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Russian Math. Surveys 31:2 (1976), 71-138From Uspekhi Mat. Nauk 31:2 (1976), 69-134

HERMITIAN K-THEORY. THE THEORYOF CHARACTERISTIC CLASSES AND

METHODS OF FUNCTIONAL ANALYSIS

A. S. Mishchenko

This paper gives a survey of results on Hermitian ^-theory over the last ten years. The mainemphasis is on the computation of the numerical invariants of Hermitian forms with the help of therepresentation theory of discrete groups and by signature formulae on smooth multiply-connectedmanifolds.

In the first chapter we introduce the basic concepts of Hermitian A'-theory. In particular, wediscuss the periodicity property, the Bass-Novikov projections and new aids to the study of ΛΓ-theoryby means of representation spaces. In the second chapter we discuss the representation theory methodfor finding invariants of Hermitian forms. In §5 we examine a new class of infinite-dimensionalFredholm representations of discrete groups. The third chapter is concerned with signature formulaeon smooth manifolds and with various problems of differential topology in which the signatureformulae find application.

Contents

Introduction 72Chapter 1. General concepts of algebraic and Hermitian ΑΓ-theory . . 75

§ 1. Elementary concepts of A-theory 75§ 2. Periodicity in Hermitian A'-theory 83§ 3. Algebraic and Hermitian ΛΓ-theory from the point of view of

homology theory 87Chapter 2. Hermitian ΛΓ-theory and representation theory 89

§ 4. Finite-dimensional representations 89§ 5. Infinite-dimensional Fredholm representations 97§ 6. Fredholm representations and bundles over classifying spaces . 104

Chapter 3. Characteristic classes of smooth manifolds and HermitianAT-theory 112

§ 7. Hirzebruch formulae for finite-dimensional representations . 113§ 8. Algebraic Poincare complexes 117§ 9. Hirzebruch formulae for infinite-dimensional Fredholm

representations 126§10. Other results on the application of Hermitian ΛΓ-theory . . . 129

References 134

71

72 A.S. Mishchenko

Introduction

One of the classical problems of algebra is that of classifying Hermitian(skew-Hermitian) quadratic forms and their automorphisms. Hermitian K-theory, the basic concepts of which were first worked out in [14], studiesunimodular quadratic (Hermitian and skew-Hermitian) forms and their auto-morphisms over an arbitrary ring Λ with involution up to "stable" equiva-lence, which is somewhat weaker than ordinary isomorphism (see § § 1, 2).Unimodularity means that the matrix of the quadratic (Hermitian or skew-Hermitian) form has an inverse. Λ-modules are supposed to be free orprojective. The stable equivalence classes of forms over a given ring Λ withinvolution form a group with respect to direct sum, which is denoted byK%(A) in the Hermitian case and by KSQ(A) in the skew-Hermitian case. Inclassical number theory one considers integral or rational forms, withΛ = Ζ or Q, forms over the ring of integral elements of some algebraicnumber field or over such a field itself. The groups of equivalence classesof forms over rings of numbers were first studied by Witt and are knownas Witt groups.

For a long time now, many authors have used number-theoretic resultson integral quadratic forms in the theory of smooth simply-connected mani-folds (Milnor [48], Rokhlin [79] and others). In 1966, in papers byNovikov and Wall ([52], [13]) it was shown that the problems of modify-ing even-dimensional multiply-connected manifolds with fundamental groupπ (technically very important in the method of classifying manifolds due toNovikov and Browder ([51])) leads to the need to compute the groupsK'Q(A) and KSQ(A), the analogues of the Witt group for the group ringΛ = Ζ[π], whose involution has the form g -»· o(g)g~l, where α: π -> {±l}is the "orientation automorphism". In its most general form, this result wasobtained by Wall [4]. Novikov [52] has examined the fundamental caseof the free Abelian group π = Z s, which arose in the proof of thetopological invariance of the Pontryagin in classes; he also pointed to thedeep connection between an algebraic problem concerning Hermitian Λ-forms and the problem of classifying all homotopically invariant expressionsin the Pontryagin—Hirzebruch classes (see also the survey [78]; its firstversion was a preprint for the International Congress of Mathematicians atMoscow in 1966; it was published later, with some additions, in 1970). In1968 Wall showed that the problem of modifying odd-dimensional manifoldsrequired the study of classes of "stably equivalent" automorphisms ofHermitian (skew-Hermitian) forms over the group ring Λ = Ζ[π] of thefundamental group π of the original manifold (for precise definitions, see§1). This work was published in detail in 1970 ([78]). The classes ofstable equivalence of automorphisms of Hermitian (skew-Hermitian) formsconstitute commutative groups with respect to direct sums, which aredenoted by K\(A) and Kf(A). Wall in his papers calls all these objects

Hermitian K-theory. The theory of characteristic classes and methods of functional analysis 73

"I-groups", where L0(ir) = **(Λ), ^ ( π ) = Α^(Λ), Ι 2 ( π ) = Af(A), L3(v) = Af (Λ).'For group rings Λ = Ζ[π] these groups have been given the name of the"Wall groups" of π. In 1968—69 Shaneson in [5], developing an idea ofBrowder [57], computed from purely geometric considerations these Wallgroups for the case, very important in topology, of the free Abelian groupπ = Zs (for the idea, see §10). However, Shaneson's result was non-effective:it remained completely unclear how one could find by algebraic operationsfrom a given Hermitian form the complete system of its stable invariants.Finally, in 1969 in [13], the present author and Gel'fand used transition tothe character group to study Hermitian forms over the group ring Ζ[π],where π = Ζ", reducing their study to that of quadratic forms over thering C*(Tn) of functions on the torus.1 In this paper, it was shown thatthe Hermitian A"o-groups over the ring Λ = C(X) of complex-valuedfunction coincide with the ordinary groups K°(X) in homotopy theory ofthe classes of stable equivalence of vector bundles (see §4). The connectionbetween Hermitian forms and ordinary ΑΓ-theory, constructed from vectorbundles, for the ring of functions C*(X), and also differential-topologicalarguments for the group ring Ζ[π], led to the idea that it was necessaryto establish for all rings Λ with involution a single homological "HermitianΑΓ-theory" Kh

n(A), possessing the property of "Bott periodicity" if 2-torsionis neglected. From the point of view of Hermitian A'-theory, periodicitymeans that the skew-Hermitian (Hermitian) groups KSQ(\) should coincide,in reverse order, with the Hermitian (skew-Hermitian) groups AT^A). Theproblem of constructing such a theory was first solved by Novikov in 1970,using ideas of the Hamiltonian formalism (see [14]), for the groupKh ® ZfVi] (see also §2 of this survey). Novikov proved homologyproperties of Hermitian AT-theory and established a link between ATQ andΚψ, clearly constructing analogues of the so-called Bass projections, whichare similar to the suspension homomorphism in homotopy theory.

In particular, for the group π = Zs, these Bass—Novikov projectionsenable us to describe in an algebraically effective way the invariants ofHermitian (skew-Hermitian) forms and automorphisms. A precise constructionof Hermitian ΑΓ-theory, with the 2-torsion taken into account, based on theideas of [14], was given by Ranicki in [65] and [66]. Sharpe [80]developed Milnor's approach to define and study the group ΑΓ2 (A).Using ideas of Quillen [67], Karoubi later developed in [73], [74], [75]a homotopy approach to the construction and proof of the 4-periodicityof the Hermitian Kh ® Z[Vi\-theory (see §3).

Quadratic forms over the ring Λ = Ζ[π], where π is a free Abelian group (a lattice in R") arise in thetheory of a solid body as the square matrix of interacting separate parts (atoms) of the lattice, η = 1, 2, 3.In this theory Λ-forms always lead to forms over the ring of functions C*{Tn), using characters; but thereis interest in the classification relative to orthogonal transformations or in eigenvalues as functions of apoint of the torus (phonon spectra).

74 A. S. Mishchenko

We have already mentioned above the connection between the Hermitian^-groups for the group ring Λ = Ζ[π] of an infinite group π and integralsof the rational Pontryagin—Hirzebruch classes over certain cycles. A con-nection between the ordinary signature τ(Λ/4*) of the integral quadratic formof the intersection of cycles on H2k(M4k) and characteristic classes wasfound by Rokhlin, Thorn, and Hirzebruch back in the early '50's (see[36]-[38]) . This link is called the "Hirzebruch formula" and takes theform (Lk(Mn), [Mn]> = r(Mr), where Lk is the Hirzebruch polynomial of thePontryagin classes. Thanks to this formula, for example, were discoveredthe smooth structures on spheres (Milnor [49]). In the middle of the 60'sanalogues of this formula were discovered in the theory of multiply-connected manifolds, when the homotopy invariance of the integrals of thePontryagin—Hirzebruch classes along special cycles (intersections of cycles)of codimension 1) were established by Novikov in a number of specialcases in 1965 ([51], [52]) and were then completely proved in papers byRokhlin, Kasparov, Hsiang. and Farrell ( [53]-[55]) . By 1970 it wascompletely clear that the problem of finding all "multiply connectedanalogues of the Hirzebruch formula" or all homotopically invariantrelations on the rational Pontryagin classes of multiply-connected manifoldsreduce completely to the problem of Hermitian Κ -theory. The author (see[46] and §8 of this survey) found a new homotopy invariant of multiply-connected manifolds, which generalizes the ordinary signature: to each closedmanifold M" there corresponds an element of Hermitian Κ* <8> Ζ[Vi] -theory,which is invariant also with respect to the bordisms Ω* £(ττ, 1) that preservethe fundamental group. Thus, there arises a homomorphism φ:Ω*(Χ"(π, 1)) -*• K*(A) <8> Z[Vi]. All homotopically invariant expressions inthe Pontryagin classes reduce to purely algebraic invariants on the Hermitian^-groups ΚΪ(Α), where Λ = Ζ[π]. If [Mn] G « „ ( £ ( * , 1)) is the bordismclass of the manifold in question, then according to a conjecture ofNovikov ([52], [78]) there must exist a purely algebraic homomorphism("Chern character") σ: ΚΊ(λ) ® Z[Vi] -»• H*(ir; Q), A = Ζ[ττ], and a"generalized Hirzebruch formula" {σ°φ[Μη\, χ>= (Lk(Mn),D\jj*(x)),where χ ΕΗ*(π; Q), π = π^Μ), ψ: Μη -»• Κ(π, 1) is the natural map, Lk theHirzebruch polynomial, and D the Poincare duality operator.

However, for non-Abelian infinite groups π there were no such methodsfor finding invariants of Hermitian Α-forms or indeed methods of definingthe homomorphism σ. In an important paper [26], Lusztig, starting fromthe methods of Atiyah-Singer and using a continuous analogue of aconstruction of the author [46], developed a new analytical approach tothese problems: he worked with the finite-dimensional representations of πand the families of elliptic operators (or elliptic complexes) associated withthem. In particular, he gave a new proof of the theorem already mentionedof the homotopy invariance of the integrals of the rational Pontryagin—Hirzebruch classes along intersections of cycles of codimension 1 and

Hermitian K-theory. The theory of characteristic classes and methods of functional anaylsis 75

obtained some particular results for individual cycles, dual to the cohomologycycles of a non-commutative fundamental group π, provided that it can beregarded as a discrete subgroup of a semisimple Lie group (§7).

The author, in [16] and [17] has applied methods of functional analysisand the theory of infinite-dimensional representations to the problem ofcomputing the Hermitian Λ^-groups. In particular, the "Fredholm represent-ations" studied in [17] enable us to find a number of new invariants ofHermitian forms over the group ring (see §6) and for a wide class ofgroups to give a complete classification of homotopically invariant express-ions in the rational characteristic classes (see §9). To the class of groupsstudied in [17] there belong the fundamental groups of all manifolds ofnon-positive curvature. For this class of groups π the following theoremis completely proved: 1) The scalar product (L k (M n )x, [Μη ]) is homo-topically invariant for every cohomology class χ that is obtained from theimage of the map Mn -> Κ(π, 1); here Lk(Mn) is the Hirzebruch polynomialof the Pontryagin classes and Mn is a closed manifold. 2) For a manifoldwith fundamental group π there are no other homotopically invariantexpressions in the rational Pontryagin classes.

Note that the Fredholm representations of the ring of functions C*(X)were used by Kasparov, realizing an idea of Atiyah [18], for an explicitconstruction of the ΑΓ-theory of homologies (see [20] and §5).

Very recently Solov'ev has studied discrete groups of algebraic groupsover a locally compact local field. An example of such a field is the fieldof /?-adic numbers. For these algebraic groups, Tits and Bruhat have proposedthe construction of a contractible complex, the so-called Bruhat—Titsstructure, on which the algebraic group acts. The factor space by the actionof a discrete subgroup without elements of finite order is an Eilenberg—MacLane space and has a number of remarkable properties. In particular,for such complexes we can construct analogues of the de Rham complexof exterior differential forms and produce an ample set of Fredholmrepresentations, which allow us to establish the homotopy invariance ofexpressions in the Pontryagin-Hirzebruch classes, provided that the funda-mental groups are isomorphic to discrete subgroups of algebraic groups.

In conclusion we note that for integral group rings of a finite group πa number of strong results have recently been obtained [81], [82] concern-ing the computation of Hermitian ΑΓ-groups, which we do not discuss inthe survey.

CHAPTER 1

GENERAL CONCEPTS OF ALGEBRAIC AND HERMITIAN K-THEORY

§ 1. Elementary concepts of AT-theory

There are many monographs on stable algebra, studying the topic from a

76 Λ. S. Mishchenko

purely algebraic point of view. We mention here the book by Bass [ 1 ] andthe lectures of Swan [2].

1. Modules over a ring Λ. The basic objects of our study will be modulesover some associative ring with unity. As a rule, we shall denote the ring byΛ. In most cases the base ring Λ will be furnished with a supplementarystructure — an involutory antiautomorphism, that is, a linear map*: Λ -*• A such that for any λ, X t, λ2 €Ε Λ,

(1.1) (λχλ2)* = λ*λ?, (λ*)* = λ.

A ring with involution arises in a natural way when one studies thehomotopy properties of multiply-connected topological spaces, in particular,smooth or piecewise linear manifolds, topological or homology manifolds.If X is a topological space and -n = Κχ(Χ) its fundamental group, then inmany problems one has to study the homology of X with coefficients inthe group ring Λ = Z[7r], taken as a local system of coefficients. Thegroup ring Λ = Ζ[π] of π has a natural involution satisfying (1.1) anduniquely determined by the condition g* = g~l (g £ π). Thus, let A be anarbitrary associative ring with involution, and Μ a finitely-generated leftΛ-module. We denote by M* the group of Λ-homomorphisms of theΑ-module Μ to A: M* = HomA(M, A). The group M* is equipped withthe structure of a left Λ-module by setting (λα) (m) = a(m)X*(a £ Μ*, m G Μ, λ G A). If β: Μ -*• Ν is a Λ-homomorphism of Λ-modules, then we denote by β* the homomorphism β*: Ν* ->· Μ*, givenby β*(α)(ηι) = α(β{ηί)) (m ε Μ, α ε TV*). This β* is a Λ-homomorphismof Λ-modules.

Although in the present section we do not consider systematicallygraded modules, we give here all the same the relevant definitions forgraded objects. By a graded Λ-module Μ we mean a left Λ-module M,together with a fixed decomposition of Μ as a direct sum of Λ-modules

Μ = ® Mk. If m ε Mk, then m is said to be homogeneous of degree k,k

deg m = k. Let TV be a second graded Α-module, Ν = ® Nk andk

α: Μ •+ Ν a Λ-homomorphism. Then a is said to be homogeneous ofdegree t if <x(Mk) C Nk + t. In that case we write deg α = t. In particular,if Μ is a graded Λ-module, then M* splits into the direct sum Μ* - ®ΜΪ,

k

where Λί| consists of all the homogeneous homomorphisms of degree (—k)from Μ to the trivially graded ring A, that is, those homomorphismsα: Μ -> A such that a{Mt) = 0 for / Φ k. Thus, the Λ-module M* is turnedin a natural way into a graded Α-module M* = © (Mf), where (M*);· = (M_j)*.

For graded Λ-modules the concept of dual homomorphism has to be adjusted.In fact, if β: Μ ^ Ν is a homogeneous homomorphism of graded Λ-modules,then by β* we denote the homomorphism β*: Ν* -*• Μ*, given by


This (3* is a homogeneous homomorphism, and

deg β* = deg β.

The operation of duality of Λ-modules is not involutory for arbitraryΛ-modules, but there is a natural homomorphism of Λ-modulesεΜ: Μ -> (Μ*)*, which is defined on homogeneous elements by the formula

*, me Μ).

The homomorphism εΜ is homogeneous and deg εΜ = 0.If Μ is a projective graded Λ-module, then εΜ is an isomorphism.2. The algebraic A"-functor. We turn now to the description of concepts

relating to algebraic and Hermitian K-theory. Suppose, as before, that Λis an associative ring, and let BM be the set of all finitely generated projectiveΛ-modules. By K0(A) we denote the Grothendieck group generated by theset Jl with the operation of direct sum of modules, in which the relation[M] = 0 is introduced for any free Λ-module Μ G &M.

Suppose next that Γ is the set of automorphisms of free finitelygenerated Λ-modules. We define the group ΑΊ (Λ) to be the factor groupof the free Abelian group generated by Γ factored by the followingrelations:

a) if (Mx, 71) and (M2, γ 2 ) are two automorphisms of Γ, then

(1.2) [Mlt Y l ] + IM» γ2] - l(M1 θ M2), (V l θ γ2)] = 0,

b) if (M, yl) and (M, y2) a r e two automorphisms of Γ for one and the

same module M, then(1.3) [M, V l ] + [Μ, γ2] _ [Μ, T lv2] = 0.

An automorphism 7 is said to be simple if [7] = 0 in ATj(A).The groups K0(A) and Kl(A) defined above are one of the basic objects

of study of algebraic ΑΓ-theory. On the other hand, these groups turn upin various problems of algebraic topology, as the domain of values ofimportant homotopic and topological invariants of spaces and manifolds.The most important example of such an invariant is the Whitehead torsionfor the homotopic equivalence of two chain complexes.

«n our exposition the group ^ 0 ( Λ ) and K1(A) will play an auxiliary rolein the development of Hermitian .^-theory, and we now turn to the studyof its concepts.

3. The Hermitian A -functor. Suppose again that we are given an assoc-iative ring Λ with involution. We consider a free finitely generated Λ-moduleM. Let a: M* ->• Μ be an isomorphism of Λ-modules satisfying the relation

(1.4) ε^α* = (-1)*α

78 A. S. Mishchenko

(as we have defined the homomorphism eM for graded modules, we supposehere that Μ has the null grading).

When Κ is even, we say that we are given a Hermitian form a and whenk is odd, a skew-Hermitian form a. Thus, a (skew-)Hermitian form over Λis a pair (M, a) consisting of a free Λ-module Μ and an isomorphism αsatisfying (1.4). For brevity we speak henceforth of Hermitian forms,distinguishing Hermitian and skew-Hermitian forms only when confusionwould otherwise arise. Two Hermitian forms (Mx, a.x) and (M2, a 2 ) aresaid to be equivalent if there is an isomorphism )3: Ml ->• M2 such that<x2 = βοίχβ*, that is, such that the following diagram is commutative:

(1.5) a i t ,U-

With each Hermitian form (M, a) we can associate a matrix with entriesin Λ. To do this we fix in Μ a free basis (ml, . . . , ms) over Λ. Let(mf, . . . , m*) be the dual basis for M*, so that mfirrij) = δ,·.-, whereδ;;· = 1 if / = / and δ,·.- = 0 if i Φ j . Then the homomorphism a: M* -*• Μdetermines the matrix A consisting of elements of Λ whose rows are thecoordinate elements of a.{mf) in the basis (mx, . . . , ms):

(1.6) a ( m f ) = 2 A o · ^ .3 = 1

The condition (1.4) in terms of A = || λ/;|| then takes the following form:

(1.7) λ& = (-1)%, .

