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CEJM 3(4) 2005 766­793

K ­theory from the p oint of view of C ­algebras and Fredholm representations
Alexandr S. Mishchenko


Departments of Mathematics and Mechanics, Moscow State University, Leninskije Gory, 119899, Moscow, Russia

Received 3 January 2005; accepted 1 July 2005 Abstract: These notes represent the sub ject of five lectures which were delivered as a minicourse during the VI conference in Krynica, Poland, "Geometry and Topology of Manifolds", May, 2-8, 2004. c Central European Science Journals. All rights reserved. Keywords: Vector bund le, K-theory, C*-algebras, Fredholm operators, Fredholm representations MSC (2000): 19L, 19K, 19J25, 55N15

1

Intro duction

In the second half of the last century, research commenced and developed in what is now called "non-commutative geometry". As a matter of fact, this term concentrates on a circle of problems and to ols which originally was based on the quite simple idea of re-formulating topological properties of spaces and continuous mappings in terms of appropriate algebras of continuous functions. This idea lo oks very old (it go es back to the theorem of I.M. Gelfand and M.A. Naimark (see, for example [1]) on the one-to-one correspondence between the category of compact topological spaces and the category of commutative unital C -algebras), and was developed by different authors both in the commutative and in the non-commutative cases. The first to clearly pro claim it as an action program was Alain Connes in his bo ok "non-commutative geometry" [2]. The idea, along with commutative C -algebras (which can be interpreted as algebras of continuous functions on the spaces of maximal ideals), to also consider non-commutative
E-mail: asmish@mech.math.msu.su


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algebras as functions on a non-existing "non-commutative" space was so fruitful that it allowed the joining together of a variety of metho ds and conceptions from different areas such as topology, differential geometry, functional analysis, representation theory, asymptotic metho ds in analysis and resulted in mutual enrichment by new properties and theorems. One of classical problems of smo oth topology, which consists in the description of topological and homotopy properties of characteristic classes of smo oth and piecewiselinear manifolds, has been almost completely and exclusively shaped by the different metho ds of functional analysis that were brought to bear on it. Vice versa, attempts to formulate and to solve classical topological problems have led to the enrichment of the metho ds of functional analysis. It is typical that solutions of particular problems of a new area lead to the discovery of new horizons in the development of mathematical metho ds and new properties of classical mathematical ob jects. The following notes should not be considered as a complete exposition of the sub ject of non-commutative geometry. The lectures were devoted to topics of interest to the author and are indicative of his point of view in the sub ject. Consequently, the contents of the lectures were distributed as follows: (1) Topological K ­theory as a cohomology theory. Bott perio dicity. Relation between the real, the complex and the quaternionic K ­theories. (2) Elliptic operators as the homology K ­theory, Atiyah homology K ­theory as an ancestor of K K ­theory. (3) C ­algebras, Hilbert C -mo dules and Fredholm operators. Homotopical point of view. (4) Higher signature, C ­signature of non-simply connected manifolds.

2

Some historical remarks on the formation of non-commutative geometry

2.1 From Poincare duality to the Hirzebruch formula.
The Pontryagin characteristic classes, though not homotopy invariants, are nevertheless closely connected with the problem of the description of smo oth structures of given homotopy type. Therefore, the problem of finding all homotopy invariant Pontryagin characteristic classes was a very actual one. However, in reality, another problem turned out to be more natural. It is clear that Pontryagin classes are invariants of smo oth structures on a manifold. For the purpose of the classification of smo oth structures, the most suitable ob jects are not the smo oth structures but the so-called inner homology of manifolds or, using contemporary terms, bordisms of manifolds. Already L.S. Pontryagin [3] conjectured that inner homology could be described in terms of some algebraic expression of Pontryagin classes, the so-called Pontryagin numbers. He established that the Pontryagin numbers are at least invariants of inner homology [4, Theorem 3]. W. Browder and S.P. Novikov were the first to prove that only the Pon-


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tryagin number which coincides with the signature of an oriented manifold is homotopy invariant. This fact was established by means of surgery theory developed in [5], [6]. The formula that asserts the coincidence of the signature with a Pontryagin number is known now as the Hirzebruch formula [7], though its special case was obtained by V.A. Rokhlin [8] a year before. Investigations of the Poincare duality and the Hirzebruch formula have a long history, which is partly related to the development of non-commutative geometry. Here we shall describe only some aspects that were typical of the Moscow scho ol of topology. The start of that history should be lo cated in the famous manuscript of Poincare in 1895 [9], where Poincare duality was formulated. Although the complete statement and its full pro of were presented much later, one can, without reservations, regard Poincare as the founder of the theory. After that was required the discovery of homology groups (E. No ether, 1925) and cohomology groups (J.W. Alexander, A.N. Kolmogorov, 1934). The most essential was, probably, the discovery of characteristic classes (E.L. Stiefel, Y. Whitney (1935); L. Pontryagin (1947); S.S. Chern (1948)). The Hirzebruch formula is an excellent example of the application of categorical metho d as a basic to ol in algebraic and differential topology. Indeed, Poincare seems always to indicate when he had proved the coincidence of the Betti numbers of manifolds which are equidistant from the ends. But after the intro duction of the notion of homology groups, Poincare duality began to be expressed a little differently: as the equality of the ranks of the corresponding homology groups. At that time it was not significant what type of homology groups were employed, whether with integer or with rational co efficients, since the rank of an integer homology group coincides with the dimension of the homology group over rational co efficients. But the notion of homology groups allowed to enrich to expand Poincare duality by consideration of the homology groups over finite fields. Taking into account torsions of the homology groups, one obtained isomorphisms of some homology groups, but not in the same dimensions where the Betti numbers coincide. This apparent inconsistency was understo o d after the discovery of the cohomology groups and their duality to the homology groups. Thus finally, Poincare duality became sound as an isomorphism between the homology groups and the cohomology groups H k (M ; Z ) = H
n-k

(M ; Z ).

