transcendence of e and π appears in Volume 1, pp. 1-4. Hilbert’sspace-filling curve appears in Volume 3, pp. 1-2.]

[18] Einstein, A., The Meaning of Relativity, Princeton University Press, 1956.[19] Thurston, W. P., Three-Dimensional Geometry and Topology, edited by Silvio Levy, Prince-

ton University Press, 1997. [An intuitive introduction to dimension 3 by the foremost ge-ometer of our generation.]

[20] W. P. Thurston’s theorems on surface diffeomorphisms as exposited in Fathi, A., and Lau-denbach, F., and Poenaru, V., Travaux de Thurston sur les Surfaces, Seminaire Orsay,Societe Mathematique de France, 1991/1979. [In French.]

History

(concentrating on famous mathematicians).

[21] Bell, E. T., Men of Mathematics, Simon and Schuster, Inc., 1937. [The book that convincedme that mathematics is exciting and romantic.]

[22] Henrion, C., Women of Mathematics, Indiana University Press, 1997.[23] Dunham, W., Journey Through Genius, Penguin Books, 1991.

Supporting Textbooks

- Topology

[24] Munkres, J. R., Topology, a First Course, Prentice-Hall, Inc., 1975. [The early chaptersexplain the basics of topology that form the prerequisites for the latter half of this book. Thelater chapters contain rather different views of some of the later theorems in our book.]

[25] Massey, W. S., Algebraic Topology: An Introduction. Springer-Verlag, New York -Heidelberg-Berlin, 1967 (Sixth printing: 1984), Chapter I, pp. 1-54. [A particularly nice

introduction to covering spaces.][26] Hatcher, A., Algebraic Topology , Cambridge University Press, 2001. [A very nice introduc-

tion to algebraic topology, a bit of which we need in Volume 2.][27] Munkres, J. R., Elements of Algebraic Topology, Addison-Wesley, 1984. [Another nice in-

troduction.][28] Alexandroff, P., Elementary Concepts of Topology, translated by Alan E. Farley, Dover

Publications, Inc., 1932. [A wonderful small book.][29] Alexandrov, P. S., Combinatorial Topology, 3 volumes, translated by Horace Komm, Gray-

lock Press, Rochester, NY, 1956.[30] Seifert, H., and Threlfall, W., A Textbook of Topology, translated by Michael A. Goldman;

and Seifert, H., Topology of 3-Dimensional Fibered Spaces, translated by Wolfgang Heil,Academic Press, 1980. [Available online.]

[31] Hurewicz, W., and Wallman, H., Dimension Theory, Princeton University Press, 1941. [SeeChapters 4, 5, and 6 of Volume 2.]

BIBLIOGRAPHY 161

- Algebra

[32] Herstein, I. N., Abstract Algebra, third edition, John Wiley & Sons, Inc., 1999. [See ourChapter 6 of Volume 1.]

[33] Dummit, D. S., and Foote, R. M., Abstract Algebra, third edition, John Wiley & Sons, Inc.,2004. [See our Chapter 6 of Volume 1.]

[34] Hardy, G. H., and Wright, E. M., An Introduction to the Theory of Numbers, fourthedition, Oxford University Press, 1960. [See our Chapter 5 of Volume 1.]

- Analysis

[35] Apostol, T. M., Mathematical Analysis , Addison-Wesley, 1957. [Good background for Rie-mannian metrics in Chapter 1 of Volume 1, and also the chapters of Volume 3.]

[36] Lang, S., Real and Functional Analysis, third edition, Springer, 1993. [Chapter XIV gives thedifferentiable version of the open mapping theorem. The proof uses the contraction mappingprinciple. See our Chapter 12 of Volume 2 for the topological version of the open mappingtheorem.]

[37] Spivak, M., Calculus on Manifolds, W. A. Benjamin, Inc., New York, N. Y., 1965. [Goodbackground for Riemannian metrics in Chapter 1 of Volume 1 and for Volume 3.]

[38] Janich, K., Vector Analysis, translated by Leslie Kay, Springer, 2001. [Good background forRiemannian metrics in Chapter 1 of Volume 1 and for Volume 3.]

[39] Saks, S., Theory of the Integral, second revised edition, translated by L. C. Young, DoverPublications, Inc., New York, 1964. [Wonderfully readable.]

[40] H. L. Royden, H. L., Real Analysis, third edition, Macmillan, 1988. [The place where wefirst learned about nonmeasurable sets.]