If (M1, ax) and (M2, <*2) are two Hermitian forms and Ax and A2 theirmatrices with respect to the bases, then the equivalence condition (1.5) canbe stated in the following way: there exists an invertible matrix Β = \\ μ;/·||with coefficients in Λ such that

(1.8) Ax = B*A2B,

when by B* = || i H we mean the matrix for which i>,y = μ; .The set of equivalence classes of Hermitian forms over Λ will be denoted

by oMh- This set splits into disjoint subsets according to the number ofgenerators of the free module on which the Hermitian form is defined, or,what comes to the same thing, the order of the matrix corresponding tothe Hermitian form. We introduce on oilh the operation of the direct sumof two Hermitian forms (M1, a,) and (M2, a2): we set

(1.9) Μ = Ml θ M2, a = αχ θ α2.

Thus, the direct sum of the two Hermitian forms (denoted by (Mx, ax) ®(M2, a2)) is the pair {M, oi) determined by (1.9). To equivalent Hermitianforms there correspond, according to (1.9), equivalent direct sums, so that


the operation of direct sum is well-defined on a/Mh.Now we are in the position to define the group K^k{A), which we call

the Hermitian ΑΓ-functor of Λ. By definition, Kh

2k{A) is the factor group ofthe free Abelian group generated by the set e#h of classes of equivalentHermitian forms, with the following relations:

(1.10) lMlt a j + IMS, a,] - l(Mlt ax) θ (M2, a2)] = 0,

(1.11) [M, a] = 0

for the Hermitian form (M, a.) whose matrix A is

To define the groups Kh

2k + l{I\.) we proceed as in algebraic AVtheory.Before defining Kh

2k+l(K) we note that according to (1.11) the Hermitianform (1.12) and also the direct sum of any number of copies of the form(1.12) determine the trivial element in Kh

2k(A). We call the Hermitian form(M, a) Hamiltonian if Μ has a basis (mx, . . . , ms, hx, . . . , hs) in whichthe matrix A of a has the form

) '

where Ε is the unit matrix of order s.Let (M, a) be a Hamiltonian form. We consider an isomorphism

7: Μ -*• Μ preserving the form a, that is, such that

(1.14) α = γαγ*.

Isomorphisms satisfying (1.14) are called Hamiltonian transformations of{Μ, α). Thus, as in the case of algebraic /^-theory, we can define the directsum of Hamiltonian transformations (Ml, α 1 ? yx) and (M2, «2» Ύ2) bysetting

(1.15) (Mlt a l t yj Θ (M2, a 2, ya) = ((M1 θ Μ2), Κ Θ a,), (T l θ γ2)).

Let Fh be the set of Hamiltonian transformations of Hamiltonian forms.The group Kh

2k + i{\) is defined as the factor group of the free Abeliangroup generated by Yh, with the following relations:

a) if {Μγ, OLX, 7,), (M2, a 2 , 72) are two Hamiltonian transformations,then

(1.16) [Mi, a l 5 T l ] + [M s, ο,, γ2] - l{Mlt otj, γχ) θ (Μ2, α,, γ,)] = 0,

b) if (M, a, 7 t ) , (M, a, y2) a r e two Hamiltonian transformations of one

and the same Hamiltonian form (M, a), then

(1.17) [M, a, vJ + [M, a, y2) - [Μ, α, γ ι Τ ϊ ] = 0,

<> [(ο ,.0-)]™»·4. Other versions of Hermitian ΑΓ-theory. We have introduced the simplest

80 A. S. Mishchenko

version of ίΓ-theory related to Hermitian forms over a ring Λ. In fact,however, in various problems we have to consider several modifications ofthe groups ^ ( A ) that are better adapted to one problem or another.

We introduce in this subsection the most important versions of HermitianA"-theory, describing their domain of applicability on the one hand andtheir interconnections on the other.

a) Historically, the first groups of Hermitian A"-theory appeared indifferential topology in the description in algebraically invariant terms ofthe obstructions to the possibility of some modifications or other of smoothmanifolds. The corresponding groups were denoted by Lk{\) and called theWall groups (after the author who first studied them — see [3], or, for adetailed exposition, the book [4]).

A Hermitian form (M, a.) on a free Λ-module Μ can be regarded as abilinear function

(1.18) β("Ϊ!, m2) = «"'KJfmj) (?%, m2 6 M).

The function β takes its values in Λ. The condition (1.17) for a to be(skew-) Hermitian implies for β that

(1.19) β Κ , m2) = (_1)*(β(τη2, m,))* K , m2 ζ Μ).

Such bilinear functions arise in smooth multiply-connected manifolds asintersection indices of cycles realized by embedded submanifolds. The inter-section indices of cycles are closely connected with the indices of self-intersection of embedded submanifolds, and the algebraic expression of thisconnection is embodied in the definition of the Wall groups.

For let Λ' denote the group

(1.20) Λ' = Λ/{ν - (-l) f tv*: ν £ Λ}.

We consider a function y: Μ -*• Λ' satisfying the following conditions:

(1.21) γ(?ηχ + m2) = γ(?%) + γ(^ 2 ) + β ^ , m2),

(1.22) f>(m, m) = y{m) + (-i)*(y(m))*,

(1.23) y(lm) = λγ(τ/ι)λ* (λ ζ Λ).

All these equations are to be understood in the following way: we have tochoose representatives in the coset y(m), carry out all operations on therepresentative in A, and then associate with it the corresponding coset.

A collection of functions (0, 7) satisfying (1.19) and (1.21)—(1.23), andsuch that the homomorphism α in (1.18) is a Hermitian form, is called aquadratic form over A. Quadratic forms also admit the operation of directsums. For if (Mlt βι, γ , ) and (M2, |32, 72) are two quadratic forms, thenwe set Μ = Mx © M2, β = βχ ® βι and we define 7 by the relation (1.21),where we suppose that 7 coincides with 7,· (/ = 1, 2) on M(.

A Hamiltonian quadratic form is a direct sum of forms (Μ, β, y) of thefollowing kind: the Λ-module Μ has two generators, the matrix A of the


bilinear form β is

A - l ° Μ

with respect to the basis (m1, m2), and y is determined with respect tothis basis by the formulae y{mx) = y(m2) = 0. Thus, we define the groupL2k(A) on the set of quadratic forms as the factor group of the freeAbelian group generated by the quadratic forms, factored by relationsanalogous to those for ^ ^ ( Λ ) .

The groups L2k+l(A) are defined with the help of automorphisms of freeΛ-modules preserving the pair of functions (β, γ), which specify a Hamilton-ian quadratic form.

b) The second version of the Wall group is bound up with the study ofobstructions to the construction of maps of smooth manifolds up to simplehomotopy equivalence. In this case it is supposed that every free Λ-moduleis furnished with a class of equivalent bases, a pair of bases for Μ beingsaid to be equivalent if the transition from one to the other is determinedby a simple automorphism α: Μ -> Μ, that is, [a] = 0 in the group K^A).

Then we distinguish the class of those quadratic forms (Μ, β, y) for whichthe bilinear form β is determined by a simple isomorphism a: M* -*· Μ inany of the equivalent bases of Μ and the corresponding dual basis for M*.Correspondingly, two quadratic forms (M 1 ; |3j, 7 t ) and (M2, β2, y2) aretaken to be equivalent if there is a simple isomorphism δ: Μχ -*• M2

sending one quadratic form into the other.

The class of automorphisms preserving Hamiltonian quadratic forms canalso be restricted to the subclass consisting of only simple automorphisms.

The corresponding analogues of Hermitian .^-theory are also called theWall groups and denoted by Ls

k(A).c) In these three versions of Hermitian X-theory we have observed a

number of restrictions in the definitions. In the first place, we have fixedthe class of Λ-modules on which the forms are considered. Up to now thisclass has been either the free Λ-modules or the free Λ-modules with abasis. Secondly, we have fixed the class of forms to be considered. We havethe cases of Hermitian forms, quadratic forms, and simple quadratic forms.Thirdly, and finally, we have fixed an equivalence relation between forms,that is, we have defined which pairs of forms are to be regarded asequivalent and which forms as equivalent to zero.

Analogous restrictions have been observed in defining the classes ofautomorphisms preserving a Hamiltonian form.

Other versions of Hermitian A"-theory have also been treated in the liter-ature. We mention the most important of these, indicating the class ofmodules and of forms and the equivalence relation in each of these versions.A useful class of Λ-modules to consider in a number of problems (see §2)is the class of projective Λ-modules. The forms are then the so-called (simple)

82 A. S. Mishchenko

Hermitian forms, that is, those bilinear functions β on Μ of the kind (1.18)for which there exists a function 7 satisfying (1.21)—(1.23). However, thefunction 7 itself is not fixed and is not considered. We denote the corres-ponding groups by Kh

k{A) (Ks

k(A)). Finally, we can take as trivial the forms(M, a), where Μ = Ν θ iV* and the morphism

is written in matrix form as

Such forms are called projective-Hamiltonian.5. Relations among the different versions of Hermitian K-theoty. Despite

the abundance of versions of Hermitian .K-theory, the differences betweenthem are not very important. Roughly speaking, if we ignore in HermitianΛΓ-theory elements of order two, then it seems that all the versions are iso-morphic to each other. As an illustration we consider the previously defined^-functors Kh

k{A), Kh

k(\), Ks

k(A), Lk(A) and Ζ,'(Λ). It is quite clear thatthe groups Lk(A) and Kh

k{A) can be obtained from the form definingLs

k(A) by "forgetting" the fine structure and going over to a coarserstructure.

Thus, we have natural homomorphisms

Li{A)-+Lk(A) + KUA)\ S \

KUA)-+KUA)These homomorphisms connecting all these groups, with the exception,perhaps, of Kh

k{A), have a common property: the kernel and cokernel ofeach of the homomorphism consists of elements of order a power of 2.One can give more precise estimates on the orders of elements in the kerneland cokernel. In this connection see, for example, [5], [6], [7]. Thus,the map Ls

k(A) -» Lk(A) can be included ([5]) in an exact sequence ofgroups

...->Wh (Λ) -+ LI (Λ) — Lh (A) -> Wk.t (A) -+ . ..

The groups Wk(A) are effectively described in terms of ΑΊ(Λ) and consistof elements of order 2.

We make now a remark that is important later on. If Λ contains thenumber -|·, then the groups Lk(A), K^{A), and Α"£(Λ) simply coincide. Infact, these three groups differ from one another in that we consider inplace of the Hermitian form a the bilinear form (3, with a function γsatisfying (1.21)—(1.23) added. We show that if y G Λ, then 7 is uniquelydetermined by the properties (1.21)—(1.23). For, let β be an arbitrarybilinear form satisfying (1.19). Then we set

Hermitian K-theory. The theory of characteristic classes and methods of functional analysis

(1.24) 7(m) = !^(i»,i») (i

It is not difficult to verify that (y(m))* = (-\)ky(m) and that (1.21) (1.23)are satisfied. To see that γ is uniquely determined by (1.21)-(1.23), it issufficient to note that the ring splits into the direct sum of two subgroupsΛ = Λ+ © Λ~, consisting of the self-conjugate and skew-conjugate elements,respectively, while Λ' (1.20) is isomorphic to Λ+ or Λ", according to theparity of k.

If 7 is a function satisfying (1.21)—(1.23), then, for example, for evenk we get

(1.25) y(m) 6 Λ+, β(/η, τη) = 2y(m),

that is, γ is uniquely determined.Thus, when | £ Λ , the set of Hermitian forms oMh coincides with the set

of quadratic forms, that is, the groups Lk(A), K^(A), and K%(A) areisomorphic.

The functional methods to which the present article is dedicated areinsensitive to adjoining to Λ the element j , and therefore, from the pointof view of these methods we cannot distinguish the different versions ofHermitian A'-theory Lk(A), Kh

k(A), Kh

k(A) and Ls

k(A), Ks

k(A), respectively.Later we shall see that from the point of view of functional methods allthese versions of Hermitian A'-theory are altogether indistinguishable.

§2. Periodicity in Hermitian A'-theory

In §1 we have defined the groups Α"£(Λ) so that automatically fork = s (mod 4) the groups Α"£(Λ) and Kh

s(A) are isomorphic. This fictitiousperiodicity has nowhere been justified. As a matter of fact, we are con-cerned here with four different groups Kk(A) (k = 0, 1,2, 3), and thequestion of the interpretation of these groups as a finite segment of some4-periodic sequence of groups remains open. The incentive for consideringalgebraic A'-theory as a 4-periodic theory is founded on the remarkableproperties of topological ΑΓ-theory discovered by Bott.

Let X be a topological space and K(X) the group generated by the locallytrivial vector bundles on X. If the fibre of the bundle is a finite-dimensionalcomplex space and the structure group is the group of unitary transformations,then we denote it by KV{X). In the case of a real vector bundle and thegroup of orthogonal transformations as structure group, we denote it byKo (X). There is also a version of topological A'-theory the group ofsymplectic transformations of a vector space over the field of quaternionsas structure group. In this case the group of bundles is denoted by Ks (X).In all three versions Bott has established a formula which is now known asBott periodicity.

If (X, x0) is a space with base point, then by K°(X, x0) we denote thesubgroup of K(X) generated by the zero-dimensional elements of K(X). We

84 A. S. Mishchenko

denote by SP(X) the p-fold suspension of the space with base point

(X, x0)SpX = (Sp X X)/(X V S*>).

Here Sp is the p-dimensional sphere with base point and the symbol Vdenotes the connected sum of two topological spaces with base points. ThenBott periodicity can be written in the form of the following equalities:

(2.1) Kb{S*X) = Kh(X,x0),

(2.2) Kb(S»X) = Kh{X, x0),

(2.3) K%p(S*X) = K&p{X,x0).

Now we introduce an argument to show that in algebraic ^-theory wecan also speak of periodicity in the sense of (2.1)—(2.3).

It turns out that the groups K(X) of topological /^-theory have a purelyalgebraic description. To see this we consider the ring Λ = C(X) of allcontinuous complex-valued functions on X. Then we can associate witheach complex vector bundle £ over X the Λ-module Μ(ξ) of all continuoussections of £. If X is compact, then M(X) is a finitely-generated projectiveΛ-module. Moreover, the correspondence £ -> M(£) establishes an equivalencebetween the category of complex vector bundles and that of finitelygenerated Λ-modules and, consequently, an isomorphism of the groups KV{X)and K0(A). Generalizing the above argument to the rings of real-valuedfunctions R(X) and of quaternions Q(X), we get natural isomorphisms

KV(X) = K0(C(X)),

K0(X) = K0(R(X)),

KSP(X) = KMX)).

Therefore, we can interpret Bott periodicity solely in algebraic terms. Forthis it is more convenient to replace (2.1), (2.2), and (2.3) by relationsamong the ^-groups for the spaces X and X X Tp, where Tp is the p-dimensional torus, that is,

Tp=Slx ... xS\.

ρ times

In the case, for example, of Kv, Bott periodicity takes the following form.We set Kjj{X, x0) = K$(S'X). Then (2.1) is equivalent to the exactness ofthe sequences:

(2.4) O-+KJ} (X)-^Kh (X X S^-^Kh (X) ->0,

(2.5) 0 - * - ί Κ Ζ ) - + Ζ ϊ Κ Χ χ ί ) ) - > ^ ( Χ ) - ^ 0 ·

In truth, the sequence (2.4) is exact almost by definition, while (2.5) wouldbe exact if we could replace K$(X) by KJ/ (X). The fact that the last twogroups can change places is equivalent to Bott periodicity.

The sequences (2.4) and (2.5) can acquire a real meaning in algebraic


A"-theory. In fact, the ring of continuous functions C(X X S1) contains thesubring of Laurent polynomials C(X)[z, z'1] C C(X X S1). To see this, wehave to represent Sl as the set of complex numbers of norm 1. ThenC(X X Sl) is the completion of C(X)[z, z~l\ with respect to a suitablenorm. The method of going from an arbitrary ring Λ to the ring of Laurentpolynomials Λ[ζ, ζ" 1 ] contains no topological ingredient at all and canserve as an analogue for the passage from X to X X S1. Therefore, in alge-braic ΛΓ-theory one of the essential problems is that of establishing theconnection between the A"-groups for Λ and its Laurent extension A[z, z~1].

The analogous problem in algebraic A"-theory was solved by Bass ([63],[64], see also [1], Ch. 12; [2], 226). Roughly speaking, if Az denotesthe Laurent extension of Λ, then Ki(Az) splits into the direct sum ofthe groups K0(A) and ΑΊ(Λ) and yet another summand Nil (Λ), consistingof the elements of the form (1 + vz±l), where ν is a nilpotent homo-morphism over Λ. In many important examples the group Nil (Λ) is trivial.The decomposition of K1(AZ) into a direct sum is effected with the helpof the Bass projections

Κ,(ΑΖ) -> ^ ( Λ ) -

* K0(A) -

It turns out that there is a similar situation in the case of HermitianA"-theory. The connection between the A'-groups for the rings Λ and Az

was established by Novikov [14]. He constructed projections that decom-pose the Hermitian A'-groups over Az into the direct sum of HermitianA'-groups over Λ. However, all the constructions were carried out in the2-adic localization of the ring and the A'-groups. Later Ranicki ([65], [66])sharpened Novikov's theorem by taking account of the 2-torsion in therings and the A'-groups and constructed projections for "twisted" Laurentbundles. Note that parallel (and rather earlier in time) similar questionswere solved by purely geometric methods (see §10).

The basic idea enabling us to construct projections for HermitianA"-theory appears to lie in a new approach to the question of defining theA'-groups. If we wish to construct Ko and ΑΊ in any category with directsums, and with zero, then we must have:

a) a class of objects containing 0 and closed with respect to direct sums,b) a concept of equivalence between two objects of the class,c) a concept of a means (or process in the terminology of [14]) of

establishing equivalence. In the case of algebraic A'-theory, these objectsare the projective Λ-modules, equivalent objects are equivalent projectiveΛ-modules, and equivalence is established by means of isomorphisms. Thenthe group Α"0(Λ) is constructed as the Grothendieck groups of the categoryof projective Λ-modules factored by the free Λ-modules. The group ΑΊ(Λ)is constructed with the help of all isomorphisms of a free Λ-module. There

86 A. S. Mishchenko

is also a concept of equivalence of isomorphisms (just as, for Ko, there isthe concept of equivalence of projective modules). In exactly the same way,if we wanted to define the next group K2(A), then we should have meansof establishing the equivalence of isomorphisms, and also a concept of theequivalence of such means.

In the case of Hermitian ΛΓ-theory the objects of the basic category areprojective modules together with Hermitian or skew-Hermitian non-degenerateforms on a module.