(1 )

The crucial understanding here is that the Poincare duality is not an abstract isomorphism of groups, but the isomorphism generated by a natural operation in the category of manifolds. For instance, in a special case of middle dimension for even-dimensional manifolds (dim M = n = 2m) with rational co efficients, the condition (1) becomes trivial since H m (M ; Q) = Hom (Hm (M ; Q), Q) Hm (M ; Q). (2 )

But in the equation (2), the isomorphism between the homology groups and the cohomology groups is not chosen at will. Poincare duality says that there is the definite


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homomorphism generated by the intersection of the fundamental cycle [M ] [
M]

:H

n-k

(M ; Q)-Hk (M ; Q).

This means that the manifold M gives rise to a non-degenerate quadratic form which has an additional invariant -- the signature of the quadratic form. The signature plays a crucial role in many problems of differential topology.

2.2

Homotopy invariants of non-simply connected manifolds.

This collection of problems is devoted to finding the most complete system of invariants of smo oth manifolds. In a natural way the smo oth structure generates on a manifold a system of so-called characteristic classes, which take values in the cohomology groups with a different system of co efficients. Characteristic classes not only have natural descriptions and representations in differential geometric terms, but their properties also allow us to classify the structures of smo oth manifolds in practically an exhaustive way mo dulo a finite number of possibilities. Consequently, the theory of characteristic classes is a most essential to ol for the study of geometrical and topological properties of manifolds. However, the system of characteristic classes is in some sense an over-determined system of data. More precisely, this means that for some characteristic classes their dependence on the choice of smo oth structure is inessential. Therefore, one of the problems was to find out to what extent one or other characteristic class is invariant with respect to an equivalence relation on manifolds. The best known topological equivalence relations between manifolds are piece-linear homeomorphisms, continuous homeomorphisms, homotopy equivalences and bordisms. For such kinds of relations one can formulate a problem: which characteristic classes are: a) combinatorially invariant, b) topologically invariant, c) homotopy invariant. The last relation (bordism) gives a trivial description of the invariance of characteristic classes: only characteristic numbers are invariant with respect to bordisms. Let us now restrict our considerations to rational Pontryagin classes. S.P. Novikov has proved (1965) that all rational Pontryagin classes are topologically invariant. In the case of homotopy invariance, at the present time the problem is very far from being solved. On the other hand, the problem of homotopy invariance of characteristic classes seems to be quite important on account of the fact that the homotopy type of manifolds seems to be more accessible to classification in comparison with its topological type. Moreover, existing metho ds of classification of smo oth structures on a manifold can reduce this problem to a description of its homotopy type and its homology invariants. Thus, the problem of homotopy invariance of characteristic classes seemed to be one of the essential problems in differential topology. In particular, the problem of homotopy invariance of rational Pontryagin classes happened to be the most interesting (and probably the most difficult) from the point of view of mutual relations. For example, the importance of the problem is confirmed by that fact that the classification of smo oth structures on a manifold by means the Morse surgeries demands a description of all


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homotopy invariant rational Pontryagin classes. In the case of simply connected manifolds, the problem was solved by Browder and Novikov who have proved that only signature is a homotopy invariant rational Pontryagin number. For non-simply connected manifolds, the problem of the description of all homotopy invariant rational Pontryagin classes which are responsible for obstructions to surgeries of normal mappings to homotopy equivalence, turned out to be more difficult. The difficulties are connected with the essential role that the structure of the fundamental group of the manifold plays here. This circumstance is as interesting as the fact that the description and identification of fundamental groups in finite terms is impossible. In some simple cases when the fundamental group is, for instance, free abelian, the problem could be solved directly in terms of differential geometric to ols. In the general case, it turned out that the problem can be reduced to the one that the so-called higher signatures are homotopy invariant. The accurate formulation of this problem is known as the Novikov conjecture. A positive solution of the Novikov conjecture may permit, at least partly, the avoidance of algorithmic difficulties of description and the recognition of fundamental groups in the problem of the classification of smo oth structures on non-simply connected manifolds. The Novikov conjecture says that any characteristic number of kind signx (M ) = L(M )f (x), [M ] is a homotopy invariant of the manifold M , where L(M ) is the full Hirzebruch class, x H (B ; Q) is an arbitrary rational cohomology class of the classifying space of the fundamental group = 1 (M ) of the manifold M , f : M -B is the isomorphism of fundamental groups induced by the natural mapping. The numbers signx (M ) are called higher signatures of the manifold M to indicate that when x = 1 the number sign1 (M ) coincides with the classical signature of the manifold M . The situation with non-simply connected manifolds turns out to be quite different from the case of simply connected manifolds in spite of the fact that C. T. C. Wall had constructed a non-simply connected analogue of Morse surgeries. The obstructions to such kinds of surgeries do es not have an effective description. One way to avoid this difficulty is to find out which rational characteristic classes for non-simply connected manifolds are homotopy invariant. Here we should define more accurately what we mean by characteristic classes for non-simply connected manifolds. As was mentioned above, we should consider only such invariants for a non-simply connected manifold as a) can be expressed in terms of the cohomology of the manifold and b) are invariants of nonsimply connected bordisms. In other words, each non-simply connected manifold M with fundamental group = 1 (M ) has a natural continuous map fM : M -B which induces an isomorphism of fundamental groups (fM ) : = 1 (M )-1 (B ) = .