References from our Paper on Hyperbolic Geometry in Flavors ofGeometry (reprinted here as our Volume 3, Chapter 2)

[41] Flavors of Geometry, edited by Silvio Levy, Cambridge University Press, 1997.[42] Alonso, J. M., Brady, T., Cooper, D., Ferlini, V., Lustig, M., Mihalik, M., Shapiro, M.,

Short, H., Notes on word hyperbolic groups, Group Theory from a Geometrical Viewpoint:21 March — 6 April 1990, ICTP, Trieste, Italy, (E. Ghys, A. Haefliger, and A. Verjovsky,eds.), World Scientific, Singapore, 1991, pp. 3–63.

[43] Benedetti, R., and Petronio, C., Lectures on Hyperbolic Geometry, Universitext, Springer-Verlag, Berlin, 1992. [Expounds many of the facts about hyperbolic geometry outlined inThurston’s influential notes.]

[44] Bolyai, W., and Bolyai, J., Geometrische Untersuchungen, B. G. Teubner, Leipzig andBerlin, 1913. (reprinted by Johnson Reprint Corp., New York and London, 1972) [Historicaland biographical materials.]

[45] Cannon, J. W., The combinatorial structure of cocompact discrete hyperbolic groups, Geom.Dedicata 16 (1984), 123–148.

[46] Cannon, J. W., The theory of negatively curved spaces and groups, Ergodic Theory, SymbolicDynamics and Hyperbolic Spaces, (T. Bedford, M. Keane, and C. Series, eds.) OxfordUniversity Press, Oxford and New York, 1991, pp. 315–369.

[47] Cannon, J. W., The combinatorial Riemann mapping theorem, Acta Mathematica 173(1994), 155–234.

[48] Cannon, J. W., Floyd, W. J., Parry, W. R., Squaring rectangles: the finite Riemann mappingtheorem, The Mathematical Heritage of Wilhelm Magnus — Groups, Geometry & Special

162 BIBLIOGRAPHY

Functions, Contemporary Mathematics 169, American Mathematics Society, Providence,1994, pp. 133–212.

[49] Cannon, J. W., Floyd, W. J., Parry, W. R., Sufficiently rich families of planar rings,preprint.

[50] Cannon, J. W., Swenson, E. L., Recognizing constant curvature groups in dimension 3,preprint.

[51] Coornaert, M., Delzant, T., Papadopoulos, A., Geometrie et theorie des groupes: les groupes

hyperboliques de Gromov, Lecture Notes 1441, Springer-Verlag, Berlin-Heidelberg-NewYork,1990.

[52] Euclid, The Thirteen Books of Euclid’s Elements, Vol. 1-3, 2nd Ed., (T. L. Heath, ed.)Cambridge University Press, Cambridge, 1926 (reprinted by Dover, New York, 1956).

[53] Gabai, D., Homotopy hyperbolic 3-manifolds are virtually hyperbolic, J. Amer. Math. Soc.7 (1994), 193–198.

[54] Gabai, D., On the geometric and topological rigidity of hyperbolic 3-manifolds, Bull. Amer.Math. Soc. 31 (1994), 228–232.

[55] Ghys, E., de la Harpe, P., Sur les groupes hyperboliques d’apres Mikhael Gromov, Progressin Mathematics 83, Birkhauser, Boston, 1990.

[56] Gromov, M., Hyperbolic groups, Essays in Group Theory, (S. Gersten, ed.), MSRI Publi-cation 8, Springer-Verlag, New York, 1987. [Perhaps the most influential recent paper ingeometric group theory.]

[57] Hilbert, D., Cohn-Vossen, S., Geometry and the Imagination, Chelsea Publishing Company,New York, 1952. [College level exposition of rich ideas from low-dimensional geometry withmany figures.]

[58] Iversen, B., Hyperbolic Geometry, London Mathematical Society Student Texts 25, Cam-bridge University Press, Cambridge, 1993. [Very clean algebraic approach to hyperbolic ge-ometry.]

[59] Klein, F., Vorlesungen uber Nicht-Euklidische Geometrie, Verlag von Julius Springer, Berlin,1928. [Mostly algebraic development of non-Euclidean geometry with respect to Klein andprojective models. Beautiful figures. Elegant exposition.]