Two Hermitian forms (M, a) and (M1, a') are taken to be equivalent ifthere is an isomorphism:

(2.6) β: Μ->- Μ',such that a = βαβ*. The trivial objects are the projective-HamiltonianHermitian forms. The classes of equivalent Hermitian forms generate thegroup K*lk(A). To obtain the group Kh

2k + l{A) we choose as objects theisomorphisms of the projective-Hamiltonian form (2.6). For these iso-morphisms we have to define an equivalence relation. Let Μ = Mp © Mx

be a direct sum decomposition of the Λ-module M, in which the

Hamiltonian Hermitian form α has the matrix A = I, n/fc 0 ) · The sub-

modules Μ and Mx are such that a is identically zero on each of them,

while the composite map

is an isomorphism. Such submodules are called Lagrangian planes. EveryHamiltonian transformation (2.6) sends a Lagrangian plane into anotherone. In particular, if j3 is a Hamiltonian transformation, then the imageβ(Μρ) of Μ also is a Lagrangian plane. Moreover, each Lagrangian planeL C Μ ® Mx is the image of some Hamiltonian transformation β:

(2.7) L = β(Μρ).

The transformation β in (2.7) is determined not uniquely, but up to com-position with a Hamiltonian transformation γ leaving Μ invariant. Byanalogy with the Hamiltonian formalism in dynamical systems we say thatthere is an elementary transfer process from β to βγ. Another type ofelementary transfer process from β to a transformation β' consists in adjoin-ing to Μ - Μ ® Μχ two additional coordinates

Μ - ν Μ Θ A(ps+1) θ A(xs+1)

and then changing coordinates by

>2 gx ί Ρ'+1 = Ps+i — Σ

^ 1This operation also has an analogue in the Hamiltonian formalism if thenew coordinate xs+1 is regarded as the time coordinate. One can alsoiterate this, regarding xs+1 as a many-dimensional vector.

Then two Hamiltonian transformations (or, what comes to the samething, two Lagrangian planes) are said to be equivalent if there is a


composition of elementary processes sending the one into the other. Theequivalence classes of transformations form the group K\k+l{K). The nextstage consists in choosing as fundamental objects processes taking one fixedLagrangian plane (say, Μ C M) to another fixed Lagrangian plane L C M.Here L can be chosen so that it projects along Mx onto a direct summand.It turns out that the composition of elementary operations sending Μ toL is completely determined by the matrix ψ in (2.8), which in this case isa non-degenerate matrix of the other sign of symmetry. Consequently, wecan introduce an equivalence relation between two compositions of ele-mentary operations if the corresponding skew-Hermitian forms are equivalent.In this way we succeed in interpreting the groups Kh

Ak+2(A.) from thepoint of view of algebraic ^Γ-theory for Hermitian forms as distinguishingthem in the category of equivalences of elements of K\k + l{A).

The approach we have given has a direct analogue in homotopy theory.For let (X, Y) be the set of continuous maps of a space X to a space Υrespecting base point. Two maps φ and ψ are regarded as equivalent if theyare homotopic: ψ ~ ψ. We denote the classes of homotopic maps by[X, Y]o. Then homotopies between two fixed maps form a new categoryin which we can introduce an equivalence relation (namely, homotopy).We denote the classes of equivalent homotopies by [Χ, Υ] ι. Carrying on,in this way, we can construct sets [X, Y] k for each k > 0. In topology,one can make the remarkable observation that the sets [X, Y]k can beincluded in homology theory (under certain natural restrictions) and thismeans that one can apply the apparatus of homological algebra for com-puting these sets.

As a matter of fact, Hermitian X"-theory has similar properties, whichenable us to carry out computations comparing the Hermitian ^-groupsfor the ring Λ and its Laurent extension Λ ζ. If we ignore torsion, that is,if we tensor all groups with the ring of rational numbers, then we havethe following decomposition:

(2-9) KRK) = KftA) Θ K^(A).

The precise decompositions are more intricate and require that we considerin parallel the three versions of Hermitian ΛΓ-theory associated with Hermit-ian forms on projective, projective-free, or free modules over Λ ([65]).

§3. Algebraic and Hermitian X~-theory from the pointof view of homology theory

1. Classifying spaces. The method of defining Hermitian ^-theory in §2points to the possibility of describing the ^-groups as homotopy invariantsof certain classifying spaces. The first such description was proposed byQuillen ([67]). We consider the space BGL(A). Since the derived group ofGL(A) is equal to its second derived group, there is a map

88 A. S. Mishchenko

f: BGL(A) -*• BGL(A)+ under which the fundamental groupπλ (BGL(A)) = GL(A) is mapped onto its Abelianization, and the integralcohomologies are isomorphic. Moreover, BGL(A)+ is universal with respectto these properties. Then the space BGL(A)+ is uniquely determined by thering Λ and has functional properties.

Quillen has proposed defining the /if-groups in the following way:

(3.1) KtA = nt{BGL(A)+).

The previously known groups KQA, Κ{Α, K2A, agree with the definition(3.1). This definition is convenient from the homotopy point of view. However,whether it is mathematically natural remains unclear.

At the same time Volodin proposed another definition, closer in spiritto the point of view of the processes explained in §2 ([68], [70]). Withthe help of elements of GL(A) one can construct a simplicial complex. Itsvertices are the elements (matrices) of GL(A). The one-dimensionalsimplexes are the elementary matrices, and so on. The homotopy groups ofthe resulting space (after shifting the dimension by 1) turn out to be thegroups Kj(A). The merit of the space introduced by Volodin lies in thefact that its homotopy properties clearly reflect the homotopy structureof the group of pseudo-isotopies of a multiply-connected manifold. Forexample, the group ir0(PISO(M)) is expressed in terms of K2(A), T^I{PISO{M))

in terms of A^3(A), and so on ([72]).

Later the homotopy equivalence of the spaces defined by Quillen andby Volodin was established ([69]).

There is still another definition of the algebraic ^-groups, basedprecisely on the analogy of the Laurent extension of Λ in algebra andmultiplication by the circle in topology ([71]). In the case of regularrings, at least for / < 2, the groups £,·(Λ) agree with the other definitions.

2. Periodicity in Hermitian ^-theory. For Hermitian ΑΓ-theory Karoubihas also considered Quillen's classifying space ([73]). In the case of aregular ring Λ, he constructs versions of Hermitian ίΓ-theory that areneeded for the passage to the Laurent extension of the rings, U"(A),Vn{A). Relying on ideas of Novikov, he establishes 4-periodicity for thehomotopy groups of the classifying spaces ([74], [75]).

Moreover, from the point of view of the homotopy groups of classifyingspaces one can clarify the deviation from 4-periodicity by taking account of theelements of order a power of 2. Just as in topological ίΓ-theory, theperiodicity of the ΑΓ-groups is established by multiplying a generator w4

into the group ATQ(A). It turns out that actually multiplication by w4 isnot an isomorphism. There is a generator u_4 e K*L4(A) such thatw4*w-4 = 4 G K'o(A). Consequently, after 2-adic localization, « 4 becomesinvertible. For comparison we may point out that in real topologicalΑΓ-theory there is an analogous picture for studying the 4-periodicity ofthe theory (see [76]).


CHAPTER 2

HERMITIAN ΑΓ-THEORY AND REPRESENTATION THEORY

§4. Finite-dimensional representations

We turn now to the discussion of functional methods for studying theinvariants of Hermitian ^Γ-theory. Our main problem is as follows.

Let Λ be an associative ring with involution and (M, a) a Hermitianform over Λ. We have to find effective methods of finding invariants of(M, a) that distinguish inequivalent forms. In §3, as a matter of fact, sucha complete set of invariants was given for Hermitian forms for one specialclass of rings, namely, group rings of free Abelian groups. The invariants of(M, a) are the signatures of the numerical quadratic forms that are theimages of some chain of Bass-Novikov projections.

1. Symmetric representations. We consider another way of finding numer-ical invariants of the Hermitian form (M, a) over Λ, based on the study ofthe representations of Λ. Let ρ: Λ ->• Mat(/, C) be a symmetric repres-entation of Λ in the ring Mat(/, C) of complex matrices of order / (see[8], 221). In Mat(/, C) we fix the involution of ordinary complex con-jugation of a matrix and then

(4.1) ρ(λ*) = (ρ(λ))* (λ 6 Λ).

By means of a symmetric representation ρ of Λ we can associate with eachHermitian form (M, a) a well-defined numerical Hermitian form on a com-plex vector space. Namely, let V be the complex /-dimensional vector spaceon which the ring Mat(/, C) acts. This action can be specified in terms ofa basis υχ . . . υ; for V, If V is furnished with a Hermitian metric in whichthe vectors (i>i . . . υ;) are orthonormal, then the involution on Mat(/, C)corresponds to the operation of duality for the linear operators on V.Using p, we turn V into a left Λ-module. To do this we set

(4.2) λν = ρ(λ)ν (λ 6 Λ, veV).

Let M be a free Λ-module with a finite number of generators (mi . . . ms).We consider the tensor product of the Λ-modules Μ and V. Since it isnecessary for the tensor product that one of the modules has a left-and the other a right-module structure, we arrange that a right-modulestructure is assigned to V by the formula

(4.3) νλ = ρ(λ*)(ν) (λ ζ Λ, ν ζ V).

It is not difficult to verify that the right action of Λ on V satisfies all theproperties of right action of a ring. Thus, we set

(4.4) Μp = Μ ®AV,

that is, Mp is the factor group of the tensor product Μ <8> V by therelations:

(4.5) km <g> ν = m <S> ρ(λ*)ΐ> (m ζ Μ, ν 6 V, λ £ Λ).

90 A. S. Mishchenko

Notice that the tensor product Mp has lost the structure of a Λ-module.On the other hand, Μ admits the structure of an action by any ring whoseaction commutes with the action of Λ on one of its factors. Such a ring,for example, is the field of complex numbers C whose action on V commuteswith the action of Mat(7, C), by definition. Consequently, Μ is a complexvector space, of dimension I's.

Moreover, if β: Μ -*• Λ is a module homomorphism, then we get a homo-morphism

(4.6) β ® 1 : Mp->Ap = F.

Thus, there is a natural homomorphism

(4.7) 6: M*-^Hom c (M p , V).

By a direct computation it can be verified that δ is a left Λ-module homo-morphism. In this way, a left Λ-module structure is inherited byHomc(A/;, V) from the left Λ-module structure on V. On the other hand,the Λ-module Hom^M,, V) is isomorphic to (Mt)* <g>c V. Therefore, weget a natural Λ-module homomorphism:

(4.8) δ: M*-+{Me)*®V.

Next, by analogy with (4.6) we have the homomorphism

(4.9) δ ®1: (M*)p-+(Mt®cV)®AV.

Now we study the space {M* ®CV) ®K V = (Mp)* ®C(V <8>Λ V). If theimage of ρ(Λ) is the whole matrix algebra Mat(/, C), then (V <8>Λ V) isisomorphic to the one-dimensional space C. In general, we can at leastassert that the Hermitian metric on the complex space V well-defines acomplex-valued linear function defined on V ®A V by the formula

(4.10) v(i>! ® v2) = {vv ya).

Thus, we can construct the composite of homomorphisms

(4.11) % (M*)P + (MP)·,

(4.12) δ: (1 ® γ)(δ®1) .

If Μ is a free Λ-module, then δ is an isomorphism. We now return to theHermitian form (M, a). Here a: M* ->· Μ is a Λ-module isomorphism, andby definition (1.7), ε^α* = (-l) f ca. We multiply a tensorially by theΛ-module V and obtain the homomorphism

(4.13) a: (a ® 1): (M*)p-+Mp,

We then set(4-14) ap: (M P )*->M P ,

(4.15) βρβϊ.Ι-1.

The homomorphism (4.14) satisfies a relation analogous to (1.17),


(4.16) β £ ρ α ? = ( - 1 ) * « ρ ,

that is, it is a (skew-) Hermitian form over C with complex conjugation asinvolution.

Our definition of the numerical Hermitian form (Mp, ap), constructedfrom (M, ex) and the symmetric representation ρ of Λ, does not refer toany basis in Μ or V. If in Μ we fix a free basis (m, . . . ms) and in V abasis (υ ι . . . V/), we can distinguish in their tensor product Mp the basis{(nij ® Vj), 1 < i < s, 1 < / < /}. Then in this basis the matrix of(Mp, ap) takes the following simple form. Let

(4.17) A = H Κι IN λ ο € Λ*be the matrix of (M, a) in the basis (m1 . . . ms). We consider the matrixAp of order l-s, split into square blocks of order / each (altogether s X sblocks), each block being equal, respectively, to p(Xi/) S Mat(/, C):

(4.18) Ap =

This Ap is the matrix of (Mp, ap) in the basis{(m; <8> vj), 1 < / < s, 1 < / < / } . The verification that ^4p is Hermitianand non-degenerate is trivial.

2. Signatures of Hermitian forms. The invariant definition of the numericalHermitian form (Mp , α ) from (M, a) and ρ shows that if two forms(Μι, αϊ) and (M2, ot2) are equivalent, then so are the forms (Mlp, alp)and (M2p, a 2 p ) . Moreover, if (M, a) is Hamiltonian, then so is (Mp, (xp).In the same way, if pj and p 2 are equivalent representations of Λ and(M, a) is a Hermitian form, then the forms (M , a ) and (Mp , ap ) areequivalent. Therefore, ρ induces a homomorphism of Hermitian A'-theories:

(4.19) signp: Kh

2k (A)-+Kh

2k (C).

Moreover, for equivalent representations ρ ι and p 2 the homomorphisms(4.19) coincide:

(4.20) signPi = sigiip2.

The notation signp is justified by the fact that the group AT^iC) ofnumerical Hermitian forms is isomorphic to the group of integers and theunique invariant of a numerical Hermitian form is its signature. We callsignp(M, a) the signature of (M, a) with respect to p.

Thus, every symmetric representation of Λ gives us a way of obtaininga numerical invariant for Hermitian forms over Λ. The larger the stock ofsymmetric representation of Λ we have available, larger will be a priorithe number of numerical invariants of Hermitian forms we can get. Indeed,in some cases, by running through the whole set of finite-dimensionalsymmetric representations of Λ we obtain a complete set of numericalinvariants for Hermitian forms over Λ, that is, with the help of theinvariants (4.19) we can distinguish any pair of inequivalent Hermitianforms over Λ.

92 A. S. Mishchenko

Before considering the class of rings Λ for which the finite-dimensionalsignatures are a complete set of invariants of Hermitian forms we make anumber of remarks.

Let Λ be a ring with involution and ρ: Λ -*• Mat(/, C) a symmetricrepresentation. Then ρ induces a representation of the tensor productΛ ®z C of Λ with the field of complex numbers. Thus, with the help ofsignatures with respect to symmetric representations we can constructinvariants, as a matter of fact, not of Hermitian forms over Λ but of theirimages in K\k{A ®z C) under the homomorphism

(4.21) Kh

2k(A)-+Kh

2h(A®zQ,

induced by the change of rings

(4.22) A + A ® Z C .

Consequently, we can from the very outset suppose that our ring Λ is analgebra over C.

We suppose next that Λ splits into the direct sum of two ringsA = A] ® Λ2, that is, Λ has a self-conjugate projection μ in the centre ofΛ, μ(Λ) = Λ 1 ; (1 - μ)(Λ) = Λ2. The projection μ is the identity on At

and (1 - μ) the identity on Λ 2. Then Kh

2k{A) is isomorphic to the directsum of groups Kh

2k(Ax) ® K\k{A2). For if A = || λ(/·|| is the matrix of aHermitian form over Λ, then the matrices

A2 = II (1 - μ)λΜ II, ( 1 - μ ) λ υ € Λ , ,

define Hermitian forms over At and Λ2 respectively. Conversely, if Ax

and A2 are the matrices of Hermitian forms over Λ! and Λ2, respectively,then A = Α ι + A2 is the matrix of a Hermitian form over Λ.

Thus, a complete set of invariants of Hermitian forms over Λ = Λ! Φ Λ2

is the union of complete sets of invariants of Hermitian forms over Αχand Λ2.

Finally, a third remark concerns a normed ring Λ with involution. This(see [8]) is a ring Λ with involution which is at the same time a normedspace, such that the following relations hold:

(4.23) | 1 | = i, | λ μ | . < ς | λ | · | μ | (λ, μ 6 Λ),

(4.24) Ι λ* I = Ι λ Ι (λ £ Λ).

A complete normed ring Λ is called a Banach ring (or Banach algebra).For normed rings Λ it is natural to limit oneself to consider only

bounded symmetric representations, that is, homomorphisms

(4.25) ρ: Λ -*-Mat(Z, C),

such that(4.26) H ρ(λ) H < C | λ | (λ 6 Λ),

for some constant C. Let Λ denote the set of all representations (4.25)satisfying (4.26) with one and the same constant C. In this case, in fact, we


obtain invariants not of Λ itself, but of its completion Λ, which is aBanach ring. Moreover, our results are completely unchanged if we changethe norm on Λ preserving the continuity of all the representations in M·Among all these norms there is a greatest one. We set

(4.27) || λ || = sup || ρ(λ) || (λ ζ Λ).

Then(4.28) || λ | K C | λ | (λ 6 Λ).

Consequently, || λ || is the greatest norm among all ring norms for whichthe representations of the set J? are bounded.On the other hand, withrespect to the norm (4.27), Λ satisfies the relation

(4-29) || λλ* || = || λ | |2 (λ ζ Λ),

that is, the completion of Λ in the norm (4.27) is a C*-algebra (see[10]).

In the case of the group ring Λ = C[TT] of a discrete group π, therepresentations provide us with a certain absolute norm relative to whicheach symmetric representation ρ of Λ is continuous. For if the existenceof a supremum (4.27) for arbitrary normed rings is guaranteed by (4.26),then in the case of a group ring Λ we obtain the existence of a supremumfrom other properties.

Thus, for a group ring Λ we set:

(4.30) || λ || = sup || ρ(λ) || (λ 6 Λ),ρ

where the supremum is taken over all symmetric representations of Λ. If/ G π is any element, then according to (1.3)

(4.31) p(/)* = p(/*) = p^-1)

Consequently,(4.32) II p(/) || < 1

for any symmetric representation p. Then if λ = Σ Xjfj (/} S π, xt Ε C), we

have the inequality

(4.33) Ι | ρ ( λ ) | | < Σ | * ί Ιi

for all symmetric representations ρ of the ring. Hence the supremum(4.30) exists and

(4.34) Ι|λ||<2ΐ^Ι·

Summarising what has been said above we can state the followingconclusion.

In the study of the invariants of Hermitian forms with the help ofrepresentation theory we may suppose without loss of generality that thebase ring Λ is a C*-algebra.

3. Two examples of group rings. In conclusion we quote two examples

94 A. S. Mishchenko

of group rings Λ for which we can obtain, with the help of the theory offinite-dimensional representations, a full set of invariants of Hermitian forms.

The first example is that of the group ring Λ = €[π] of a finite groupπ.

This ring splits into the direct sum of matrix algebras

( Λ = Θ Λ;,

A£ = Mat(Zf, C),

and the projections

(4.36) p,: A^Ai

form a complete set of pairwise inequivalent irreducible symmetricrepresentation of Λ. Then

(4.37) * i ( A ) = θ *2*(Λ,).