(3 )

Then the bordism of a non-simply connected manifold M is the singular bordism [M , fM ] (B ) of the space B . Hence, from the point of view of bordism theory, a rational characteristic number for the singular bordism [M , fM ] is a number of the following form
([M , fM ]) = P (p1 (M ), . . . , pn (M ); fM (x1 ), . . . , fM (xk )) , [M ] ,

(4 )


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where pj (M ) are the Pontryagin classes of the manifold M , xj H (B ; Q) are arbitrary cohomology classes. Following the classical paper of R.Thom ([14]) one can obtain the result that the characteristic numbers of the type (4) form a complete system of invariants of the group (B ) Q. Using the metho ds developed by C. T. C. Wall ([13]) one can prove that only higher signatures of the form
sign x (M ) = L(M )fM (x); [M ]

(5 )

may be homotopy invariant rational characteristic numbers for a non-simply connected manifold [M ].

3

Top ological K ­theory

3.1 Locally trivial bundles, their structure groups, principal bundles
Definition 3.1. Let E and B be two topological spaces with a continuous map p : E -B . The map p is said to define a local ly trivial bund le if there is a topological that for any point x B there is a neighborho o d U x for which the p-1 (U ) is homeomorphic to the Cartesian pro duct U â F . Moreover, it is the homeomorphism preserves fibers. This means in algebraic terms tha U â F - p-1 (U ) E p p U = U B is commutative where : U â F -U, (x, f ) = x is the pro jection onto the first factor. The space E is called total space of the bund le or the fiberspace, the space B is called the base of the bund le, the space F is called the fiber of the bund le and the mapping p is called the projection. One can give an equivalent definition using the so-called transition functions: Definition 3.2. Let B and F be two topological spaces and {U } be a covering of the space B by a family of open sets. The system of homeomorphisms which form the commutative diagram (U U ) â F - (U U ) â F (U U ) = (U U



space F such inverse image required that t the diagram



(6 )


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and satisfy the relations


= id, for any three indices , , on the intersection (U U U ) â F (7 )





= id for each .

By analogy with the terminology for smo oth manifolds, the open sets U are called charts, the family {U } is called the atlas of charts, the homeomorphisms U â F - p-1 (U ) E U = U



(8 )

B

are called the coordinate homeomorphisms and the are called the transition functions or the sewing functions. Two systems of the transition functions , and define isomorphic lo cally trivial bundles iff there exist fiber-preserving homeomorphisms
U â F - U â F

h

U

=

U

such that
- = h 1 h .

(9 )

Let Homeo (F ) be the group of all homeomorphisms of the fiber F . Each fiberwise homeomorphism : U â F -U â F, (1 0 ) defines a map : U -Homeo (F ), So instead of


(1 1 )

a family of functions


: U U -Homeo (F ),

can be defined on the intersection U U and having values in the group Homeo (F ). The condition (7) means that (x) = id, (x) (x) (x) = id. x U U U .


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and we say that the co chain { } is a cocycle. The condition (9) means that there is a zero-dimensional co chain h : U -Homeo (F ) such that (x) = h-1 (x) (x)h (x), x U U . Using the language of homological algebra the condition (9) means that co cycles { } and { } are cohomologous. Thus the family of lo cally trivial bundles with fiber F and base B is in one-to-one correspondence with the one-dimensional cohomology of the space B with co efficients in the sheaf of germs of continuous Homeo (F )­valued functions for the given open covering {U }. Despite obtaining a simple description of the family of lo cally trivial bundles in terms of homological algebra, it is ineffective since there is no simple metho d for calculating cohomologies of this kind. Nevertheless, this representation of the transition functions as a co cycle turns out to be very useful because of the situation described below. First of all, notice that using the new interpretation, a lo cally trivial bundle is determined by the base B , the atlas {U } and the functions { } taking values in the group G = Homeo (F ). The fiber F itself do es not directly take part in the description of the bundle. Hence, one can at first describe a lo cally trivial bundle as a family of functions { } with values in some topological group G, and thereafter construct the total space of the bundle with fiber F by additionally defining an action of the group G on the space F , that is, defining a continuous homomorphism of the group G into the group Homeo (F ). Secondly, the notion of lo cally trivial bundle can be generalized and the structure of bundle made richer by requiring that both the transition functions and the functions h not be arbitrary but take values in some subgroup of the homeomorphism group Homeo (F ). Thirdly, sometimes information about lo cally trivial bundle may be obtained by substituting some other fiber F for the fiber F but using the `same' transition functions. Thus, we come to a new definition of a lo cally trivial bundle with additional structure -- the group where the transition functions take their values, the so-called the structure group. Definition 3.3. A lo cally trivial bundle with the structure group G is called a principal G­bundle if F = G and the action of the group G on F is defined by the left translations. Theorem 3.4. Let p : E -B be a principal G­bund le. Then there is a right action of the group G on the total space E such that: 1) the right action of the group G is fiberwise, 2) the homeomorphism -1 transforms the right action of the group G on the total space into right translations on the second factor. Using the transition functions it is very easy to define the inverse image of the bundle. Namely, let p : E -B be a lo cally trivial bundle with structure group G and the collec-


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tion of transition functions : U -G and let f : B -B be a continuous mapping. Then the inverse image f (p : E -B ) is defined as a collection of charts U = f -1 (U ) and a collection of transition functions (x) = (f (x)). Another geometric definition of the inverse bundle arises from the diagram E = f (E ) E â B - E B


(1 2 )

=

B



- B

f

where E consists of points (e, b ) E âB such that f (b ) = p(e). The map f : f (E )-E is canonically defined by the map f . Theorem 3.5. Let : E -E (1 3 ) be a continuous map of total spaces for principal G­bund les over bases B and B . The map (13) is generated by a continuous map f : B -B if and only if the map is equvivariant (with respect to right actions of the structure group G on the total spaces).