[60] Kline, M. Mathematical Thought from Ancient to Modern Times, Oxford University Press,

New York, 1972. [A 3-volume history of mathematics. Full of interesting material.][61] Lobatschefskij, N. I., Zwei Geometrische Abhandlungen, B. G. Teubner, Leipzig and Berlin,

1898. (reprinted by Johnson Reprint Corp., New York and London, 1972) [Original papers.][62] Mosher, L., Geometry of cubulated 3-manifolds, Topology 34 (1995), 789–814.[63] Mosher, L., Oertel, U., Spaces which are not negatively curved, preprint.[64] Mostow, G. D., Strong Rigidity of Locally Symmetric Spaces, Annals of Mathematics Studies

78, Princeton University Press, Princeton, 1973.[65] Poincare, H., Science and Method, Dover Publications, New York, 1952. [One of Poincare’s

several popular expositions of science. Still worth reading after almost 100 years.][66] Ratcliffe, J. G., Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics 149,

Springer-Verlag, New York, 1994. [Fantastic bibliography, careful and unified exposition.][67] Riemann, B., Collected Papers, Kendrick Press, Heber City, Utah, 2004. [English translation

of Riemann’s wonderful papers][68] Swenson, E. L., Negatively curved groups and related topics, Ph.D. dissertation, Brigham

Young University, 1993.[69] Thurston, W. P., The Geometry and Topology of 3-Manifolds, lecture notes, Princeton Uni-

versity, Princeton, 1979. [Reintroduced hyperbolic geometry to the topologist. Very excitingand difficult.]

[70] Weyl, H., Space—Time—Matter, Dover, New York, 1922. [Weyl’s exposition and develop-ment of relativity and gauge theory which begins at the beginning with motivation, philoso-phy, and elementary developments as well as advanced theory.]

BIBLIOGRAPHY 163

Further Technical References (arranged by chapter)

For the entirety of Volume 2

[71] Newman, M. H. A., Elements of the Topology of Plane Sets of Points, Cambridge UniversityPress, 1939.[A good alternative introduction to the topology of the plane.]

- Volume 1, Chapter 1

[72] Feynman, R., The Character of Physical Law, The M.I.T. Press, 1989, p. 47.[All ofFeynman’s writing is fun and thought provoking.]


[73] Gilbert, W. J., and Vanstone, S. A., An Introduction to Mathematical Thinking, PearsonPrentice Hall, 2005. [The place where I learned the algorithmic calculations about theEuclidean algorithm. See our Chapter 2.]



[74] Reid, C., Hilbert, Springer Verlag, 1970. [A wonderful biography of Hilbert, with an extendeddiscussion of the Hilbert address in which he stated the Hilbert problems. See our Chapter4.]


[75] Apostol , T. M., Calculus , Volume 1, Blaisdell Publishing Company, New York, 1961. [Theplace where I first learned areas by counting. See our Chapter 5.]


[76] Hilton, P., and Pedersen, J., Approximating any regular polygon by folding paper, Math.Mag. 56 (1983), 141-155. [Method for approximating many angles algorithmically by paper-folding.]

[77] Hilton, P., and Pedersen, J., Folding regular star polygons and number theory Math. Intel-ligencer 7 (1985), 15-26. [More paper-folding.]

[78] Burkard Polster, Variations on a Theme in Paper Folding, Amer. Math. Monthly 111 (2004),39-47. [More paper-folding approximations to angles. See Chapter 6 and the impossibility oftrisecting an angle.]

164 BIBLIOGRAPHY


[79] Wagon, S., The Banach-Tarski Paradox, Cambridge University Press, 1994.[A wonderfulexposition of the Hausdorff-Banach-Tarski paradox, without the emphasis on the graph ofthe free group. See our Chapter 7.]

- Volume 1, Chapter 7; Volume 3, Chapter 1.

[80] Coxeter, H. S. M., and Moser, W. O., Generators and Relations for Discrete Groups, secondedition, Springer-Verlag, 1964. [The place where I learned that groups can be viewed as graphs(the Cayley graph or the Dehn Gruppenbild). See our Chapter 7 where we use the graph ofthe free group on two generators and Chapter 25 where we use graphs as approximationsto non Euclidean geometry.]


[81] Peano, G. , Sur une courbe, qui remplit toute une aire plane, Mathematische Annalen 36(1), 1890, pp. 157-160. [The first space-filling curve, described algebraically. See our Chapter12.]

[82] Peano, G., Selected works of Giuseppe Peano, edited by Kennedy, Hubert C., and translated.With a biographical sketch and bibliography, Allen & Unwin, London, 1973.

[83] Hilbert, D., Uber die stetige Abbildung einer Line auf ein Flachenstuck, MathematischeAnnalen 38 (3), 1891, pp. 459-460. [Hilbert gave the first pictures of a space-filling curve.See our Chapter 12.]