It only remains for us to exhibit the structure of Hermitian forms over thefull matrix algebra Mat(/, C). Let A be the matrix corresponding to aHermitian form (M, a) over Mat(/, C):

4 = 11 λ» ||, λ,, 6 Mat (Ζ, C),

We can regard A as a numerical matrix A of order Ir, split into blocksX,y. Then A is non-degenerate and Hermitian: A* = (—\)kA. Conversely, ifA is a non-degenerate Hermitian numerical matrix of order Ir, then bysplitting it into blocks each of order /, we obtain a matrix A of order swhose elements belong to Mat(7, C). Similarly, if Αχ and A2 are twomatrices over Mat(/, C) determining equivalent Hermitian forms, then theirnumerical representatives Αχ and A2 are equivalent matrices of order Ir.Conversely, if Α ι and A2 are two numerical non-degenerate matrices andΒ* Αχ Β - A2, then by splitting Αχ, A2, and Β into blocks of order /, weobtain matrices Αχ, A2, and Β with elements in Mat(/, C) for whichΒ*ΑχΒ = Α2.

From the arguments above we see that

(4.38) K\k (Mat (/, C)) = Kh

2k (C) = Z,

that is, for Hermitian forms over Mat(/, C) a complete set of invariantsconsists of a single number, the signature of the numerical representationof the Hermitian form.

Thus, we can state the following proposition:THEOREM 4.1. Let A = C[TT] be the group ring of a finite group π

with the natural involution, and let {p,·} (1 < i < s) be a complete systemof irreducible unitary representations of π. Then

(4.39) Kh

Zk(A)^Z\


and the projection on the ith component in the free Abelian group Zs canbe identified with the signature relative to the representation p;·:

sigiip.: #2Α(Λ)-»-Ζ.

As a second example we consider the group ring A = C[Z'] of the freeAbelian group Z1 of rank /. In the preceding section we have explainedthat in place of Λ we should consider its completion with respect to acertain norm induced by the system of symmetric representations of thering.

Instead of considering the set of all symmetric representations, we limitourselves for a while to the class of one-dimensional unitary representations,that is, to the characters of Zl. Obviously, the set of characters of the freeAbelian group Z' is isomorphic to the Cartesian product of / copies of thecircle, that is, to the torus Tl of dimension /. As coordinates on Tl wecan take the sets of complex numbers (zx . . . Zj) of modulus 1. Let{αχ . . .«;} be a free basis for Z', and a £ Z1 any element,

ι(4.40) a=\]a?i.

Then from a we can construct a function on T!:_ ι

(4.41) a{zu ...,zn) = z(a)=l]^i.The formula (4.41) gives us a symmetric homomorphism of A = C[Z']into the ring C(T!) of continuous functions on T!:

(4.42) χ: A^C(Tl)

(the involution on C(Tl) is defined as complex conjugation on the values ofa function). Then the norm on A, given by (4.27) on the set of charactersof Z', is the same as the uniform norm for continuous functions on Tl.Consequently the completion Λ of A in this norm coincides with the ringC[T!] or, more precisely, the homomorphism (4.42) extends to a ringisomorphism

(4.43) χ: A-+CIT1].

The norm of A induced by the uniform norm on C(T') need not a prioribe the greatest norm in which any symmetric representation of A is con-tinuous, since in (4.27) the supremum is taken not with respect to allrepresentations of A, but only with respect to the characters of Z'. Butnonetheless it can be established that the norm in A defined by (4.30)with respect to all representations of A is equivalent to the uniform normof C(r ' ) . For if ρ: Λ -> Mat(s, C) is a symmetric representation, then thematrices pia^) . . . p{at) are unitary and commute pairwise. Consequently,in some other basis all these matrices reduce to diagonal form, that is, ρis the direct sum of one-dimensional representations. Therefore, the norm(4.30) can be estimated in terms of the norm (4.27) constructed from theone-dimensional representations.

96 A. S. Mishchenko

Thus, the study of the invariants of Hermitian forms over Λ = C[Z']reduces with the help of representation theory to the study of theinvariants of the ring of continuous functions C(T!).

We can generalize the problem and consider the class of continuousfunctions C{X) over an arbitrary topological (compact) space ([13], and[14], 271—273). In this case a Hermitian form (M, a) is represented bya Hermitian matrix

(4.44) A = || K) II- Κ = λ·« (1 < i, j < s),

whose elements are complex-valued functions λ,·,· on X. We can interpretthe matrix-valued function (4.44) as a continuous family of non-degenerateHermitian forms on the trivial vector bundle I X C . We fix a pointχ £ X. Then on the fibre χ Χ Cs we obtain an individual non-degenerateHermitian form. We split χ Χ Cs into the orthogonal sum of two subspacesV* and V~ on which our Hermitian form is positive definite and negativedefinite, respectively. Then the union of the subspaces V* for all χ ε Χforms a subbundle

(4.45) ξ + = U Vi

of the trivial bundle X X Cs. Similarly, the union of the subspaces V~forms a subbundle

(4.46) ξ_= U V~x,χζΧ

and(4.47) E + 9 t = X x C s .

Thus, with the help of (4.45) and (4.46)) we associate with each Hermit-ian form {M, a) two vector bundles £+(M, a) = £+ and ξ^(Μ, a) = £_ overX. It is quite clear that the construction of £+ and £_ does not depend onthe choice of a free basis in Μ (that it, does not depend on the choiceof A in the class of equivalent Hermitian matrices).

This construction admits a converse, that is, if we are given two finite-dimensional bundles ξ+ and £_ over our spaces and a unitary isomorphism

(4.48) φ: ξ + θ ξ_->Χ X C,

then we can construct a Hermitian form (M, a) over C{X). To do this weassign to each fibre χ Χ Cs the Hermitian form that is equal to theHermitian metric on the fibre φ(£+)χ of the subbundle φ(ζ+), while on thefibre φ(ζ-)χ of the subbundle φ(ξ^) it is equal to the Hermitian metricwith the opposite sign and leaves the two subspaces φ(ζ+)χ and φ(ζ-)χ

orthogonal. As a result we have a continuous family of Hermitian formson the bundle I X C1 or, what comes to the same thing, a non-degenerateHermitian matrix over C(X).

If we replace the isomorphism (4.48) by another unitary isomorphism φ',then by our construction we obtain a Hermitian matrix over C{X), which isequivalent to the original one.


So we have, in fact, established the following theorem:THEOREM 4.2. Let X be a compact topological space and C(X) the

ring of continuous complex-valued functions on X, with complex conjuga-tion as involution. The map

(4.49) (Μ, α) -»- |+(M, a) - dim g.(M, a)

induces an isomorphism(4.50) x:Kh

2h(C(X))-^K(X)

of the group Kh

2k(£.(X)) with the topological K-functor of complex vectorbundles over X.

4. Real representations. We make a few remarks concerning applications ofreal representations. It stands to reason that just as in the case of complexrepresentations from a real representation we can construct without anychange signatures of Hermitian forms. A natural change of Λ in this casewould be to replace it by A ® z R and to complete it in to the norm oftype (4.30) constructed from the set of all the real representations. However,in the case of the free Abelian group Zl we cannot interpret the completionof Λ = R[Z'] as the ring of continuous functions of some topologicalspace. It seems more natural in this case to consider in the character groupT! the involution of complex conjugation and the anticomplex involution onall the bundles considered over Tl. This approach of reducing all objectsover the real field to analogous objects over the complex field can be foundfor bundles in [15] and for Hermitian forms in [13].

§5. Infinite-dimensional Fredholm representations

1. Definition of Fredholm representations. In §4 we have considered theconstruction of invariants of Hermitian forms with the help of finite-dimensional symmetric representations of a ring Λ in the full matrix ringMat(/, C). We have shown that in the case of the group ring Λ = C[ir] ofa finite group π we can construct in this way a complete set of invariantsof Hermitian forms. In fact, the finite-dimensional rings are the only classof rings for which finite-dimensional signatures form a complete set ofinvariants of Hermitian forms.

For example, even in the case of the group ring of a free Abelian groupthe finite-dimensional representations do not give us new invariants ofHermitian forms compared with the ordinary signature of numerical Hermit-ian forms. For if π = Z' and ρ: Ζ' -*• U(n) is a unitary representation, thenthere is a continuous family of homomorphism pt: Zl -*• U(n) ( 0 < t < 1)with p 0 - p, px = E. Since the signature of a Hermitian matrix does notchange for nearby Hermitian forms, for an arbitrary Hermitian form (M, a)over A = C[Z'] the function signp.(M, a) is constant. Consequently, thesignature signp (M, a) is the same as sign (M, a) relative to the homomor-phism of Zl into the trivial group.

98 A. S. Mishchenko

Thus, the problem arises naturally of searching for a class of represent-ations of Λ with the help of which we might be able to construct a morecomplete set of invariants of Hermitian forms.

In the present section we consider such a class of infinite-dimensionalrepresentations, which furnish us with new invariants of Hermitian forms(see [16], [17]). We call them Fredholm representations of the ring. Withthe help of Fredholm representations we can construct invariants not onlyof Hermitian forms, but also of other objects, such as classifying spaces fordiscrete groups and elliptic operators on compact manifolds. We give theseexamples at the end of the section despite the fact that at a first glancethey do not appear to be related to the matter being studied.

We now turn to the precise definition of Fredholm representations.We fix two Hubert spaces Hi and H2. Let ργ and p 2 be two symmetric

representations of Λ in the rings of bounded operators on Hx and H2,respectively. Next, let F: Hx -> H2 be a Fredholm operator satisfying thefollowing condition:

(5.1) For each λ Ε.Λ the operator Fpx{\) - p2(X)F is compact.The triple ρ = (p 1 ( F, p2) consisting of p 1 ; p2, and F and satisfying

(5.1) is called a Fredholm representation.Although from the point of view of representation theory it is more

natural to call ρ = ( p 1 ; F, p2) a wreath product of the two representationsρ ι and p2, all the same we call it a Fredholm representation. The thoughtbehind this nomenclature is that in all future applications the pair ofrepresentations pj and p2 related by the Fredholm operator F embodiesthe formal difference of px and p 2 , that is, the virtual representation

Pi ~ Pz-We generalize the definition of Fredholm representations to the case of

Fredholm complexes. Let Ht{\ < / < s) be Hubert spaces andFj(\ < i < 5 — 1) be bounded operators

(ο.Ζ) ΐΐι—>Η2-^... >HS,

such that the FjFi_l are compact. Then (5.2) is said to be a Fredholm com-plex if there are also operators G,(2 < i < s):

(5.3) Hi^H2¥-...^Hs,

such that the operators

(5.4) Fi-fii + GwFt-i

are compact. Now just as for Fredholm operators, so also for Fredholmcomplexes there is a well-defined index of a Fredholm complex, not de-pending on homotopies of the operators F, in the class of Fredholm com-plexes, nor on varying the F,· by a compact component. Moreover, for theFj forming a Fredholm complex we can choose compact summands Kt

such that the new operators F/ = F,· + K{ form a genuine complex, thatis, F/ F/+ 1 = 0. Here the homology of the newly formed complex


(5.5) Hl^H2^i...^UHs

is finite-dimensional, and the index of the Fredholm complex (5.2) is thealternating sum of the dimensions of the homology groups of the complex(5.5).

Suppose now that p ; (l < i < s) are symmetric representations of Λ inthe rings of bounded operators on the spaces i/ ;(l < / < s).

We say that we are given a Fredholm complex of representations of Λif the following condition is satisfied:

(5.6) For each λ £ Λ the operators FjP^X) - pi+l(F{) are compact.2. Signatures of Hermitian forms. It turns out that with the help of

Fredholm representations (or of Fredholm complexes of representations) ofa ring Λ we can construct numerical invariants of Hermitian forms of thetype of signatures of finite-dimensional representations.

We turn at once to the construction of these invariants. We consider forsimplicity the case of a Fredholm representation ρ = (p 1 ( F, p 2)·

Let (M, a) be an arbitrary Hermitian form over Λ, and A the matrix of(M, a) with respect to some free basis for the Λ-module M,

By analogy with §3 we consider the matrix

(5.7) Λ , = II P( (λ;;) II (1 = 1,2).

The elements of the matrix (5.7) are operators acting on the Hubert spaceHj. Therefore, (5.7) can be interpreted as the expression of a boundedoperator acting on the direct sum of s copies of //;. We denote thisoperator by

Moreover, we set

(5.9) F=\\F6U\\ (1

Then we obtain the following diagram:

t -I A.

P 2(5.10)

in which the operator FAp- ApF is compact. It is quite clear from theHermitian nature of A and the symmetry of pI that the operator (5.8) isHermitian, invertible, and bounded. From spectral theory it follows thatthe Hubert space {Ht)

s splits uniquely into the orthogonal sum of two sub-

100 A. S. Mishchenko

spaces by means of projections commuting with A ,

(5.H) {Hl)s = m®m,

and A is represented in the form of a matrix corresponding to the decom-

position (5.11),

< 5 · 1 2) + ** %then the operators A\ and Al are positive.

We represent F also in matrix form corresponding to the decomposition(5.11)

(5.13) Μ££)·The condition for the diagram (5.10) to commutate up to compactoperators can be expressed as follows:

(5.14) FiA

(5.15) FzA

(5.16) F3A

(5.17) F kA\-

where e/Γ denotes the space of compact operators.From (5.15) and (5.16) it follows that F2 and F3 are compact. Thus,

when we change the operator (5.13) by a compact component, we find thatthe operator represented by the matrix

0 Fj

is Fredholm, or, what comes to the same thing, that F t and F 4 areFredholm operators. We then get an invariant of the Hermitian form(M, a.) from the equality

(5.18) signp(M, cc) = indexi^ — index^4.

Now (5.18) does not depend on the choice of a basis in M, and when(M, a) is Hamiltonian, this invariant vanishes:

(5.19) signp(M, a) = 0.

Next, for a direct sum of Hermitian forms we have

(5.20) signp ((Mt, at) θ (M2, az)) = signp (Mu cc4) + signp (M2, a2),

which means that, with (5.19), we have defined a homomorphism

(5.21) signp: Kh

ik (Λ) -> Z.

In the case of skew-Hermitian forms the construction of the invariant-sign (M, ex) is completely analogous. The only difference is the decompos-ition of the Hilbert space (5.11) for which the operators are representedin the form


where A] and A] are positive operators.

Thus, for each Fredholm representation ρ we have constructed a homo-

morphism

(5.23) signp: K\h (Λ) -> Z.

So far we have considered the construction of the signature of Hermitian

forms for a Fredholm representation of the ring Λ. If we are now given a

Fredholm complex ρ of representations of Λ of the form (5.5), then with

its help we can associate with each Hermitian form (M, a) a numerical

invariant by a process analogous to (5.8)—(5.18). We denote this invariant

by signp(M, a).

We make two remarks concerning the behaviour of the signature of a

Fredholm representation ρ = (p 1 ( F, p 2 ) of Λ. Let ί be a compact

operator and F' = F + K. Then the triple p' = (pj, F', p 2 ) is also a

Fredholm representation, and

(5.24) signp(M, a) = signp< (M, a).

Thus, from the point of view of signatures of Hermitian forms we can

regard ρ and p' as equivalent.

The second remark concerns sufficient conditions for signp (M, a) = 0

to hold for any Hermitian form (M, a). Suppose that the Fredholm

representation ρ = (p1, F t , p 2 ) is such that F is invertible and the

operators Fpj(X) - p2(X)F are small in norm in comparison with F~l.

Then signp (M, a) = 0. The precise formulation of the sufficient condition

is the following: Let pt = ( p u , Ft, p2t) be a continuous family of Fred-

holm representations, parameterized by a numerical parameter /. Suppose

that the operators Ft are invertible and that for any λ G Λ,

(5.25) 1ΐΐη||/ϊΙϊ-1ρ2(λ-1)/;'ίρ1(λ)ϋ = 0.ί->·οο

Then signp(M, a) = 0 for any Hermitian form (M, a).

The condition we have given is not invariant, but it is useful in a

priori estimates of signatures.

3. Connection with finite-dimensional representations. Suppose that a

Fredholm representation ρ = (pl, F, p 2 ) satisfies a rather stronger condition

than (5.1), namely, for any λ ε Λ

(5.26) FPl(X) - p2(k)F = 0.

Then the kernel and cokernel of F are p r and p2-invariant finite-dimensional

spaces. We denote the representations of Λ on the kernel and cokernel of

F by p\ and p 2 , respectively. Then

(5.27) signp (M, a) = signPi (Μ, α) —signp-(M, a).

The formula (5.27) justifies the point of view that when the Fredholm

operator ρ satisfies (5.26), then it should be identified with the formal

difference of p\ and p 2 .


In particular, in the case of the group ring Λ = C[n] of a finite groupπ each Fredholm representation is equivalent to the formal difference offinite-dimensional representations in the sense of the remark in §5.2.Namely, if ρ = (p 1 ( F, p 2 ) is a Fredholm representation, then there is acompact operator Κ such that by setting F' = F + Κ, ρ = (ply F',p2), weobtain a Fredholm representation satisfying (5.26). To see this, we set

Then 8ζπ

F' = τ τ Γ Σ p z ( g M > F p i (g)> Pz {h~i] F>pi { h ) = F ' ·gin

Thus, in the case of the group ring of a finite group, the Fredholmrepresentations, as was to be expected (see §4.3, Theorem 4.1), do notprovide us with signatures of invariant Hermitian forms that are new incomparison with those derived from finite-dimensional representations.

4. Elliptic operators for Fredholm representations. Apparently, Fredholmrepresentations were first introduced by Atiyah [18] in the study of pro-blems concerning the index of elliptic operators on a compact closedsmooth manifold.

We consider a smooth compact closed manifold X, the Hubert space ofsquare-summable functions L2{X) on X, and a pseudo-differential operatorD: L2(X) -*• L2(X) of order zero. The ring of continuous functionsΛ = C{X) acts on L2(X), and the commutator Βφ - φϋ (φ e Λ) is apseudo-differential operator of order (—1) [19] and, consequently, compact.In the case of an elliptic operator D we obtain in this way a Fredholmrepresentation ρ of Λ on the Hubert space L2(X), p = {ρχ,ΰ, ρχ), where px

is a representation of Λ = C(X) on Η = L2(X) for which Ρι(φ)ψ = φψ.Consequently, in describing the homotopy invariants of an elliptic

operator D and, in particular, its index, we can raise the problem of find-ing invariants of a Fredholm representation of the ring of continuousfunctions Λ = C{X) on X.

It turns out that with every Fredholm representation ρ of Λ = C(X)we can associate an element κ(ρ) S K0(X), where K0(X) is the group ofthe generalized homology theory, dual to K-theory, constructed on thebasis of complex vector bundles. The elements χ of K0(X) can be identifiedwith the natural transformations of the further K°(X X *) to the functionK°(*\ that is, with the homomorphisms x(Y): K°(X X Y) -+ K°(Y), suchthat for any continuous map φ: Υχ -> Υ2 the following diagram commutes:

K°(Xx Yi) —ΥΛ

(1ΧΦ)* ι ι

κ« (χ χ y.) —ΥΛ κ° (y2)Thus, to determine an element κ(ρ) €Ξ .&OW it is sufficient to associatewith the Fredholm representation ρ of Λ = C(X) the homomorphisms


x(p)(Y): K°(X X Y) -* K°(Y). Let ξ e K°(X X Y) be a finite-dimensional

bundle over the space XX Y. We denote by Yy(i) the space of continuous

sections of £ over the subspace XX {y} (y S Y). The space 1 (2;) is a

finitely generated projective module over Λ = C(X). Then D induces the

Fredholm operator

(5.28) Fy=\®D: TB(t)®AH-+Ty{t)®AH

up to a compact component. So we obtain a continuous family of Fred-

holm operators Fy, parametrized by the points of Y, and operating fibre-

wise on the bundle Μ with the Hilbert fibre Yy{£) ®Λ Η over Y. The

family of Fredholm operators F determines (see [15]) an element of

K°(Y). So we have completely determined an element κ(ρ) e KQ(X) from

the Fredholm representation ρ of Λ = C(X).