3.2 Homotopy properties, classifying spaces
Theorem 3.6. The inverse images with respect to homotopic mappings are isomorphic bund les. Therefore, the category of all bundles with structure group G, B ndlsG (B ) forms a homotopy functor from the category of CW-spaces to the category of sets. Definition 3.7. A principal bundle p : EG -BG is called a classifying bundle iff for any CW­space B there is a one-to-one correspondence B ndlsG (B ) [B , BG ] generated by the map : [B , BG ]-B ndlsG (B ), (1 5 ) (f ) = f (p : EG -BG ). Theorem 3.8. The principal G­bund le, pG : EG -BG is a classifying bund le if al l homotopy groups of the total space EG are trivial: i (EG ) = 0, 0 i < . (1 7 ) (1 6 ) (1 4 )


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3.3 Characteristic classes
Definition 3.9. A mapping : B ndlsG (B )-H (B ) is called a characteristic class if the following diagram is commutative B ndlsG (B ) - H (B ) f f


(1 8 )

B ndlsG (B ) - H (B ) for any continuous mapping f : B -B , that is, is a natural transformation of functors. Theorem 3.10. The family of al l characteristic classes is in one-to-one correspondence with the cohomology H (BG ) by the assignment (x)( ) = f (x) for x H (BG ),


(1 9 )


f : B -BG ,

= f (p : EG -BG ).

3.4 Vector bundles, K ­theory, Bott periodicity
Members of the special (and very important) class of lo cally trivial bundles are called (real) vector bundles with structure groups GL(n, R) and fiber Rn . The structure group can be reduced to the subgroup O (n). If the structure group O (1) can be reduced to the subgroup G = S O (1) then the vector bundle is trivial and is denoted by 1. Similar versions arise for other structure groups: 1) Complex vector bundles with the structure group GL(n, C) and fiber Cn . 2) Quaternionic vector bundles with structure group GL(n, K) and fiber Kn , where K is the (non-commutative) field of quaternions. All of them admit useful algebraic operations: 1. Direct sum, = 1 2 , 2. tensor pro duct, = 1 2 , 3. other tensor operations, HOM (1 , 2),k ( ). etc. Let be a vector bundle over the field F = R, C, K. Let ( ) be the space of all sections of the bundle. Then (1) = C (B ) -- the ring of continuous functions with values in F . Theorem 3.11. The space ( ) has a natural structure of (left) C (B )­module by fiberwise multiplication. If B is a compact space, then ( ) is a finitely generated projective C (B )­ module. Conversely, each finitely generated projective C (B )­module can be presented as a space ( ) for a vector bund le . The property of compactness of B is essential for ( ) to be a finitely generated mo dule.


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Definition 3.12. Let K (X ) denotes the abelian group where the generators are (isomorphism classes of ) vector bundles over the base X sub ject to the following relations: [ ] + [ ] - [ ] = 0 (2 0 )

for vector bundles and , and where [ ] denotes the element of the group K (X ) defined by the vector bundle . The group defined in Definition 3.12 is called the Grothendieck group of the category of all vector bundles over the base X . To avoid confusion, the group generated by all real vector bundles will be denoted by KO (X ), the group generated by all complex vector bundles will be denoted by KU (X ) and the group generated by all quaternionic vector bundles will be denoted by KS p (X ). Let K 0 (X, x0 ) denote the kernel of the homomorphism K (X )-K (x0 ): K 0 (X, x0 ) = Ker (K (X )-K (x0 )) . Elements of the subring K 0 (X, x0 ) are represented by differences [ ]-[ ] for which dim = dim . The elements of the ring K (X ) are called virtual bundles and elements of the ring K 0 (X, x0 ) are virtual bundles of trivial dimension over the point x0 . Now consider a pair (X, Y ) of the cellular spaces, Y X . Denote by K 0 (X, Y ) the ring K 0 (X, Y ) = K 0 (X/ Y , [Y ]) = ker(K (X/ Y )-K ([Y ])) where X/ Y is the quotient space where the subspace Y is collapsed to a point [Y ]. For any negative integer -n, let K
-n

(X , Y ) = K 0 (S n X , S n Y )

where S n (X ) denotes the n­times suspension of the space X: S n X = (S n â X )/(S n X ). Theorem 3.13. The pair (X, Y ) induces an exact sequence K 0 (Y , x0 )-K 0 (X, x0 )-K 0 (X, Y )- -K -K . . . -K
-1 -2 -1 -2 -1 -2

(Y , x0 )-K (Y , x0 )-K

(X, x0 )-K (X, x0 )-K

(X, Y )- (X, Y )- . . . (X, Y )- . . . (2 1 )

...................................................
-n

(Y , x0 )-K

-n

(X, x0 )-K

-n

...................................................

Consider a complex n­dimensional vector bundle over the base X and let p : E -X be the pro jection of the total space E onto the base X . Consider the space E , as a new base space, and a complex of vector bundles 0-0 -1 -2 - . . . - n -0,
0 1 2









n-1

(2 2 )


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where = p is the inverse image of the bundle , k is the k -skew power of the vector bundle and the homomorphism k : k -
k +1



is defined as exterior multiplication by the vector y E , y x , x = p(y ). It is known that if the vector y x is non-zero, y = 0, then the complex (22) is exact. Consider the subspace D ( ) E consisting of all vectors y E such that |y | 1 with respect to a fixed Hermitian structure on the vector bundle . Then the subspace S ( ) D ( ) of all unit vectors gives the pair (D ), S ( )) for which the complex (22) is exact on S ( ). Denote the element defined by (22) by ( ) K 0 (D ( ), S ( )). Then, one has the homomorphism given by multiplication by the element ( ) : K (X )-K 0 (D ( ), S ( )) . (2 3 )

The homomorphism (23) is an isomorphism called the Bott homomorphism. In particular the Bott element K 0 (S2 , s0 ) = K -2 (S0 , s0 ) = Z generates a homomorphism h: K -n (X, Y )-K -(n+2) (X, Y ) (2 4 ) which is called the Bott periodicity isomorphism and hence forms a perio dic cohomology K ­theory.