[84] G. Polya, Uber eine Peanosche Kurve, Bull. Acad. Sci. Cracovie, A, 1913, pp. 305-313.[Polya’s triangle-filling curve. See our Chapter 12.]

[85] Lax, P. D., The differentiability of Polya’s function, Adv. Math., 10, 1973, pp. 456-464.[Lax recommends the non-isosceles triangle in Polya’s construction since it simplifies thedescription of the path followed to the point represented by a binary expansion. See ourChapter 12.]


[86] Mandelbrot, B., The Fractal Geometry of Nature, W. H . Freeman & Co, 1982. [Mandelbrotsuggests the use of Hausdorff dimension as a means of recognizing sets that are locallycomplicated or chaotic. He defines these to be fractals. See our Chapter 13.]

[87] Falconer, K. J., The Geometry of Fractal Sets, Cambridge University Press, 1985. [Seereference [86] and our Chapter 13.]

[88] Devaney, R. L., Differential Equations, Dynamical Systems, and an Introduction to Chaoswith Morris Hirsch and Stephen Smale, 2nd edition, Academic Press, 2004; 3rd edition,

Academic Press, 2013. [See reference [84] and our Chapter 13.]

- Volume 2, Chapter 8 and 11

[89] Moore, R. L., Concerning upper semi-continuous collections of continua, Trans. Amer.Math. Sc. 27 (1925), pp. 416-428. [Moore shows that his topological characterization of theplane or 2-sphere allows him to prove his theorem about decompositions of the 2-sphere. Seeour Volume 2, Chapters 8 and 11.]

BIBLIOGRAPHY 165

[90] Wilder, R. L., Topology of Manifolds, American Mathematical Society, 1949 . [Our proof ofthe topological characterization of the sphere is primarily modelled on Wilder’s proof, withwhat we consider to be conceptual simplifications. See our Chapter 8.]


[91] Rado, T., Uber den Begriff der Riemannschen Flache, Acts. Litt. Sci. Szeged 2 (1925), pp.101-121. [The first proof that 2-manifolds can be triangulated. See our Chapter 20.]


[92] Andrews, Peter, The classification of surfaces, Amer. Math. Monthly 95 (1988), 861-867l[93] Armstrong, M. A., Basic Topology, McGraw-Hill, London, 1979.[94] Burgess, C. E., Classification of surfaces, Amer. Math. Monthly 92 (1985), 349-354.[95] Francis, George K., Weeks, Jeffrey R., Conway’s ZIP proof, Amer. Math. Monthly 106

(1999), 393-399.


[96] Rolfsen, D., Knots and Links, AMS Chelsea, vol 346, 2003. [See our Chapter 22.]

For the entirety of Volume 3, see the references above taken from our article inFlavors of Geometry, beginning with reference [41].


[97] Misner, C. W., and Thorne, K. S., and Wheeler, J. A., Gravitation, W. H. Freeman andCompany, 1973.

- Volume 3, Chapters 4 and 5

[98] Abelson, H., and diSessa, A., Turtle Geometry, MIT Press, 1986. [The authors use the pathsof a computer turtle to model straight paths on a curved surface.]

For additional informationand updates on this book, visit

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This is the second of a three volume collection devoted to the geometry, topology, and curvature of 2-dimensional spaces. The collection provides a guided tour through a wide range of topics by one of the twentieth century’s masters of geometric topology. The books are accessible to college and graduate students and provide perspective and insight to mathematicians at all levels who are interested in geometry and topology.

The second volume deals with the topology of 2-dimensional spaces. The attempts encountered in Volume 1 to understand length and area in the plane lead to examples most easily described by the methods of topology (fluid geometry): finite curves of infinite length, 1-dimensional curves of positive area, space-filling curves (Peano curves), 0-dimensional subsets of the plane through which no straight path can pass (Cantor sets), etc. Volume 2 describes such sets. All of the standard topological results about 2-dimensional spaces are then proved, such as the Fundamental Theorem of Algebra (two proofs), the No Retraction Theorem, the Brouwer Fixed Point Theorem, the Jordan Curve Theorem, the Open Mapping Theorem, the Riemann-Hurwitz Theorem, and the Classification Theorem for Compact 2-manifolds. Volume 2 also includes a number of theorems usually assumed without proof since their proofs are not readily available, for example, the Zippin Characterization Theorem for 2-dimensional spaces that are locally Euclidean, the Schoenflies Theorem characterizing the disk, the Triangulation Theorem for 2-manifolds, and the R. L. Moore’s Decomposition Theorem so useful in understanding fractal sets.

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