Kasparov [20] has found conditions under which two Fredholm repres-

entations pl and p 2 of Λ = C(X) lead to one and the same element of

K0(X). If κ(ρ) = 0, then ρ can be obtained by means of the following

operations:

a) homotopies in the class of Fredholm representations,

b) the addition of direct summands p' = (p\, F, p'2) in which F is

invertible and satisfies (5.26).

With the help of the invariant κ(ρ) of a Fredholm representation ρ of

Λ = C(X) one can give a more transparent description of the Atiyah—Singer

formula [21] for the index of an elliptic operator. Let ρ = (p 1 ( D, p 2 )

be a Fredholm representation of Λ on the Hilbert space Η = L2(X), con-

structed from the elliptic pseudo-differential operator D of order zero. Let

αφ) G K°{T*X) be the element of K°C{T*X) determined by the symbol

of D. Since the cotangent bundle T*X is a quasicomplex manifold, there is

Poincare duality, that is, an isomorphism 3: K°C(T*X) -» KQ(T*X) = K0(X).

Here K® denotes the "compact ίΓ-functor", that is, the relative groups for

the one-point compactification of the base. Then from the Atiyah—Singer

formula we get the following relation

(5.29) da(D) --= x(p l t D, p2).

The Atiyah—Singer formula itself for the index of an elliptic operator

can be obtained as an analogue of the direct image for elliptic operators.

If φ: Χι ->· X2 is a continuous map of compact manifolds and

ρ = (pl, Fx, p 2 ) a Fredholm representation of Aj = C{XY), then by apply-

ing the change of rings φ*: Λ2 = C(X2) -• Λι = C(X2) we get a Fred-

holm representation φ*(ρ) = (φ*ρχ, F, φ*ρ2) of the ring A2. Here we have

the relation

(5.30) φ*(κ(ρ)) = κ(φ*(ρ)).

In particular, if the manifold X2 consists of a single point, X2 = pt, then

(5.31) x(<P*(p)) = index F.

Consequently, applying (5.29) we come to the formula for the index of D:


(5.32) index Ζ) = <p^{da(D)) = φ. (a(D)),

φ,: Ka

c{T*X) -+K°(Vt) = Ζ

is the direct image in A"-theory ([22]). The derivation of the traditionalAtiyah—Singer formula in terms of the characteristic classes of the elementsof o(D) is by now standard routine in AT-theory.1

§6. Fredholm representations and bundles over classifying spaces

In §5 we have described how with the help of the Fredholm repres-entations of a ring Λ we can find signature invariants of Hermitian formsover Λ and, in case Λ = C(X) is the ring of continuous functions on amanifold X, invariants of elliptic operators.

If Λ is the group ring of a discrete group π, Λ = C[TT], then with thehelp of Fredholm representations we can describe the homotopy invariantsof the classifying space Βπ of π, namely, vector bundles over Bit.

1. Finite-dimensional representations and bundles. The connection betweenvector bundles over a classifying space BIT and finite-dimensional represent-ations of 7Γ was considered earlier in papers of Atiyah and Hirzebruch [22],[23]. With each finite-dimensional representation p: ir -*• Mat(«, C) of πwe can associate a finite-dimensional complex bundle % over Βπ. To dothis we consider the universal covering space over BIT:

φ: En -*-Βπ.

The group π acts freely on Επ, and if Βπ is a simplicial complex, then Επalso has a simplicial structure, the projection φ is a simplicial map, and theaction of π is also simplicial.

The representation ρ provides a linear action of π on the Euclideanspace V = C". We consider the direct product Επ Χ V with the diagonalaction, that is, g(x, y) = (g(x), p(g)y) (x S Επ, y £ V). We can constructthe following commutative diagram of maps:

Εκ X V —^-> (Επ Χ V)/n

Επ —Φ-> Βπ

Here (Επ X V)/ir denotes the space of orbits of the action of π onΕπ X V, \jj is the projection onto the first factor, ψ(χ, y) = x, φ and φ arethe natural projections onto the space of orbits, and ψ is the map of theorbit space induced by ψ. Thus ψ is a locally trivial bundle with fibre V.For if χ G Βπ is any point, then there is a neighbourhood U of χ suchthat the inverse image φ~ι (U) splits into the disjoint union of neighbourhoods

1 It is equivalent to a proof of the formula (5.29).

Hermitian K-theory. The theory of characteristic classes and methods of functional analysis JO5

g *= "•}, φ~ι U = U gW, and φ homeomorphically maps each compon-

ent gW onto U. Then ψ homeomoφhically maps ι/Γ1 (gW) - gW X F ontothe set φ~ι(υ)-

Consequently, ψ is a locally trivial bundle. Since the action of π on Vis linear, the transition functions of ψ also are linear, hence ψ is a vectorbundle. We denote by ξρ the bundle ψ: {Επ Χ Κ)/π -»• Βπ. We also denoteby ζρ the induced element of Κ(Βπ).

It is not difficult to verify that

Thus, the correspondence ρ -> ξρ induces a homomorphism of the ringΜ 00 of virtual representations of π to the ring Κ(Βπ):

(6.1) ξ: Μ{η)

It is natural to investigate the question in what cases % is an isomorphismor to describe the image and kernel of ξ.

When π is a compact Lie group, Atiyah and Hirzebruch ([22], [23]) haveproved that ker ξ - 0, while the image Im ξ is dense in the ring Κ(Βπ),equipped with the topology induced by the finite-dimensional skeletons ofΒπ (see also [24]).

2. Fredholm representations and bundles. Just as in the case of thesignature invariants of Hermitian forms, so finite-dimensional representationsof infinite discrete groups provide a small stock of bundles from the groupΚ(Βπ). There are only individual results, describing the image of (6.1) fordiscrete subgroups of the real symplectic group Sp(2«, R). For example, itis shown in [25] that if π C Sp(2«, R) is a discrete torsion-free group andif the homogeneous space Sp(2«, R)/TT is compact, then the inclusion homo-morphism HP(B Sp(2n, R)) ->• Hp (Βπ) is an epimorphism for p<(n + 2)1 A.For an application of this algebraic result to obtain signature invariants ofHermitian forms, see Lusztig [26].

With the help of Fredholm representations it becomes possible toobtain a wide class of vector bundles over Βπ.

We describe the corresponding geometrical construction of the bundleξρ £ Κ(Βπ) for a Fredholm representation ρ = (p l f F, p2) of a discretegroup π. In this case it is convenient to represent the elements of Κ(Βπ)not as linear combinations of vector bundles, but as continuous families{Fx} of Fredholm operators, parametrized by the points χ of Βπ (see[15]). If we are given a continuous family {Fx} (X € X) acting from oneHubert space Hl to another H2 (where X is some topological spaces), andif the dimension of the kernel dim Ker Fx of Fx is a locally constantfunction, then the element of K(X) corresponding to {Fx} is the differ-ence of the two finite-dimensional bundles U Ker Fr and U Coker Fr.

sxHere, U Ker Fx is to be understood as a subspace of the direct product

xex


Χ Χ Η χ, and U Coker Fx as the factor space X X H2. Thus, it will be

convenient for us to interpret the family of Fredholm operators as a homo-morphism of trivial bundles with Hubert fibres

(6.2) XxH^XxH,

The diagram (6.2) is commutative, and over each χ €Ξ Χ the map F is aFredholm operator of Hubert spaces. It is quite clear that if in place oftrivial bundles with Hubert fibre we consider arbitrary bundles 38 χ and 38 2

with Hubert fibres over X and a bundle homomorphism F: 381-*~38i,which is a Fredholm operator on each fibre, then we equally well cometo an element of K(X), since each locally trivial bundle with Hubert fibreis isomorphic to a trivial bundle (see, for example, [15]).

We return now to the case when the base I is a classifying space Βπ.Let 38 x and 3C 2 be two bundles with Hubert fibres Η χ and H2, respectively,and let F: 38\ -+M2 be a Fredholm bundle homomorphism on each fibre.We denote by 3SX and 382 the inverse images of 381 and 38 2 under theprojection φ: Επ -*• Βπ of the universal covering of Βπ. Then π acts freelyboth on En and on each of the bundles 38 χ and 38 %. Also, the homo-morphism F induces a homomorphism F: 38Ί-ν382, that is equivariantunder the action of π. Conversely, if G: 38r —>- 38a, is an arbitrary equi-variant homomorphism then it is always induced by some homomorphismF: 38 x -»-<$? 2.

Therefore, to specify an element of Κ(Βπ), it is enough to give an equi-variant homomorphism G: 38x -*-382 that is a Fredholm operator oneach fibre, for some pair of bundles 38 χ and 38 2 with Hubert fibre andfree action by π on the base space.

Let ρ be a Fredholm representation of π, that is, a triple ρ =(ρι, F, p2)consisting of two unitary representations on Hilbert spaces Hy and H2,and a Fredholm operator F: Ηχ -*• H2 such that for each g ε π the operatorPi(g)F - Fpxig) is compact. We have to associate with ρ an element £p

of Κ(Βπ). For this purpose we construct over Επ two equivalent bundles3SX and 38 2 with Hilbert fibre and an equivariant homomorphismG: Μχ-^Μζ that is a Fredholm operator on each fibre.

For the bundles 38χ and 38 2 we take the trivial bundles38-χ = Επ Χ Ηχ, and 382 = Επ Χ Η2, with diagonal action of π, that is,g(.x, y) = (gx, Pk(g)y) (* G Eir> y e Hk). Then the map G can be regardedas a continuous, equivariant family of Fredholm operators {Gx} (x S Επ).Equivariance of {Gx} means that

(6.3) p2(g)Gx = G<f*pi(g),

We look for an equivariant family {Gx} such that for any point χ e Επ(6.4) the difference F - Gx is a compact operator.


If G'x is another family satisfying (6.3) and (6.4), then by settingGx = Gx + tG'x we get a homotopy between {Gx} and{G^.) in the classof families satisfying (6.3) and (6.4). Consequently, both {Gx} and {Gx}define one and the same element of Κ(Βπ), which we denote by %p.

It only remains for us to establish the existence of at least one family{Gx}, satisfying (6.3) and (6.4). To do this we represent Βπ as asimplicial complex and Επ as a simplicial complex with simplicial actionof π. We construct {Gx} by an inductive process on the dimension of theskeletons of Επ. The zero-dimensional skeleton (.Επ)0 consists of a discreteset of vertices and splits into the union of pairwise disjoint orbits of theaction of π. Let A C (Επ)° be some orbit, A = U ga, where a £ A is

one of the vertices of A. If at a G A we are given an operator Ga, thenat the other points b £ A the operator Gb is uniquely determined by thecondition (6.3) of equivariance of the family by the formula

(6.5) Gb =

where g G π is an element of π such that b = ga. Thus, to determine afamily Gx(x €E (£π)°) satisfying (6.3) and (6.4) it is enough to choose apoint a in each orbit A and to set Ga = F. At all other points of thezero-dimensional skeleton the family is defined by (6.5).

Then (6.3) is automatically fulfilled and (6.4) follows from the definitionof a Fredholm representation.

Suppose now as an inductive hypothesis that we are given a family {Gx}on the &-dimensional skeleton (En)k, satisfying (6.2) and (6.3). We considerthe set of all (k + 1 )-dimensional simplexes of Επ. Now π, acting freely onΕπ, permutes its simplexes leaving none of them invariant. Therefore, theset of all (k + l)-dimensional simplexes can be split into the disjoint unionof "orbits" and in each orbit all the simplexes can be obtained as imagesof a given simplex of the "orbit" with the help of the action of π. LetAk + 1 be one of the (k + 1 )-dimensional simplexes of Επ. If the family{Gx} is given for each χ G Δ * + 1 , then to get an equivariant family on thewhole orbit we have to use (6.5). If the point χ G Ak + l belongs to theboundary dAk + 1 of Δ* + 1 , then (6.5) gives us an identity, by the inductivehypothesis. Therefore, it is sufficient to extend the family of operators{ Gx }, which is defined by the inductive hypothesis on the boundary3Δ* + 1 , to a family defined on the whole simplex ΔΛ + 1 with (6.3) pre-served. Since (6.3) holds on 3Δ* + 1 and the space of compact operators iscontractible (as a linear space), the required extension of {Gx} to Δ* + 1 ,and hence to its whole orbit, always exists.

This completes the construction of the element %p £ Κ{Βπ) for a Fred-holm representation.

NOTE. The reader should be warned against trying to prove.that £p istrivial, on the basis of (6.3). The fact is that (6.3) is not invariant under


a change of coordinates in the fibres of the bundles &βχ and Μ%. Thereforeajxivializiation oiSBx and a^2 over Βπ induces over Επ a trivialization of•Mx andS8 2 in which, generally speaking, (6.3) is not satisfied.

3. Modification of the construction of a bundle. The construction des-cribed above can be generalized to the case of continuous families ofFredholm representations. Let ρ = {px} - {plx, Fx, p2x} be a continuousfamily of Fredholm representations, parametrized by the points of sometopological space X. Then by applying the construction of subsection 2 weget a continuous family of Fredholm operators {Gry}, parametrized by thepoints of the direct product Χ Χ Βπ, that is, an element £p G K(X X BIT).If we ignore the torsion in K(X Χ Βπ), this group can be identified withthe zero grading of the tensor product K(X Χ Βπ) « [K*(X) ® Κ*(Βπ)] 0 ,that is, £p splits into the sum

?P = Σ «ft <8> Rft,h

where the {ak} form a basis for K*(X), and βΙ( £ Κ*(Βπ). Consequently,the family ρ of Fredholm representations gives us a whole set of homotopyinvariants $k, lying in Κ*(Βπ). Moreover, if we can guarantee that for somepoint χ of a closed subspace Υ of X the family { Gx y} consists of invert-ible operators, then we can define £p as an element of the relative groupK(X Χ Βπ, Υ Χ Βπ).

As a condition for the invertibility of a family of operators {Gx y] wecan take the following:

(6.6) \\F-1p2(g)FPl{g-l)-l \\<l, for any g G π.Then in the construction of {Gxy}wv can add to (6.3) the condition

(6.7) || F-*Gx,y - 1 | | < 1,

guaranteeing the invertibility of operators of (G }.Next, we can start not from a Fredholm representation ρ = (plt F, p 2 )

of π, but from a Fredholm complex (5.2) of representations of π. In thiscase we have to construct a continuous equivariant family of Fredholmcomplexes {Gix} (1 < / < s - 1, χ Ε Επ}, such that

(6.8) The operators F{ — Gjx are compact for each point χ G Επ.In the case of a continuous family ρ = {px} (x G X) of Fredholm

representations we can repeat almost word for word the construction of£p €E K(X Χ Βπ) and also write down a condition on ρ for £p to bedefined as an element of the relative group K(X Χ Βπ, Υ Χ Βπ).

Finally, if ρ = (pj, Fi} p 2 ) and ρ = (pi, F', p2) are two Fredholmrepresentations of π such that F' - F is compact, then ξρ = ζρ·.

4. The construction of Fredholm representations for a particular class ofgroups. As we have indicated in subsection 2 above, we are interested inthe question how large the class of bundles over Βπ is that we can con-struct with the help of Fredholm representations. A priori, this class maybe larger than that of bundles obtained from finite-dimensional bundles.


We notice to. begin with that, by analogy with the signatures of Hermit-ian forms, in the case of a Fredholm representation ρ = (p l f F, p 2 )satisfying (5.26) rather than (5.1), the element £ G Κ{Βπ) coincides withthe difference £p. — %p*, where p\ and p'2 are representations of π on thekernel and cokernel of F, respectively. Taking into account that for finitegroups π each Fredholm representation can be changed into one satisfying(5.26) by adding a compact operator to the representation, we find thatfor a finite group π the classes of bundles over Βπ constructed with thehelp of Fredholm representations and of finite-dimensional representationsare identical.

In any case, if we wish to describe the class of bundles over Βπrepresentable in the form £p for some Fredholm representation ρ of π,then we must present an explicit construction of a Fredholm representation.

There is an important class of infinite groups π for which such a geo-metrical construction of Fredholm representations is possible. This class ofgroups is defined by the following property:

(6.9) The space Βπ is homotopically equivalent to a compact Riemannianmanifold with a metric of non-positive curvature in every two-dimensionaldirection.

In particular, this class of groups includes finitely generated free Abeliangroups, the fundamental groups of compact two-dimensional surfaces, anddiscrete groups of motions of homogeneous spaces of semisimple non-compact Lie groups.

Let X be a complete Riemannian manifold with metric of non-positivecurvature, πχ(Χ) = π, homotopically equivalent to Βπ. Then its universalcovering X is diffeomorphic to Euclidean space, and moreover, there is onX an equivariant metric of non-positive curvature. We consider the spaceT*X of the cotangent bundle of X, which we denote by Y. In this spacethe metric on X induces a natural metric such that in the universal coverY, the group π preserves the metric.

Thus, we have the commutative diagram

Υ —'-+ Υ

η ι*χ — > χ

Next, we consider on X the exterior power Ak(cT*X) of the complexific-ation of the cotangent bundle. We set £fc = \jj*<p*(Ak(cT*X)). The points ofΥ are pairs (3c, y), (3c Ε X), where y is a cotangent vector to X at 3c.

The vectors of the bundle ξΛ split into linear combinations of exteriorproducts of the form yl /\ • . . f\ yk G £fc, where yl, . . . , yk are cotangentvectors to X.

We consider the complex of bundles(6.10) Ιο > Ιι > . . . > t7i,


where the homomorphisnis aQ, at, . . . , an_l are the operators of exteriormultiplication by covectors of the form (y + /ω(χ)) at (x, y) S Ϋ. Hereoo(x) is a section of the cotangent bundle (that is, a differential form) ofX. Note that π acts freely on X and Y, and also on the bundles£o, · • • , ζη· Moreover, from the point of view of this action the operatorof exterior multiplication on the covector y at (x, y) is equivariant, andthe commutator of a;- and the action of g G π is the operator of exteriormultiplication by the covector (cj(gx) - g*oj(x:)).

If the covector (y + /ω(χ)) at (x, -y) Ε Υ is different from zero, thenthe complex (6.10) is exact, that is, has trivial homology.

We denote by ψι(^·) the "direct image" of £,-, that is, the bundle withHubert fibre whose fibre is constructed at each point y £ Υ as the ortho-gonal sum of the finite-dimensional fibres of £;- at all the inverse images ofy. A unitary action of π (which also respects fibres) is induced in theinfinite-dimensional bundle i//i(£y-), and the homomorphisms a0, . . . ,αη^γ

induce homomorphism of the "direct images", so that we obtain a newcomplex

(6.11) ψ, (ξυ) —•% ψ, (ξ,) -^U . . . — ^ ψ, (En).

The complex (6.11) is the required family of Fredholm complexes ofrepresentations, provided that by a suitable choice of ω(χ") we can guar-antee that the conditions in the definition of Fredholm representationshold.