3.5 Relations between complex, symplectic and real bundles
Let G be a compact Lie group. A G­space X is a topological space X with continuous action of the group G on it. The map f : X -Y is said to be equivariant if f (g x) = g f (x), g G. Similarly, if f is a lo cally trivial bundle and also equivariant then f is called an equivariant lo cally trivial bundle. An equivariant vector bundle is defined similarly. The theory of equivariant vector bundles is very similar to the classical theory. In particular, equivariant vector bundles admit the operations of direct sum and tensor pro duct. In certain simple cases the description of equivariant vector bundles is reduced to the description of the usual vector bundles. The category of G­ equivariant vector bundles is go o d place to give consistent descriptions of three different structures on vector bundles -- complex, real and symplectic. Consider the group G = Z2 and a complex vector bundle over the G­space X . This means that the group G acts on the space X . Let E be the total space of the bundle and let p : E -X be the pro jection in the definition of the vector bundle . Let G act on the total space E as a fiberwise operator which is linear over the real numbers and anti-complex over complex numbers, that is, if G = Z2 is the generator then (x) = (x), C, x E . (2 5 )


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A vector bundle with the action of the group G satisfying the condition (25) is called a K R­bund le . The operator is called the anti-complex involution. The corresponding Grothendieck group of K R­bundles is denoted by K R(X ). Below we describe some of the relations with classical real and complex vector bundles. Proposition 3.14. Suppose that the G­space X has the form X = Y â Z2 and the involution transposes the second factor. Then the group K R(X ) is natural ly isomorphic to the group KU (Y ) and this isomorphism coincides with restriction of a vector bund le to the summand Y â {1}, 1 G = Z2 , ignoring the involution . Proposition 3.15. Suppose the involution on X is trivial. Then K R ( X ) KO ( X ) . (2 6 )

The isomorphism (26) associates to any K R­bund le the fixed points of the involution . Proposition 3.16. The operation of forgetting the involution induces a homomorphism K R(X )-KU (X ) and when the involution is trivial on the base X this homomorphism coincides with complexification c : KO (X )-KU (X ). Moreover, the pro of of Bott perio dicity can be extended word by word to K R­theory: Theorem 3.17. There is an element K R(D 1,1 , S 1,1) = K R homomorphism given by multiplication by : K Rp,q (X, Y )-K R is an isomorphism. It turns out that this scheme can be mo dified so that it includes another type of K ­theory ­ that of quaternionic vector bundles. Let K be the (non-commutative) field of quaternions. As for real or complex vector bundles, we can consider lo cally trivial vector bundles with fiber K n and structure group GL(n, K ), the so called quaternionic vector bundles. Each quaternionic vector bundle can be considered as a complex vector bundle p : E -X with additional structure defined by a fiberwise anti-complex linear operator J such that J 2 = -1, I J + J I = 0, where I is fiberwise multiplication by the imaginary unit. More generally, let J be a fiberwise anti-complex linear operator which acts on a complex vector bundles and satisfies J 4 = 1, I J + J I = 0. (2 8 )
p-1,q -1 -1,-1

(pt) such that the (2 7 )

(X.Y )


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Then, the vector bundle can be split into two summands = 1 2 both invariant under the action of J , that is, J = J1 J2 such that
2 2 J1 = 1, J2 = -1.

(2 9 )

Hence, the vector bundle 1 is the complexification of a real vector bundle and 2 is a quaternionic vector bundle. Consider a similar situation over a base X with an involution such that the operator (28) commutes with . Such a vector bundle will be called a K RS ­bund le . Lemma 3.18. A K RS ­bund le is split into an equivariant direct sum = 1 2 such that J 2 = 1 on 1 and J 2 = -1 on 2 . Lemma 3.18 shows that the Grothendieck group K RS (X ) generated by K RS ­bundles has a Z2 ­grading, that is, K R S (X ) = K R S 0 (X ) K R S 1 (X ). It is clear that K RS0 (X ) = K R(X ). In the case when the involution acts trivially, K RS1 (X ) = KQ (X ), that is, K R S ( X ) = KO ( X ) KQ ( X ) where KQ (X ) is the group generated by quaternionic bundles.

q

-8 Z

-7

-6

-5

-4 Z u Z
O

-3

-2 Z2 h2 0

-1 Z2 h 0

0 Z u2 = 4 Z u

K

O

= Z

2

0

0 Z2 h2

0 Z2 h

0

K

Q

u

2

0

0
and KQ .

Fig. 1 A list of the groups K

4

Elliptic op erators as the homology K ­theory, Atiyah homology K ­theory as an ancestor of K K ­theory

4.1 Homology K ­theory. Algebraic categorical setting
A naive point of view of homology theory is that the homology groups dual to the cohomology groups h (X ) should be considered as h (X ) = Hom (h (X ), Z).
def

(3 0 )


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This naive definition is not go o d since it gives a non-exact functor. A more appropriate definition is the following. Consider a natural transformation of functors Y : h (X â Y )-h (Y ) (3 1 )

which is the homomorphism of h (Y )­mo dules and for a continuous mapping f : Y -Y gives the commutative diagram h (X â Y ) - h (Y ) (id â f ) f


(3 2 )

h (X â Y ) - h (Y ) Let be the family of all natural transformations of the type (31, 32). The functor h (X ) defines a homology theory.


4.2 PDO
Consider a linear differential operator A which acts on the space of smo oth functions of n real variables: A : C (Rn )-C (Rn ). and is presented as a finite linear combination of partial derivatives A=
||m

a (x)

|| . x

(3 3 )

Put a(x, ) =
||m

a (x) i|| .

The function a(x, ) is called the symbol of a differential operator A. The operator A can be reconstructed from its symbol as A = a x, 1 i x .

Since the symbol is a polynomial with respect to the variables , it can be split into homogeneous summands a(x, ) = am (x, ) + am-1 (x, ) + · · · + a0 (x, ). The highest term am (x, x) is called the principal symbol of the operator A while whole symbol is sometimes called the ful l symbol.


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Proposition 4.1. Let y = y (x) be a smooth change of variables. Then, in the new coordinate system the operator B defined by the formula (B u)(y ) = (Au (y (x)))
x=x(y )

is again a differential operator of order m for which the principal symbol is bm (y , ) = am x(y ), y (x(y )) . x (3 4 )

The formula (34) shows that the variables change as a tensor of valency (0, 1), that is, as components of a cotangent vector. The concept of a differential operator exists on an arbitrary smo oth manifold M . The concept of a whole symbol is not well defined but the principal symbol can be defined as a function on the total space of the cotangent bundle T M . It is clear that the differential operator A do es not depend on the principal symbol alone but only up to the addition of an operator of smaller order. The notion of a differential operator can be generalized in various directions. First of all, notice that (Au) (x) = F-x (a(x, ) (Fx- (u)( ))) , (3 5 ) where F is the Fourier transform. Hence, we can enlarge the family of symbols to include some functions which are not polynomials. Then the operator A defined by formula (35) with non-polynomial symbol is called a pseudodifferential operator of order m (more exactly, not greater than m). The pseudo differential operator A acts on the Schwartz space S. This definition of a pseudo differential operator can be extended to the Schwartz space of functions on an arbitrary compact manifold M . Let {U } be an atlas of charts with a lo cal co ordinate system x . Let { } be a partition of unity subordinate to the atlas of charts, that is, 0 (x) 1, (x) 1, supp U .


Let (x) be functions such that supp U , (x) (x) (x). Define an operator A by the formula A(u)(x) =


(x)A ( (x)u(x)) ,

(3 6 )

where A is a pseudo differential operator on the chart U (which is diffeomorphic to Rn ) with principal symbol a (x , ) = a(x, ). In general, the operator A depends on the choice of functions , and the lo cal co ordinate system x , uniquely up to the addition of a pseudo differential operator of order strictly less than m.


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The next useful generalization consists of a change from functions on the manifold M to smo oth sections of vector bundles. The crucial property of the definition (36) is the following Proposition 4.2. Let a : (1 )- (2 ), b : (a )- (3 ) be two symbols of orders m1 , m2 . Let c = ba be the composition of the symbols. Then the operator b(D )a(D ) - c(D ) : (1 )- (3 ) is a pseudodifferential operator of order m1 + m2 - 1. Proposition 4.2 leads to a way of solving equations of the form Au = f (3 7 )

for certain pseudo differential operators A. To find a solution of (37), it suffices to construct a left inverse operator B , that is, B A = 1. Usually, this is not possible, but a weaker condition can be realized. Condition 4.3. a(x, ) is invertible for sufficiently large | | C . The pseudo differential operator A = a(D ) is called an el liptic if Condition 4.3 holds. If A is elliptic operator than there is an (elliptic) operator B = b(D ) such that AD - id is the operator of order -1. The final generalization for elliptic operators is the substitution of a sequence of pseudo differential operators for a single elliptic operator. Let 1 , 2 , . . . , k be a sequence of vector bundles over the manifold M and let
1 2 0- (1 )- (2 )- . . . - (k )-0

a

a

ak

-1

(3 8 )

be a sequence of symbols of order (m1 , . . . , mk-1 ). Suppose the sequence (38) forms a complex, that is, as as-1 = 0. Then the sequence of operators 0- (1 ) - (2 )- . . . - (k )-0
a1 (D )

(3 9 )

in general, do es not form a complex because we can only know that the composition ak (D )ak-1(D ) is a pseudo differential operator of the order less then ms + ms-1 . If the sequence of pseudo differential operators forms a complex and the sequence of symbols (38) is exact away from a neighborho o d of the zero section in T M then the sequence (39) is called an el liptic complex of pseudo differential operators.


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4.3 Fredholm operators
The bounded operator K : H -H is said to be compact if any bounded subset X H is mapped to a precompact set, that is, the set F (X ) is compact. If dim Im (K ) < then K is called a finite-dimensional operator. Each finite-dimensional operator is compact. If limn Kn - K = 0 and the Kn are compact operators, then K is again a compact operator. Moreover, each compact operator K can be presented as K = limn Kn , where the Kn are finite-dimensional operators. The operator F is said to be Fredholm if there is an operator G such that both K = F G - 1 and K = GF - 1 are compact. Theorem 4.4. (1) dim Ker F index F = F. (2) index F = (3) there exists Let F be a Fredholm operator. Then < , dim Coker F < and the image, Im F , is closed. The number dim Ker F - dim Coker F is cal led the index of the Fredholm operator dim Ker F - dim Ker F , where F is the adjoint operator. > 0 such that if F - G < then G is a Fredholm operator and index F = index G, (4) if K is compact then F + K is also Fredholm and index (F + K ) = index F. (5) If F and G are Fredholm operators, then the composition F G is Fredholm and index (F G) = index F + index G. The notion of a Fredholm operator has an interpretation in terms of the finitedimensional homology groups of a complex of Hilbert spaces. In general, consider a sequence of Hilbert spaces and bounded operators
0 1 0-C0 -C1 - . . . -Cn -0.

(4 0 )

d

d

dn-

1

(4 1 )

We say that the sequence (41) is a Fredholm complex if dk dk-1 0, Im dk is a closed subspace and dim (Ker dk /Coker dk-1) = dim H (Ck , dk ) < . Then the index of Fredholm complex (41) is defined by the following formula: index (C, d) =
k

(-1)k dim H (Ck , dk ).