The commutator of Aj and the action of g S π can be described by adiagonal matrix in which each term is the operator of exterior multiplic-ation by the covector (co(gx) - #*ω(χ)), and χ S X runs through the orbitof the action of π. Therefore, for the commutator to be compact it issufficient that

(6.12) lim \\a(gx) — g*o)(z)|j = 0.X-*-oo

For the complex (6.11) to be Fredholm at each y G Υ it is sufficientthat on each orbit the homology groups of the complex (6.10) are trivialeverywhere, with the exception of finitely many points of each orbit, thatis, that ω(χ) vanishes only at finitely many points of each orbit, and thatthe norm || ω(χ~) || is bounded below at infinity.

We construct such a function OJ(X). By the duality induced by theRiemannian metric it is sufficient to produce a vector field with similarproperties. For this purpose we fix an initial point x~0 and consider the(unique) geodesic γ(χ) beginning at x 0 and ending at a point x. Let ω(χ)be the tangent vector to γ(χ) at x, whose length is equal to //(I + /),where /is the length of γ(χ). Then ω(χ) = 0 only at the pointχ = x 0 , lim || ω(χ) II = 1, and since the curvature is non-positive condition

(6.12) is satisfied.

Hermitian K-theory. The theory of characteristic classes and methods of functional analysis H I

So we have produced a family ρ = {py} (y S 7) of Fredholm complexesof representations of π. Moreover, there is a compact subspace Ζ C Υ suchthat at the points y G Υ \ Ζ, ρ satisfies an additional condition of type(6.7) so that £p can be defined as an element of the group KC(Y X X) ofbundles with compact support.

The Chern character of £p (see [17] can be computed by the followingformula:

(6.13) c h £ p = y ; c a i ® bt.

where at G H*(H; Q) is a basis for the cohomology groups, bt G H*{X, Q)is the dual basis by Poincare duality, and aai G H*(Y; Q) are the imagesin the cohomology groups under the Thorn isomorphism.

Thus, with the help of a single family of Fredholm representations ρ weobtain for groups π satisfying (6.9) invariants bt, which form a basis forthe cohomology groups of Βπ.

5. The construction of individual Fredholm representations. In the preced-ing subsection we have presented for one class of groups π a family ofFredholm representations, and with the help of it we have been able todescribe the cohomology of Βπ (or, modulo torsion, the bundles over BIT).

However, the question remains open what bundles over Βπ can bedescribed with help of Fredholm representations, not indirectly, but directlyas bundles of the type ξ . It turns out that, starting from a family ofFredholm representations, we can construct new individual representations,whose invariants generate the invariants of the original family. It isappropriate to state a precise proposition.

THEOREM 6.1 [27]. Let ρ = {px} (x Ε X) be a family of Fredholmrepresentations of a group π, £ Ε K(X Χ Βπ) the bundle constructed insubsection 2, and ch£p = Σ ak (g> bk, where the elements ak G Η (Χ; Q)

form a basis for the cohomogy groups and bk G Η*(Βπ; Q). Then we canfind Fredholm representations pk of π such that

(6.14) ch|P f t = lhbh (λκφΟ).

Theorem 6.1 explains the idea behind studying families of Fredholmrepresentations. For example, from this theorem and the construction ofsubsection 4 it follows that, in the case of a group π satisfying (6.9),"almost all" the bundles can be constructed with the help of Fredholmrepresentations of π. Here the term "almost all" means that the bundles oftype ξρ generate a subgroup of finite index in Κ(Βπ). In one particularcase, π = Ζ Χ Ζ, this corollary to Theorem 6.1 was first established bySolov'ev [29].

To prove Theorem 6.1 we need an additional geometrical construction,which is precisely followed through in the cases of families of finite-dimensional representations of π.


We consider a topological space X and a finite-dimensional bundle ηover X on which ττ acts linearly. The action of π can be regarded as afamily ρ = {px} (ί ε I ) of finite-dimensional representations of it on thefibre of η. This family ρ generates a (finite-dimensional) bundleζρ € AT(JT X 2?π). Suppose that X is a compact closed manifold. Then πacts linearly on the space of sections Γ(η <g> ξ) of the tensor product ofη with an arbitrary bundle ζ and also on the completions //S(T? ® f) ofΓ(η ® ξ) in the Sobolev norm. We consider two bundles ζ χ and f2 and theelliptic pseudodifferential operator

(6.15) D: Γ(η® i j — r f t ® ζΒ),

which is a Fredholm operator on the Sobolev spaces

(6.16) D: H\x\ Θ ζχ) -+Hs{r) ®· ζ2).

The commutator of Z) and the action of g €Ξ π is an operator of smallerdegree, and so is a compact operator on the Sobolev spaces.

Thus, if we denote by pi and p 2 two representations of π acting onΗ\η <g> $Ί) and Η\η ® f 2), then the triple ρ = (ρ 1 ; A p 2 ) is a Fredholmrepresentation.

The computation of £- G Κ(Βπ) can be made with the help of theAtiyah—Singer theorem on the index of an elliptic operator, or moreprecisely, its generalization to the case of families [28]. In particular, ifthe symbol of D has the form 1 ® ο: φ*(η ® ζι) ~* φ*(ν ® £2)* whereφ: Τ*Χ -*• X is the projection, then

(6.17) ^ = -Ψΐ(ξΡ®σ)·

Here ψ: Τ*Χ Χ Βπ -> BIT is the projection, σ G K(T*X) the elementdefined by the symbol σ and ψ\ the "direct image" in ΑΓ-theory. ChoosingD suitably, we can easily prove Theorem 6.1 by using theorems on realizingrational homology classes of X with the help of singular manifolds.

In the case of infinite-dimensional Fredholm representations, the situationis technically somewhat more complicated, but nevertheless remains analogous.

CHAPTER 3

CHARACTERISTIC CLASSES OF SMOOTH MANIFOLDS ANDHERMITIAN A:-THEORY

The theory of Hermitian forms and Hermitian /iT-theory have deep appli-cations in the study of the invariants of smooth of piecewise-linearmanifolds. It would be difficult to give a historical survey of all theseapplications. We note only the chief aspects of the profound connectionbetween Hermitian ΛΓ-theory and the theory of smooth manifolds. Thisis above all the problem of classifying smooth structures of a given homo-topy type, and the homotopy invariant characteristic classes of smoothmanifolds. In this chapter we give a survey of the problems concerned


with relations for characteristic classes of smooth manifolds that arise inthe application of representation theory to the computation of characteristicclasses. These relations are known as Hirzebruch formulae.

§7. Hirzebruch formulae for finite-dimensional representations

1. Simply-connected manifolds. The classical Hirzebruch formula is con-cerned with a compact closed 4A>dimensional oriented manifold, say X.

We consider the cohomology group H2k (X; R) with real coefficients. Itis a finite-dimensional vector space on which there is defined a non-degenerate symmetric bilinear form (x, y) - (xy, [X]), x, y E.H2k(X; R).Here [X] denotes the fundamental class of X, [X] G H4k(X\ Ζ). Thesignature of this form is called the signature of X and is denoted by r(X).So we obtain an integral-valued function τ(Χ), defined on the set of all4k-dimensional oriented compact closed manifolds.

It is not difficult to see that for a disjoint sum X = Xx U X2 signaturesare additive: τ(Χ) - τ(Χγ) + τ(Χ2), and if the manifold is the boundary ofsome (4k + l)-dimensional oriented manifold W, X = dW, then the signa-ture T(X) of X is trivial. This means that r(X) determines a linear functionon the group Q,4lc of oriented bordisms. In bordism theory it is known thatevery linear function on the group of bordisms is a characteristic number,that is, there exists a characteristic class (say, L4k) such that

(7.1) x{X)=2*(Lth(X),[X]).

The equality (7.1) is usually called the Hirzebruch formula with anexplicit description of the class L4k. To define the characteristic class L4k

we denote by L the sum

(7.2) L= Σ L4 h.ft=0

Then in terms of the Wu generators ([30] ^[32]) the characteristic class(7.2) is

The characteristic class (7.2) or (7.3) is called the Hirzebruch class.There are at least two distinct proofs of the Hirzebruch formula (7.1).

One of these is based on the computation of the ring of oriented cobord-isms and the verification of the Hirzebruch formula on multiplicativegenerators. For if X = Xl X X2, then τ(Χ) = τ(Χχ) τ(Χ2). On the otherhand, it follows from (7.3) that L(X) = L(XX) L(X2). Thus, if the formulais true for two manifolds Xx and X2, then it is also true for X = Χγ Χ X2.

The second proof is based on the representation of the signature τ(Χ)of X as the index of an elliptic operator on X and the calculation of thisindex by means of the Atiyah—Singer formula ([31]). To represent τ(Χ)

H4 A. S. Mishchenko

as the index of an elliptic operator it should be observed that according tode Rham theory the real cohomology of X is isomorphic to the de Rhamhomology, that is, the cohomology of the complex of exterior differentialforms of X. Let Ω7(Χ) denote the space of exterior differential forms ofdegree / on X, and d: Ω ;(Χ) -»• Ω / + 1 (Χ) the operator of exterior differ-entiation. Then we obtain the de Rham complex

Ω° (Χ) Λ Ω ' ( Χ ) Λ . . . Λ Ωη (Χ)

(η = dim X, d1 = 0). On the space Ω(Χ) = e Sl'(X) of all exterior differ-/

ential forms there is the operation of exterior multiplication, which inducesa ring structure on the de Rham homology groups.

In particular, to determine the signature τ(Χ) of X (dim X = 4k) weconsider the bilinear form on the cohomology of middle dimensionH2k{X; R) induced by the equality

(7.4) (α, β>= j a A P , α, β ξ Ω ^ Χ ) .

Suppose that a Riemannian metric is given on X. It induces on each ofthe spaces Ώ,'(Χ) a scalar product

(7.5) (α, β) (α, β 6 Ω*(Χ)).

Then the bilinear form (7.4) can be reduced to (7.5) by means of someoperator*

(7.6) (α, β) = (a, * β).

Let τ(α) = ii{J-l)+2k *(cx) (a e Ώ.ΚΧ)). It is easily verified that r 2 = 1.Consequently, the space Ω(Χ) of all forms splits into the direct sum oftwo eigenspaces Ω(Χ) - Ω+ ® Ω~ for the involution r. Let δ be theoperator formally adjoint to the operator of exterior differentiation withrespect to the scalar product (7.5), and let D = d + δ.

It is easy to verify that DT = ~rD, that is, D maps Ω+ into Ω" and Ω~into Ω+. After this it remains to observe that

(7.7) index D+ = τ(Χ).

2. Multiply-connected manifolds. For a multiply-connected manifoldX, dim 4k, πχ(Χ) - π, we can define a whole series of signature invariantsusing the finite-dimensional unitary representations of the fundamentalgroup. We consider for this purpose a unitary representation ρ of π on afinite-dimensional space V. We denote by Η'{Χ', V) the cohomology groupsof X with coefficients in the local system V. Although we cannot intro-duced ring structure on the groups H'{X; V), we can define a bilinear formwhich, so to speak, can be constructed from X with the help of thecohomology product and from V with the help of the scalar productφ: V ® V ->· R on V:


(α, β) = (φ(ο ® β), [Χ]) (α, β ζ Η*(Χ; V)).

Then on the group of middle dimension H2k(X; V) there is a non-degeneratesymmetric bilinear form whose signature we denote by τρ(Χ). In the case ofthe trivial representation of π on R the signature τ (X) obviously coincideswith the ordinary signature of the manifold defined in subsection 1. Justas for the ordinary signature we can prove that the numbers are invariantunder, the special bordisms in which the fundamental group of the "film"is isomorphic to the fundamental group of the boundary.

In the language of bordism theory, this property can be formulated inthe following way. Let X be a smooth compact closed manifold andπι(Χ) = π. Then (see [33]) there exists a continuous map φχ: X -> Βπ ofX to the classifying space Βπ, unique up to homotopy, inducing an iso-morphism of fundamental groups (φχ)*: π1(Χ) -> π1(Βπ) = π. If W is thefilm bounding X, dW = X, and if π1 (W) is naturally isomorphic to πι (Χ)then ψχ extends to a continuous map <pw : W -*• Bit. Thus, τ AX) is definedby the integral-valued function

(7.8) τ ρ : Ώ*(Βπ) ^ Ζ

on the group of oriented bordisms of the classifying space Βπ. (For thedefinition of bordism see [34]).

Each numerical function on the group of bordisms can be described bymeans of characteristic classes ([35]). In particular, for the functions (7.8)there are characteristic classes a: of oriented manifolds and cohomologyclasses bt £ Η*(Βπ) such that

(7.9) τρ(Χ) = 2(ΜΧ)φ*(&/), [*]>•

The signature τ (X) has a number of algebraic properties which enable usto obtain a priori information about the right-hand side of (7.9). Firstly,τ (X) is an additive function on Ω*(2?π), that is,

(7.10) τρ(Χ1) + τ ρ ( Χ 2 ) = τ ρ ( Χ 1 υ Χ 2 ) .

Next, if Υ is a simply-connected oriented manifold, then

('•11) τ ρ ( Χ χ 3 - ) = τ ρ ( Χ ) . τ ( Υ ) .From (7.10) and (7.11) it follows that the right-hand side of (7.9) has onlyone component and that a(X) - L(X). For to determine the classes at{X)and bj it is enough to check the formula on some additive basis for thebordism group Ω*(5π). Moreover, since the left-hand side of (7.9) also isan integer, it is enough to check the same formula on an additive basis ofthe group Ω*(Βπ) <8> Q, where Q is the field of rational numbers. Such abasis can be chosen in the following way. Let Ω-ίΤ be the group of framedbordisms ([34]). Then there is the following isomorphism:

(7.12) Ω.(5π)®<ί«Ωί '0Βπ)®Ω 1 > ® Q.


Let {{Yj, ψγ.)} be a basis for Ώ/*(Βπ). Then every bordism(Χ, φχ) €ΞΩ*(5π) can be represented as a linear combination

(7-13) (Χ, ψχ) = Σ(ΖίΧΥ>, φγ.),

where the Zy- G Ω* are simply-connected manifolds. The left-hand side of(7.9) takes the form

(7.14) τρ(Χ) = Σ τ ( Ζ ; ) τ ρ ( Υ , , <py).j J

On the right-hand side of (7.9) we may suppose without loss of generalitythat the classes {£/} form a basis for the cohomology Η*(Βπ), dual to thebasis {(.Υ/, φγ.)}- Then

(7.15) τρ (X) = 2 (a, (Zj Χ Υ,) φ γ (b3), [Ζ, χ Yj]).j J

Since the tangent bundle to Yj is trivial, a;(Zy Χ 7;·) = α;·(Ζ;·). Consequently,

(7.16) τρ(Χ) = 2i

Setting Zj - 0 for / Φ I we get

(7.17)

Taking into account the classical Hirzebruch formula (7.1) we obtain forany simply-connected manifold Zh and therefore, in general, for anymanifold Zh the relation

(7.18)- 1

Turning to (7.15) we come to the simpler relation

f τ ρ (X) = 22h 2 (L (Zl Χ Υ;) τ ρ (Υ,, cpy ) φ* (6,), [Ζ, Χ Υ,]> =(7.19) { ' • ' ;

I = 22fe(L(X)cp£(6), [Χ]),where b = Σ τ (Yh ψγ^ι.

It remains only to establish the value of the cohomology classb G Η*(Βπ), which depends only on p.

Unfortunately, the bordism method fails here. Therefore, we have to usethe other method described in subsection 1 to compute the classicalsignature of manifolds with the help of a suitable choice of ellipticoperator. The first such calculation was carried out (in a somewhat moregeneral setting) by Lusztig [ 26 ] .

In contrast to the classical signature, we have to consider, in place of thede Rham complex, exterior differential forms with values in a finite-dimensional bundle ψχ (ζρ) (the dimension of ξρ is the same as that of thevector space V on which IT acts). The only difficulty that has to be over-come in constructing the elliptic Hirzebruch operator in our case consistsin verifying that we can construct the operator of exterior differentiationof exterior differential forms with values in <£y(£o)- For this purpose it is

Hermitian K-theory. The theory of characteristic classesand methods of functional analysis 117

enough to remark that in a coordinate representation of φχ{ζρ) we canchoose the transition functions on the intersections of the local charts sothat they are locally constant matrix functions. All the remaining con-structions remain unchanged. It turns out that

(7.20) b = chp = c h | p .

Thus, we obtain the multiply-connected Hirzebruch formula

(7.21) xp(X) = 2^(L(X)^chlp,[X])

for a finite-dimensional unitary representation ρ of the fundamental groupof X.

The class of formulae (7.21) can be increased [26] by consideringrepresentations of π in the group of automorphisms O(m, n) of the vectorspace Vm+n, preserving a bilinear Hermitian form of type (m, n). With thehelp of such a representation ρ we can define a signature τρ{Χ) of coho-mology with coefficients in the local system of coefficients Vm+n, andalso an element ηρ €Ξ Κ(Βπ). This element ηρ is constructed as follows.Let £ be the finite-dimensional bundle constructed as in §5, from therepresentation ρ: π -> O(m, n) C GL(m + n) of π in the group of alllinear automorphism of F m + " . Then in each fibre of ξ we have a bilinearform of type (m, n), which induces a decomposition of £p into the directsum ξρ = ξ* ® |~ of two subbundles on one of which (£*) the bilinear formis positive definite, and on the other (£~) negative definite. Then bydefinition

(7.22) η ρ = IJ — ξρ·

As a result we obtain the Hirzebruch formula

(7.23) Tp(X) = 2 2 f t ( L ( Z ) c h 9 ^ p , [X]).

§8. Algebraic Poincare complexes

1. The problem of classifying smooth structures and the homologyinvariants of a manifold. The Hirzebruch formulae obtained in §7 formultiply-connected manifolds with the help of finite-dimensionalrepresentations of the fundamental group are, apart from the purelymathematical beauty of the relations, of deep value for a number ofproblems in the theory of smooth manifolds.

We mention, first of all, that the definition of the signature τ (X) of Xfor a representation ρ only depends on the homotopy properties of thecohomology ring H*(X; V), that is, it is a homotopy invariant of X. Thismeans that if ψ: Χλ -*• X2 is a continuous map of a manifold Χγ to amanifold X2 and a homotopy equivalence or, what comes to the samething, if φ induces an isomorphism of all homotopy groupsφ**: irk(Xl) ->• irk(X2) (k > 1), then the signature of X and Xt are equal,


τρ{Χ\) = τρ(Χ2), for each finite-dimensional representation ρ ofπ = ΤΓ,ΟΧ-,) = π,(ΛΓ,).

Consequently, the right-hand sides of the Hirzebruch formulae, which arecharacteristic numbers of the manifolds and hence do not a priori dependonly on the homotopy type of the manifold but also on the smoothstructure introduced on it, are in fact homotopically invariant.

Research on the homotopy invariance of characteristic classes (and alsoof the characteristic numbers derived from them) began to be developedabout 25 years ago. The first results in this direction are due to Rokhlin[36] and Thorn [37], who proved the homotopy invariance of the (unique)Pontryagin class on a four-dimensional oriented manifold. The proof wasessentially based on the formula they found for the signature (7.1) forfour-dimensional manifolds. In 1956 Hirzebruch [38] established, properly,the general formula (7.1) for arbitrary oriented 4/c-dimensional manifolds andso proved the homotopy invariance of the Pontryagin—Hirzebruch classesLAk(X), dim X = 4k.