Theorem 4.5. Let
d d

0 1 0-C0 -C1 - . . . -Cn -0

dn-

1

(4 2 )

be a sequence satisfying the condition that each dk d conditions are equivalent:

k -1

is compact. Then the fol lowing


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(1) There exist operators fk : Ck -Ck-1 such that fk+1 dk + dk-1fk = 1 + rk where each rk is compact. (2) There exist compact operators sk such that the sequence of operators dk = dk + sk forms a Fredholm complex. The index of this Fredholm complex is independent of the operators sk .

4.4 Sobolev spaces
Consider an arbitrary compact manifold M and a vector bundle . One can define a Sobolev norm on the space of the sections (M , ), using the formula u
2 s

=
R
n

u(x)(1 + )s u(x)dx,

where is the Laplace-Beltrami operator on the manifold M with respect to a nian metric. The Sobolev norm depends on the choice of Riemannian metric, of the bundle in the trivial bundle uniquely equivalent norms. Hence, the co of the space of sections (M , ) is defined correctly. We shall denote this co by Hs (M , ).

Riemaninclusion mpletion mpletion

Theorem 4.6. Let M be a compact manifold, be a vector bund le over M and s1 < s2 . Then the natural inclusion Hs2 (M , )-Hs1 (M , ) is a compact operator. Theorem 4.7. Let a(D ) : (M , 1 )- (M , 2 ) be a pseudodifferential operator of order m. Then there is a constant C such that a(D )u
s -m

(4 3 )

(4 4 )

C u s,

(4 5 )

that is, the operator a(D ) can be extended to a bounded operator on Sobolev spaces: a(D ) : Hs (M , 1 )-H
s -m

(M , 2 ).

(4 6 )

Using theorems 4.6 and 4.7 it can be shown that an elliptic operator is Fredholm for appropriate choices of Sobolev spaces. Theorem 4.8. Let a(D ) be an el liptic pseudodifferential operator of order m as in (44). Then its extension (46) is Fredholm. The index of the operator (46) is independent of the choice of the number s.


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4.5 Index of elliptic operators
An elliptic operator (D ) is defined by a symbol : (1 )- (2 ) (4 7 )

which is an isomorphism away from a neighborho o d of the zero section of the cotangent bundle T M . Since M is a compact manifold, the symbol (47) defines a triple ( (1 ), , (2 )) which in turn defines an element [ ] K (D (T M ), S (T M )) = Kc (T M ), where Kc (T M ) denotes the K ­groups with compact supports. Theorem 4.9. The index index (D ) of the Fredholm operator (D ) depends only on the element [ ] Kc (T M ). The mapping index : Kc (T M )-Z is an additive homomorphism. In addition, index (D ) = p [ ], where p : Kc (T M )-Kc (pt) = Z is the direct image homomorphism induced by the trivial mapping p : M -pt. (4 8 )

4.6 The Atiyah homology theory
The naive idea is that cohomology K ­group (with compact supports) of the total space of cotangent bundle of the manifold M , Kc (T M ), should be isomorphic to a homology K ­group due to a Poincare duality, K (T M ) K (M ). This identification can be arranged to be a natural transformation of functors D : Kc ( T M ) K ( M ) , D ( ) : K (M â N ) - K (N ) D ( )( ) = index ( ) K (N ). Therefore, the homology K of triples = (H, F, ), where C (M )-B (H ) is a representat the algebra B (H ) of bounded operator F f - f F : H -H ­groups, K (M ) can be identified with the collection H is a Hilbert space, F is a Fredholm operator, : ion of the algebra C (M ) of all continuous functions to operators, such that for any function f C (M ) the is compact. If is a vector bundle on M then the (4 9 )


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space ( ) is a finitely generated pro jective mo dule over C (M ). Therefore the operator F id : H C (M ) ( )-H C (M ) ( ) is Fredholm. Hence, one obtains a natural transformation : K (M â N ) - K (N ), ( ) = index (F id ) K (N ). This definition was an ancestor of K K ­theory. (5 0 )

5

C ­algebras, Hilb ert C ­mo dules and C ­Fredholm op erators

5.1 Hilbert C ­modules
The simplest case of C ­algebras is the case of commutative C ­algebras. The Gelfand­ Naimark theorem ([1]) says that any commutative C ­algebra with unit is isomorphic to an algebra C (X ) of continuous functions on a compact space X . This crucial observation leads to a simple but very useful definition of Hilbert mo dules over the C ­algebra A. Following Paschke ([21]), the Hilbert A­mo dule M is a Banach A­mo dule with an additional structure of inner pro duct x, y A, x, y M which possesses the natural properties of inner pro ducts. If is a finite-dimensional vector bundle over a compact space X , then ( ) is a finitely generated pro jective Hilbert C (X )­mo dule. And conversely, each finitely generated projective Hilbert mo dule P over the algebra C (X ) is isomorphic to a section mo dule ( ) for some finite-dimensional vector bundle . Therefore K0 ( C ( X ) ) K ( X ) . =

5.2 Fredholm operators, Calkin algebra
A finite-dimensional vector bundle over a compact space X , can be described as a continuous family of pro jectors, that is a continuous matrix-valued function P = P (x), x X , P (x) Mat(N , N ), P (x)P (x) = P (x), P (x) : CN -CN . This means that = Im P . Here Mat(N , N ) denotes the space of N â N matrices. Hence if = ker P then = N. (5 1 ) Then 1 Nk 1 Nk H âX where H is a Hilbert space, or H X = H âX . k= k= Hence, there is a continuous Fredholm family F (x) : H -H, Ker (F ) = , Coker (F ) = 0. (5 2 )