Further research showed that the Pontryagin classes, generally speaking,are not homotopy invariants (G. Whitehead, Dold, Thorn) and for simply-connected manifolds the highest Hirzebruch class is the only homotopicallyinvariant Pontryagin class. The last assertion was proved by Browder [39]and S. P. Novikov [40] on the basis of a full study of the problem ofclassifying smooth structures of given homotopy type of simply-connectedmanifolds. In fact, they established a stronger assertion: in each homotopytype of simply-connected manifolds the Pontryagin class is a homotopyinvariant if and only if it is proportional to the highest Hirzebruch class.

In the case of multiply-connected manifolds the answer to the problemof the homotopy invariance of the characteristic classes is more compli-cated, but in one way or other it has been directly tied up with progressin the solution of the other problem: that of classifying smooth structuresof given homotopy type of a multiply-connected manifold.

We turn now to a precise description of the problem of classifying smoothstructures of given homotopy type. Let Xo be a finite CW-complex. Asmooth structure of homotopy type Xo consists of a smooth closed mani-fold X and a homotopy equivalence φ: X -»· Xo. Two smooth structures(X\, ψι) and (X2, ψ2) of the homotopy type of Xo are said to be equi-valent if there is a smooth homeomorphism

such that the diagram

χ*is homotopically commutative, that is, the maps ψι and φ2φ are homo-topic. Then the set of all smooth structures of homotopy type Xo splits


into classes of pairwise equivalent structures. We denote the set of theseclasses by tf (Xo).

Then the problem of classifying smooth structures of given homotopytype Xo consists in describing the set c/(X0) with the help of the homotopyinvariants of Xo. In particular, one of the questions consists in findingmethods of distinguishing smooth structures that are effective from thetopological point of view.

We mention to begin with that for the set e7(X0) to be non-empty it isnecessary that Xo has certain properties, namely that Xo is a Poincarecomplex· of some formal dimension n. This means that there is a cycle[Xo] ^ Hn(X0; Z) such that the intersection homomorphism

Π IXoh H*(X0; Α) -+Η,{Χ0; A)

is an isomorphism, where Λ = Ζ[π] is the group ring of the fundamental

group π = π^Χο) of Xo-One of the invariants of a smooth structure (Χ, φ) of given homotopy

type Xo is the "normal bundle of the smooth structure", that is, the ele-ment £ = (φ~1)*(ν(Χ)) S K0(X0). It is clear that if the normal bundles oftwo smooth structures are distinct, then so are the smooth structures them-selves. But on the one hand, not every element of Ko (Xo) can be realizedas the "normal bundle of a smooth structure", while on the other hand,different smooth structures may a priori have the same "normal bundle".

For an element £ €Ξ Ko (Xo) to be realized as a "normal bundle of asmooth structure" a number of conditions have to be satisfied. One suchrelation is easily written in geometric terms. Let ξ be a finite-dimensional(dim ξ = Ν) bundle over the complex Xo, and X | the Thorn complex ofξ (that is, the one-point compactification of the space of £). Suppose nowthat:

There exists a map of the (N + n)-dimensional sphere:

(8.1) χ: SN+n -»-X§ of degree 1.

Condition (8.1) is one of the requirements that have to be imposed on £for it to be the "normal bundle of a smooth structure". For, if (Χ, φ) isa smooth structure, then the map induces a bundle homomorphism

(8.2) φ : ν ( Χ ) + ξ .

Recalling that the space of the normal bundle (for sufficiently largeΝ = dim v(X)) can be regarded as a domain of the sphere SN+n, we canextend the map (8.2) to a map χ of the sphere SN+n to the one-pointcompactification A"J of the space of ξ. The map χ is transverse-regularalong the null section XQ C Χ$, and χ'Η-^ο) = %·

(8.1) has another equivalent formulation when Xo is a smooth manifold,in terms of the so-called /-functor and the Adams cohomology operations


(see [41]—[45]). It is useful for us to note only one fact: the set ofelements ξ Ε Κο(Χ0) satisfying (8.1) is a coset of a subgroup of finiteindex in Ko{X0).

However, (8.1) is by no means a sufficient condition for the existenceof a smooth structure with a given normal bundle. The only thing that canbe asserted is this: there is a manifold X and a map φ: X -*• Xo of degree1 so that the inverse image φ*{%) is isomorphic to the normal bundlev(X). But φ need not be a homotopy equivalence. Therefore, it is reason-able to ask with what modifications of X and of φ we end up with ahomotopy equivalence.

Now at this point the famous technique of modifying smooth manifoldsor surgery enters the stage. The manifold X, the map φ, and the isomorphismφ: v(X) -*• φ*(ζ) in the previous paragraph form a triple (Χ, φ, ψ), andamong them we can establish a natural bordism relation. The triple(Χ, φ, φ) is called the normal map of the manifold X to the pair(Xo, £). Using surgery we can answer the following question.

Let (Χ, ψ, φ) be a given normal map. When is there a triple {X\ φ, φ')bordant to it in which φ is a homotopy equivalence? The answer wascompletely worked out by Novikov for simply-connected manifolds [40]and by Wall for multiply-connected manifolds [3], [4] . Namely, let Ln(ir)be the Wall group of π, defined in §1.4, and let π = ΐΐ^Χ^). Then witheach triple we can associate an element Θ(Χ, φ, φ) Ε Ln(jt) satisfying thefollowing conditions:

a) Θ(Χ, φ, φ) depends only on the bordism class of the triple,b) Θ{Χ, φ, φ) = 0 if and only if there is another triple (Χ', φ', ψ')

bordant to (Χ, φ, φ) in which φ' is a homotopy equivalence.The element Θ(Χ, φ, φ) is called the obstruction to modifying the map

φ to become a homotopy equivalence. The first condition means thatΘ(Χ, φ, φ) must be completely described at least up to elements of finiteorder by means of the characteristic classes of the map φ: X -*• Xo. Infact, in the case of simply-connected manifolds and η = 4k, the groupL4lc(l) is isomorphic to the group of integers and is described by a singleintegral-valued parameter. Then

(8.3) Θ(Χ, φ, ψ) = τ(Χ) - τ(Χ0),

where τ(Χ) is the signature of the manifold.Formula (8.3) and property b) of the surgery obstruction gives in the

simply-connected case complete information on the homotopy invarianceof the rational Pontryagin classes of the manifold. For, by the Hirzebruchformula, r(X) = <L^k(X), [X]). Consequently, if φ: X -* Xo is a homotopyequivalence of two manifolds, then Θ(Χ, ψ, φ) = τ{Χ) - τ(Χ0) = 0. Therefore,LAll(X) = L4k(X0). To establish that every other Pontryagin class is nota homotopy invariant we must produce two manifolds X and Xo and ahomotopy equivalence ψ: X -* Xo such that p{X) Φ φ*ρ(Χ0). Consequently,


in terms of the problem concerning surgery it is sufficient to produce abundle £ satisfying (8.1) and also such that p(£) Φ ρ(Χ0) andΘ(Χ, φ, φ) = 0, that is, L4k(%) = L4k(X0). The latter is a problem concern-ing the algebraic properties of Ko (Xo) and has been solved positively.

2. Generalized signatures of multiply-connected manifolds. We have cometo the problem of describing the obstruction to surgery Θ{Χ, φ, ψ) G Ln(it)with the help of the characteristic classes of ψ. But before we give thisdescription, we turn our attention to the fact that in the case of simply-connected manifolds the obstruction to surgery Θ(Χ, φ, \p) can be expressedin terms of the signatures of X and Yo,

(8.4) Θ(Χ, φ, ψ) = τ(Χ) - τ(Χ0),

that is, by the homotopy invariants of the two manifolds. That the signa-ture T(X) of X is the only homotopically invariant characteristic numberfollows from (8.4).

Thus, also in the multiply-connected case we must try to express thesurgery obstruction Θ(Χ, φ, Φ) in terms of the homotopy invariants of Xand Xo. For simplicity we look for these invariants as elements of K%(A)for the group ring Λ = Ζ[1/2][π] (see [46]). For the changes that wouldhave to be made for Z[7r], see [7].

Thus, we consider a ring Λ with involution containing 1/2. Let (M, d)be a free chain complex over Λ, that is a free graded Λ-module Μ (see§ 1) and a homogeneous homomorphism d of degree (— 1)

d: Μ -> Μ, <Ρ = 0.

Then the dual complex (M*, d*) also is a chain complex. Let β:Μ*-+Μbe a homogeneous homomorphism of degree η and suppose that thefollowing conditions hold:

(8.5) a) β* = β,

(8.6) b) dp + βό* = 0,c) the homomorphism β induces an isomorphism of the

homology groups

(8.7) #(β): Η*(Μ*, d*) -> H(M, d).

The equality (8.5) must be understood as an identification of the modulesΜ and (M*)*, that is, in the form &Μ β* = β. Then we call the triple(M, d, β) an algebraic Poincare complex (APC) of formal dimension n.

Before we turn to the study of APC's, we mention that Hermitian formsand also automorphisms preserving a Hamiltonian form are particular casesof an APC or can be reduced to one. For let us consider a chain complex(M, d) such that Mf = 0 if / Φ k, η = 2k. Then the homomorphism βreduces to a map on the one term β: Μ% -*• Mk, while (8.7) means thatβ is an isomorphism. Then (8.5) on replacing the fcth grading by the zerograding goes over into condition (1.17). Thus, the pair (Mk, |3) is a (skew-)Hermitian form over A.


To interpret automorphisms in terms of APC's, we consider the APC(M, d, β), η = 2k + 1, such that Mf = 0 for / Φ k, k + 1. Then β gives usthe diagram:

Mh —-> Mk

(8.8)d*

+ 1 <

Let us forget the gradings on M. Then in place of d* in (8.8) we musttake (~\)kd*, according to (1.12). Thus, άβ + {~\ψβά* = 0 in (8.8). Now(8.5) means that (β£ + 1 ) * = flk. If we introduce on the module JW^+1 ®Mk+l

a (skew-) Hermitian Hamiltonian form by means of the matrix

(8.9)

then the homomorphism

determines a Lagrangian plane ΐηΛί^+1 ®Mk+l and hence a class of auto-morphisms preserving the form (8.9).

In the set of all APC's we can introduce a relation of bordism type. Weconsider a free chain complex (M, d) and a subcomplex (Mo, d) so thatΛ/ο C Μ is a direct summand. Let β*: Μ* -*• Μ be a homogeneous homo-morphism of degree π such that

(8.10) a) β* = β,

(8.11) b) ώβ + $d* = 0(mod M 0),

c) the homomorphism (3 induces an isomorphism of thehomology groups

(8.12) #(β): H{M*, d*) -*· H{MIMO, d).Then we call α = (Λί, Mo, d, β) an algebraic Poincare complex with bound-ary (APC with boundary), of formal dimension n.

From (8.10) and (8.11) it follows that the homomorphismάβ + βά*: Μ* ->• Mo has a unique decomposition

where /: M o -*· Μ is the inclusion and the triple (Mo> c?, j30) is an APC offormal dimension (n — 1). From (8.12) we obtain the following diagram ofexact homology sequences:

Hermitian K-theory. The theory of characteristic classes and methods of functional analysis χ 23

H(M0) -> H{M) -* H(M/M0) _» jy (ΛΓ0)

t t i tH(Po) H(P>* H(P) H(Po)

The triple (Mo, d, j30) is called the boundary of the APC with boundary aand is denoted by 9a. If a, = (M\, dt, j3t) and a2 = (M2, d2, β2) are twoAPC's, then we call their sum the APC (Μ, Θ M2, dx θ d2, β1 © β2). Ifa = (M, d, β) is an APC, then by (-a) we denote the APC (M, d, -β). Theoperation a. -*• (—a) is called change of orientation on the APC ex.

Two APC's a t and OL2 are said to be bordant if there is an APC withboundary 7 such that ax ® (~α2)

= 97. Bordism is an equivalence relation.The complete analogy with the theory of bordisms of smooth (or

piecewise-linear) manifolds is not by chance. The fact is that with eachclosed oriented smooth simplicial manifold X with fundamental groupπ = iTi(X) we can associate an APC a(X) over the ring Λ = Ζ[1/2] [π]. Todo this we take for the free chain complex (M, d) the complex of simpli-cial chains of X with local system of coefficients A, and for β the operator

where (Γ)[Χ]) is the operator of intersection with the fundamental cycle(see [47]).

The same construction can be made for a smooth oriented manifold Xwith boundary 9^. Here we have

(8.14) δσ(Χ) = σ(ΘΧ).

We must sound one note of caution. The operation (Π[ΛΤ]) depends,generally speaking, on the ordering of the zero-dimensional vertices of thesimplicial decomposition of X. Therefore, strictly speaking, the APC a(X)is not uniquely determined. But it is not difficult to prove that fordifferent orderings of the vertices of X we get bordant APC's, and (8.14)is true for coherent orderings on X and its boundary dX.

Thus, we find it convenient to introduce the following notation. Wedenote by Ω^(Λ) the set of bordism classes of APC's. The sum operationfor APC's induces a group structure on Ωη(Α).

Then all the above constructions and examples can be presented asseveral homomorphisms.

a) The group ^ ( A ) is mapped homomorphically to Ωη(Λ):

(8.15) ψ. Κ*(Α) -+ Ωη(Λ),

b) The groups of oriented bordisms Ώ,%° (Βπ) of the classifying spaceΒπ are mapped homomorphically to ΩΜ(Λ):

(8.16) σ: Ω ° (ftx)-^ Ωη (Λ).

To establish the existence of the homomorphism (8.15) corresponding to


the interpretation of Hermitian forms and automorphisms as APC's, we haveto verify that to equivalent Hermitian forms and automorphisms there corres-pond bordant APC's.

We turn now to a description of those properties of φ and a thatexplain to us the usefulness of introducing APC's. First of all, the analogybetween APC's and smooth manifolds can be taken further. In particular,the operation of elementary surgeries of smooth manifolds and of "glueingon handles" can be interpreted "on the level" of their chain complexes sothat analogous elementary operations can be defined for APC's. Theinclusion of a sphere in the smooth manifold is replaced by a homomorphismof a free Λ-module to the group of cycles of the chain complex. Theabsence of any obstruction to the construction of such homomorphisms (incontrast to the embedding of a sphere in the manifold) leads to the con-clusion that Milnor's method of killing homotopy groups ([48], [49]) ofsmooth manifolds can be applied to "killing" the homology groups of anAPC. Then every APC α is bordant to another APC a' whose homologygroups are non-zero only in the middle dimension (for η = 2k) or thetwo middle dimensions (for η = 2k + 1).

Moreover, chain-homotopically equivalent APC's are bordant.As a result we come to the following assertion.THEOREM 8.1. The homomorphism ψ: Κ%(Α) -*• Ω Λ (Λ) is an isomor-

phism.Thus, we cannot distinguish the groups K%(A) and ΩΜ(Α), that is, with

each APC of even formal dimension we can associate a Hermitian formand with each APC of odd formal dimension an automorphism preservinga Hamiltonian form. We need not use Theorem 8.1 to do this, by theway, but can use the following simple argument. Let (M, d, β) be an APC,which we present in terms of the following diagram:

(8.17)

This can be regarded as a biregular complex. Then we can construct from(8.17) a new complex in which the grading is equal to the sum of the twogradings in (8.17):

(8.18) M0+-Mi®Ml+-...^Mn®M\'<-Ml1

and the differential is equal to the sum of the differentials (see [50]) in(8.17).

The chain complex (8.8) is "self-dual" and exact. Consequently, we cansplit it symmetrically with respect to its ends into the direct sum of threecomplexes (lopping off direct components from both ends) one of whichhas the form


(8.19) 0 •«- Nk -*- N% *- 0

when η - 2k, or the form

(8.20) 0*-Nh+Mh+l®Mt+l+-Nt+-0

when η = 2k + 1. The complex (8.19) determines a Hermitian form and(8.20) a Lagrangian plane in Hamiltonian form.

Taking Theorem 8.1 into account, we may suppose that the homomor-phism (8.16), in fact, maps the group of oriented bordisms Ω^°(5π) toAT* (π):

(8.21) σ: Ωξ°(Βπ)-+ΚΪ(*)·

We denote by γ: Ln(n) -*• Κ%(π) the natural homomorphism of the Wallgroup Ln(n) to Κ^(π). Let Xo be a smooth manifold, π = TT^XQ),

(Χ, φ, ψ) a normal map of X to the pair (Xo, ξ), that is, a mapφ: X -> Xo and a bundle isomorphism ψ: v(X) -+ ψ*(ξ). The obstructionΘ(Χ, φ, \p) to modifying the map by surgery to become a homotopyequivalence lies in Ln(n).

THEOREM 8.2.

(8.22) γ(θ(Ζ, φ, ψ)) = σ(Χ) - σ(Χ0)

inTheorem 8.2 can be regarded as an analogue of (8.3) for simply-

connected manifolds. Naturally, the element o(X) is called the generalizedsignature of the manifold. The generalized signature, just like the classicalsignature, is a homotopy invariant and a bordism invariant. Consequently,σ(Χ) can be described with the help of the characteristic numbers of thebordisms of BIT, that is, the numbers of the form (ρ(Χ)ψχ (a), [X]), whereρ is a Pontryagin class and a S Η*(Βπ) a cohomology class.

With the help of Theorem 8.2 we obtain an answer to the question whatcharacteristic classes of multiply-connected manifolds are homotopyinvariants.

THEOREM 8.3. Let X be a multiply-connected manifold, π = π^Χ). Thenthe only homotopically invariant characteristic numbers are those of theform

(8.23) σα(Χ) = <£(Χ)φ*(β), [X]),

where a £ H*(Bir\ Q) and L(X) is the Hirzebruch class of X.The numbers (8.23) are called the higher signatures of X. For some

classes of groups η it has been established that all the higher signatures arehomotopy invariants. In 1965, S. P. Novikov ([51], [52]) established thehomotopy invariance of the Hirzebruch class of codimension 1 and somepartial results for other codimensions. There he conjectured1 that if a

1 See also [78].


cohomology class a is the direct product of 1-dimensional cohomologyclasses, then the corresponding higher signature oa{X) is a homotopyinvariant. In 1966, Rokhlin ([53]) proved this conjecture for two-dimensionalclasses, and later it was completely proved by Farrell and Hsiang [54] andby Kasparov [55].

The higher signatures aa{X) are functions on bordisms

(8.24) σα: O f (Ba) -* Q.

If oa(X) for some class a £ Η*(Βπ; Q) is a homotopy invariant, then thehomomorphism (8.24) splits:

(8.25) ^ /

Kn(n)

Conversely, if δ: Κ%(π) -> Q is a linear functional, then it determines ahomotopically invariant characteristic number δ(σ(Χ)) of X, which accordingto Theorem 8.3 is necessarily a higher signature aa(X) for some cohomologyclass a = αδ € H*{Bit; Q). Consequently, running through all the linearfunctionals δ: Κ%(π) -> Q, we obtain all the classes ab £ H*(Bir; Q) whosehigher signatures are homotopically invariant.

§9. Hirzebruch formulae for infinite-dimensionalFredholm representations

The generalized signature a{X) £ K%(rt) of a multiply-connected mani-fold X gives a new interpretation of the multiply-connected Hirzebruchformulae (7.21). The left-hand side τρ(Χ) of (7.21) is a homotopy invariantand, therefore, must depend only on the value of the generalized signature.A detailed analysis of the definition of τ (X) for a finite-dimensionalrepresentation ρ of the fundamental group π = it\{X) of a manifold Xshows that this number can be expressed in the following way:

(9.1) τ ρ (Ζ) = signp(a(X)),

where sign is the signature of the Hermitian form defined in §4, (4.19).Thus, the Hirzebruch formula can be written more naturally in the

following form:

(9.2) signp(a(Z)) = 22fi<Z,(X)cp£ch|p, [X]).

This way of writing the Hirzebruch formula is useful to us because boththe left- and the right-hand sides of it make sense for a wider class ofrepresentations: the infinite-dimensional Fredholm representations of π. Forif ρ is a Fredholm representation of π, then the left-hand side of (9.2) isdefined by (5.18), and the element £ on the right-hand side is defined in


§6.2. Thus, we can state the following proposition [ 1 7 ] :

THEOREM 9.1. Let X be a closed compact oriented manifold,dim X = 4k, τΤι(Χ) = π, and ρ a Fredholm representation of IT. Then wehave the Hirzebruch formula

(9.3) signp(a(Z)) = 22ft(Z(X)cpich|p, [X]),

where ψχ: X -*• Βπ is the classifying map for X.If ρ - {ργ} (y € F) is a family of Fredholm representations, then the

left-hand side of (9.2) is no longer a number, but the bundleugnp(a(X)) G K(Y), and the element £p is a bundle with base spaceΒπ Χ Υ. By applying to the left-hand side the Chern character, we can re-write the Hirzebruch formula as an equality of two cohomology classes ofH*{Y; Q):

(9.4) ch signp(a(X)) = 22i!(L(X)cp£chEp, [X]).

The proof of (9.3) for infinite-dimensional Fredholm representations doesnot repeat that of the Hirzebruch formula in the case of finite-dimensionalrepresentations. The latter is based on the observation that its left-hand sidecan be interpreted as the signature of certain cohomology groups and,consequently, can be reduced to the index of some elliptic operator. In thecase of infinite-dimensional Fredholm representations this interpretation isnot available to us, and we can only use the indirect definition of theHermitian form a(X) for X.

We mention the highlights of the proof of (9.3) for infinite-dimensionalFredholm representations. Let X be a smooth simplicial oriented closedmanifold, π = n1(X), and dim X = 4k. To construct the Hermitian formo(X) we start from the APC of simplicial chains of X with the local systemof coefficients Λ = Z[l/2] [?r]. First of all, taking account of a remark in§4, we change the ring of scalars Z[l/2] to the field of complex numbers.Therefore, let Λ = C[ir]. Then a(X) is an APC of formal dimension η - 4k.To obtain a Hermitian form, we perform on o(X) a number of elementaryoperations of the type of Morse surgeries for smooth manifolds andfactorizations by acyclic subcomplexes. The whole set of these operationscan be organized as an APC α with boundary, of formal dimension η + 1,and the boundary θα is the direct sum of the APC a(X) and a Hermitianform ao{X).

We consider now a Fredholm representation ρ = (plt F, p2) of π (and,hence, of A). The unitary representations px and p 2 act on Hubert spaces//] and H2, respectively. By analogy with (4.4) we consider the tensorproducts of a(X), a, and σο(Χ) with px and p2, respectively. In the endwe obtain two infinite-dimensional "algebraic Poincare complexes", sayα and <xp , and the Fredholm operator F defines (up to compactoperators) a Fredholm map Fa. ap -> ap . Moreover, each elementaryoperation on APC's of the type of a Morse surgery or factorization by an


acyclic complex goes over, under the tensor product, to a pair of element-ary operations compatible with the Fredholm map Fa. Therefore, it is notdifficult to verify that signp (σο(Χ)) can be computed with the help ofAPC's, by passing to the complexes (σ(Χ))Ρ] and (σ(Χ))Ρ2 and using theconstructions described by the diagrams (8.17)—(8.19).

The next stage consists in passing from the Hubert "algebraic Poincarecomplexes" (σ(Χ))ρ and (σ(Χ))ρ to some elliptic complex of differentialoperators.

Each of the unitary representations px and p 2 induces infinite-dimensional bundles My and 38* on X, and F is a family of Fredholmoperators Fx:$8-y-*$£2.Note that the bundlesMyandSS%are locally flat.We can, therefore, consider the spaces of exterior differential formsΏ/\Χ; &Sy),Q/(.X;$g2) with coefficients i n ^ a n d d ^ .

Just as in the cases of finite-dimensional bundles, we construct involutionsak\ Ω/(Χ; Mk) -»• £ln~~'XX;Mh) continuous in the appropriate Sobolevnorms.

The de Rham complex of exterior differential forms {Ώ!{Χ;Μκ)ι d} canbe mapped to the complex of simplicial cocycles (σ(Χ))* by integrating

over simplexes. This homomorphism induces an isomorphism of cohomologygroups. Then each elementary operation on (σ(Χ))ρ is matched by a

corresponding operation on {n'(X;SSh),d}. As the result of a finite num-ber of such operations we obtain an isomorphism of two Hermitian forms.On the other hand, its signature can be computed as the index of anelliptic diagram, by splitting © Ω7(Χ; $Sh) into the direct sum Ω+ ® Ω" of

/two eigensubspaces of a. and constructing the Hirzebruch operator{d + δ): Ω+ ->· Ω" corresponding to this decomposition. The index of thisdiagram (taking into account the Fredholm map Fx) by the Atiyah—Singerformula is equal to (L(X) ch£p, [X]).

The formulae (9.3) and (9.4) provide us with plenty of homotopicallyinvariant higher signatures, because the right-hand side of 9.3 is a highersignature for the cohomology class

(9.5) a = chl p·

In (9.4) we have on the right-hand side a whole family of higher signatures.For let b} £ H*(Y; Q) be a basis for the cohomology groups and letch£p = Σ α} <8> bj. Then the right-hand side of (9.4) has the form

(9.6) 2 2*Σ( ^j

Consequently, the higher signatures aa. are homotopically invariant.

The Hirzebruch formulae, together with the image (6.1) of the homo-morphism from the set of Fredholm representations i?(7r) into the groupΚ(Βπ) describe a set of higher signatures of varieties, which are homotopically

Hermitian K-theory. The theory of characteristic classes and methods of functional analysis j 29

invariant.For example, if π satisfies (5.9), then it follows from (6.13) that all the

higher signatures of a manifold with fundamental group π are homotopicallyinvariant.

Of course, the class of group π with homotopy invariance of the highersignatures for which we can obtain detailed information is much wider. Forexample, if ?r is a free product of two other groups, π = ττλ * π 2 , then theclassifying space Βπ is homeomorphic to the connected sum of theclassifying spaces, Βπ = Βπχ V 5 π 2 . Consequently, the cohomology andthe ΛΓ-functor split into direct sums:

H*(Bn, *t) = H*(Bnlt pt)® H*(Bn2, pt),

K°(Bn, *t) = K'iBTi!, pt)® K°(Bn2, pt),

and the set of Fredholm representation Μ (π) contains the direct sum91 (IT χ) ® Μ(π2)- Thus, the image of (6.1) for the group π contains thedirect sum of the homomorphism (6.1) for π, and π 2 . In particular, for afinitely generated free group π all the higher signatures are homotopicallyinvariant.There are 4/:-dimensional manifolds X involved in the Hirzebruchformulae. Therefore, the homotopy invariance of the higher signaturesoa(x) is also proved for 4A>dimensional manifolds X and %-dimensionalcohomology classes S Η*(Βπ; Q). However, similar assertions on thehomotopy invariance of the higher signatures can also be proved for otherdimensions.

Suppose that π satisfies (6.9), and, for example, dim X - 4k— 1. Weconsider the direct product X' - X X Sl of X with the circle S1. Thenthe group π' = πι(Χ Χ 5 1 ) = π Χ Ζ also satisfies (6.9). Consequently,there is a family of Fredholm representations ρ = {py} (y € Y) such thatch£p = Σ ai ® bj and ai (ΞΗ*(Βπ'), bt E.H*(Y), are bases for the cohomologygroups. The space BIT' is homeomorphic to the direct product BIT' = Βπ Χ S1.Therefore, the basis {a,·} can be represented in the form {a'-, a'. <8> υ},α'. €Ξ Η (Βπ), υ G H1^1). The Hirzebruch formula gives us the homotopyinvariance of the number oa< (X') = (L(X)<fi^-(aj <8>'u), [X'] >. SinceL(X') = L(X), with X' - XX S\ it follows that

σο ;(Χ) = <Ζ,(Χ)φ3:(α·), [X]) = {L (Χ') φ$(α} ® ν), [ Χ χ 5 ' ] ) = α ^ , ( Α " ) .

Consequently, the higher signatures aa;{X) for dim a'- = 4& - 1 are homo-topically invariant.

For the other dimensions the homotopy invariance of the higher signa-tures is established similarly.

§10. Other results on the application of Hermitian AT-theory

1. The application of bordism theory to the Hirzebruch formulae. TheHirzebruch formulae both for finite-dimensional, and for infinite-dimensional


Fredholm representations have been proved on the basis of the Atiyah—Singer theorem on the index of elliptic operators.

In one particular case - for simply-connected manifolds - there isanother way of proving the Hirzebruch formula on the basis of bordismtheory. The difficulty in applying bordism theory in the case of themultiply-connected Hirzebruch formula lies in the choice and effectivedescription of generators of the group of oriented bordisms Ω*(2?π) of theclassifying space Βπ of π. As was shown in §7, it follows from bordismtheory that the right-hand side of the Hirzebruch formula always is of theform τρ{Χ) - (L(X)a , [X]), where a G Η*(βπ) is some cohomologyclass depending on p. Consequently, the main difficulty is in proving thatap = ch£p. In this direction there have been some definite advances ([56]),which also allow us to make clear the homotopy nature of the propertyof the higher signatures to be homotopy invariants of multiply-connectedmanifold.

By some analogy with the papers of Volodin ([68], [70]) we canconstruct for Hermitian ίΓ-theory a universal space H^A), having a numberof useful properties from the point of view of Hermitian ΛΤ-theory. To dothis, we consider a ring Λ with involution. In §8 we have defined analgebraic Poincare complex over Λ and also an APC with boundary. Wecan go further in this direction and, by analogy with η-manifolds, definebundles of APC's over some simplicial complex K. The exact definition isas follows: let Κ be a simplicial complex, and denote by Κ also thecategory of all its subcomplexes and their inclusion morphisms. We considerthe functor Π that associates with each subcomplex K' C Κ the complexΠ(ϋΓ') of free Λ-modules, with U(K'C\ Κ") = Π ( ί ' ) η Π ( Ι " ) andΠ(2Τ DK") = U(K') + Π ( Ο , and if K' C K", then Π(/Ο C Yl{K") as adirect summand. Next, suppose that with each oriented simplex σ C Κthere is associated a homomorphism β(σ): Π(σ)* -»• Π(σ), satisfying theproperties of an APC with respect to the boundary Π(9σ) C Π(σ), offormal dimension dim σ. Suppose, finally, that the homomorphisms arecompatible in the following way:

(10.1) d$(e) + $(G)d= 2 β (σ').

where the summation is over all orientations of the boundary σ' C σ com-patible with that of σ. The functor Π and the homomorphism β(σ) arecalled a sheaf of algebraic Poincare complexes.

It is important to note the following assertion: let X be a simplicialsmooth oriented manifold, and Π a sheaf of APC's over X. Then the pair(Il(X), βχ), where

(10.2) β χ = Σ β (σ),dim o=dim X


is an APC.APC sheaves behave like bundles. In particular, it makes sense to speak

of the inverse image of a sheaf of APC's under simplicial maps. Eachsimplicial complex Κ has a distinguished sheaf of APC's, that of its simpl-icial chains over the ring of integers, or if π = πλ(Κ), also over the groupring Λ = Ζ[1/2] [π] of π. Then W(A) is the "universal" complex withuniversal sheaf Π Λ of APC's over it.

If /: X -»• W(A) is a simplicial map of a smooth manifold X, then fromthe inverse image / * ( Π Λ ) we construct the APC {/*(ΠΛ)(Τ), βχ} (see(10.2)) and according to Theorem 8.1, the element

(10.3) ψ-ΗΧ, /) € #η(Λ), η = dim X.

Now (10.3) depends only on the bordism class (X, f) G £ln(W(A)). Conse-quently, we obtain the homomorphism

(10.4) a: O f (W (Λ)) -> Kh

n (A).

The elements of the homotopy group nn(W(A)) determine in a natural waycertain bordisms of W(A), that is, there is a homomorphismω: πη(Κ>(Λ)) -> Ω%° (W(A)). It turns out that the composition

(10.5) αω: π # ( Λ ) ) -»-Λ5(Λ)

is an isomorphism.As we have already shown, over the classifying spaces BIT there is the

special sheaf of APC's of its simplicial chain with coefficients in the localsystem Λ = Ζ[1/2] [π]. Then we obtain the map

(10.6) ζ: Bn-*-W(A),

which induces a homomorphism ξ * : Ω^° (5π) -> Ω^° (W(A)). The com-posite map

coincides with the generalized signature (8.21)

σ: ς%>(Βπ)-+Κ*ί(π).

All the homotopically invariant higher signatures can be written in thefollowing way. Let 7: Κ^(π) -• Ζ be a linear functional. The compositemap ay. Ω,^° (W(A)) -> Ζ is described by characteristic classes of the form

ay(X, /) = (L(X)f*(ay), [X]),

where ay G H*(W(A); Q) is some cohomology class. Then the highersignature ab(X), b = $*(ay) G Η*(Βπ; Q) is homotopically invariant. Sincethe homotopy groups of W(A) are periodic of period 4, then the cohomologyring H*(W(A); Q) is described by the generators of the cohomology groupof the first four dimensions. It is convenient, in fact, to make at the outset


the change of rings Λ -*• A <g> C. Then the period of the homotopy groupsis two. Let u}^Hx(W{K ® C); Q) and vk EH2(W (Λ ® C);Q) be basesfor the cohomology groups. Then there are elements vk t E.H2l(W(A <8> C); Q)such that the cohomology ring H*(W(A ® C); Q) is isomorphic to the ringof topological polynomials

H*(W(A ® C); Q) = A(u}) ® Q K , ] .

From the generators {vk[ />i}we form the Chern polynomials wk, regard-ing the elements vk t as elementary symmetric polynomials. Then the classeswk G H*(W(A <8> C); Q) are those whose inverse images $*(wk)GH*(Bir;Q)give homotopically invariant higher signatures.

The Hirzebruch formulae have an elegant interpretation in terms of thebordisms of W(A). Observe that under the change of ringsΛ -»· Mat(«, Λ) = Λ' the spaces W(A) and W(A') are homotopically equi-valent. Therefore, a finite-dimensional representation ρ: Λ -»• Mat(«, C)induces a map p * : W(A) ->• W(C). So we obtain the following commutativediagram (n = 4k):

signp

Thena B [X]),

t h 3 t 1 S ' ch ξρ = ζ*ρ*αα> αα ζ Η* (W (C); Q).

The last equality raises the question of the existence of bundles over W(A)whose Chern character determines the classes p*aa. In fact, passing toBanach algebras, which from the point of view of representation theory,as we saw in §4, is not a restriction, we can introduce in W(A) a weakertopology. To do this we consider all free Λ-modules furnished with freebases and we express all homomorphisms with the help of matrices withelements in the algebra Λ. Then we consider two APC's as being close ifall their homomorphisms are close. We denote the weak topology ofW(A) by W(A). We obtain a continuous map

(10.7) ε: W(A) -+W(A).

It turns out that (10.7) is a weak homotopy equivalence. On the otherhand, the zero-dimensional skeleton of W(A) is no longer a discrete spacein the weak topology, but is the space F(A) of Hermitian forms over Λ.The inclusion

(10.8) ε0: F(A)


induces an isomorphism of homotopy groups in the 2-adic localization

(e0).: nh(F(A)) ® Ζ [1/2]-5. nb (W (A)) ®Z[l/2] .

Therefore, ignoring 2-torsion, we find that W(A) is weakly homotopicallyequivalent to F(A). The latter space can be interpreted as the classifyingspace BGL(A) of GL(A). We can also extract more precise informationabout the homotopy properties of (10.8). From this point of view, (10.6)is a map classifying bundles over Bit with Banach fibre Λ and with thesame representation of π in its group ring Λ.

2. Rational-homological manifolds. The Hirzebruch formulae are true fora wider class of manifolds, although the methods of proof given here aresuitable only for smooth manifolds (see [56]). Solov'ev has shown thatthe homomorphism (10.2) defines an APC when X is a simplicial rational-homological manifold. Consequently, all the assertions of §10.1, in parti-cular, the Hirzebruch formulae, are true for rational-homological manifolds.

3. Geometrical methods of computing the Wall groups. In this directionthe first work to be mentioned is apparently the paper of Browder andLevine ([57]), which is concerned with the problem of surgery of a sub-manifold of codimension 1. Later a series of papers by Farrell and Hsiang([58], [59], [54]) and then by Shaneson ([5]) solved, as a matter of fact,the problem of computing the Wall groups for a free Abelian group.

The geometrical idea (as distinct from the algebraic methods consideredin §§2 and 3) consists in the following: each element θ e Ln(ir) of theWall group can be realized as an obstruction to surgery of a normal mapto a homotopy equivalence. If π splits into the direct product π = π' Χ Ζ,we can take as a model manifold X with fundamental group π the directproduct of X' with fundamental group π' and the circle Sl:

X = Χ' χ S1.Then each normal map

(10.9)1 ψ: v

can without loss of generality be regarded as being transversal-regular alongX' C X. Thus, we obtain a new normal map

(10-10)Ψ.ψ: v ( F > > φ* (ξ),

Realizing any element 0 S Ln(n) as an obstruction θ - Θ(Υ, φ, ψ) tomodifying the map (10.9) to a homotopy equivalence, we get the element

(10.11) ?(θ) = Q(Y', φ, φ) £ £„_!<«').

(10.11) gives us the construction of a homomorphism (provided that it iswell-defined) q: Ln(ir) -+ Ln_l(ir'). The precise computation of the depend-

134 A . S. Mishchenko

ence of q{6) on the realization (10.9) shows that, in fact, by the givenmethod we can map the Wall group Ls

n(ir) of obstruction to surgeries to asimple homotopy equivalence

(10.12) q: £,»(«) ^L^n').

Using the operation of slicing the manifold X along X' (that is, by passage

to X' X /), (10.12) fits into an exact sequence

(10.13) 0 _ L'n (π') -> Ls

n (π) - * Ln_, (π') - * 0.

With the help of (10.13) it has been possible to obtain full information

about the Wall groups for free Abelian groups.

Further generalizations of (10.13) have been applied to various classes of

amalgamated products of groups. (In this context, see the papers of

Cappell [60], [61].)

Clearly, we have not been able to give a full survey of applications of K-

theory, since at the present time this area is in a state of rapid development.

The reader can find a more complete survey of the results of foreign authors

on Hermitian ^-theory up to 1972 in the collection of papers of the Seattle

conference on algebraic AT-theory [62].

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Received by the Editors 14 October 1974

Translated by I. R. Porteous

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