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And conversely, if we have a continuous family of Fredholm operators F (x) : H -H such that dim Ker F (x) = const , dim Coker F (x) = const then both = Ker F and = Coker F are lo cally trivial vector bundles. More generally, for an arbitrary continuous family of Fredholm operators F (x) : H -H there is a continuous compact family K (x) : H -H such that the new family F (x) = F (x) + K (x) satisfies the conditions dim Ker F (x) = const , dim Coker F (x) = const consequently defining two vector bundles and which generate an element [ ] - [ ] K (X ) not depending on the choice of compact family. This correspondence is actually one-to-one. In fact, if two vector bundles and are isomorphic then there is a compact deformation of F (x) such that Ker F (x) = 0, Coker F (x) = 0, that is F (x) is an isomorphism, F (x) G(H ). The remarkable fact discovered by Kuiper ([22]) is that the group G(H ) as a topological space is contractible, i.e. the space F (H ) of Fredholm operators is a representative of the classifying space BU for vector bundles. In other words, one can consider the Hilbert space H and the group of invertible operators GL(H ) B (H ). The Kuiper theorem says that i (GL(H )) = 0, 0 i < . (5 3 )

5.3 K ­theory for C ­algebras, Chern character
Generalization of K ­theory for C ­algebra A. KA (X ) is the Grothendieck group generated by vector bundles whose fibers M are finitely generated pro jective A­mo dules, and the structure groups Aut A (M ). KA (X ) are the corresponding perio dic cohomology theory. For example, let us consider the quotient algebra Q(H ) = B (H )/K (H ), the so-called Calkin algebra, where B (H ) is the algebra of bounded operators of the Hilbert space H , K (H ) is the algebra of compact operators. Let p : B (H )-Q(H ) be the natural pro jector. Then the Fredholm family F (x) : H -H generates the family F : X -Q(H ), F (x) = p(F (x)), F (x) is invertible that is F : X -G(Q(H )). So, one can prove that the space G(Q(H )) represents the classifying space BU for vector bundles. In other words, K 0 (X ) KQ(H ) (X ). =1 A generalization of the Kuiper theorem for the group GL (l2 (A)) of all invertible A operators which admit adjoint operators. Let F be the space of all Fredholm operators. Then K (X ) [X, F ]. (5 4 ) Let Q = B (H )/K be the Calkin algebra, where K is the subalgebra of all compact operators. Let G(Q) be the group of invertible elements in the algebra Q. Then one has a homomorphism [X, F ]-[X, Q],


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hence a homomorphism
1 K 0 (X )-KQ (X )

which is an isomorphism. The Chern character
chA : KA (X )-H (X ; KA (pt) Q)

is defined in a tautological way: let us consider the natural pairing
K (X ) KA (pt)-KA (X )

(5 5 )

which generates the isomorphism
K (X ) KA (pt) Q-KA (X ) Q

(5 6 )

due to the classical uniqueness theorem in axiomatic homology theory. Then, the Chern character is defined as the composition
-1 ch chA : KA (X ) KA (X ) Q-K (X ) (KA (pt) Q) -

(5 7 ) ch -H (X ; KA (pt) Q). Therefore, the next theorem is also tautological: Theorem 5.1. If X is a finite C W ­space, the Chern character induces the isomorphism
chA : KA (X ) Q-H (X ; KA (pt) Q).

5.4 Non-simply connected manifolds and canonical line vector bundle
Let be a finitely presented group which can serve as a fundamental group of a compact connected manifold M , = 1 (M , x0 ). Let B be the classifying space for the group . Then there is a continuous mapping fM : M -B such that the induced homomorphism (fM ) : 1 (M , x0 )-1 (B , b0 ) = (5 9 ) (5 8 )

is an isomorphism. One can then construct the line vector bundle A over M with fiber A, a one-dimensional free mo dule over the group C ­algebra A = C [ ] using the representation C [ ]. This canonical line vector bundle can be used to construct the so-called assembly map µ : K (B )-K (C [ ]) (6 0 )


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5.5 Symmetric equivariant C ­signature
Let M be a closed oriented non-simply connected manifold with fundamental group . Let B be the classifying space for the group and let fM : M -B , be a map inducing the isomorphism of fundamental groups. Let (B ) denote the bordism group of pairs (M , fM ). Recall that (B ) is a mo dule over the ring = ( pt ). One can construct a homomorphism : (B )-L (C ) (6 1 )

which for every manifold (M , fM ) assigns the element (M ) L (C ), the so-called symmetric C ­signature, where L (C ) is the Wall group for the group ring C . The homomorphism satisfies the following conditions: (a) is homotopy invariant, (b) if N is a simply connected manifold and (N ) is its signature then (M â N ) = (M ) (N ) L (C ). We shall be interested only in the groups after tensor multiplication with the field Q, in other words, in the homomorphism : (B ) Q-L (C ) Q. However, (B ) Q H (B ; Q) . Hence one has : H (B ; Q)-L (C ) Q. Thus, the homomorphism represents the cohomology class H (B ; L (C ) Q). Then, for any manifold (M , fM ) one has
(M , fM ) = L(M )fM ( ), [M ] L (C ) Q.

(6 2 )

Hence, if : L (C ) Q-Q is an additive functional and () = x H (B ; Q) then
signx (M , fM ) = L(M )fM (x), [M ] Q

should be the homotopy-invariant higher signature. This gives a description of the family of all homotopy-invariant higher signatures. Hence, one should study the cohomology class H (B ; L (C ) Q) = H (B ; Q) L (C ) Q and lo ok for all elements of the form ( ) = x H (B ; Q).


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5.5.1 Combinatorial description of symmetric C ­signature Here we give an economical description of algebraic Poincare complexes as a graded free C ­mo dule with the boundary operator and the Poincare duality operator. Consider a chain complex of C ­mo dules C, d:
n

C=
k =0 n

Ck , dk ,
k =1 k -1

d=

dk : Ck -C and a Poincare duality homomorphism D : C -C, They form the diagram

deg D = n.

Cn - C

n C0 1 C1 2 · · · - Cn - - D0 D1 D

d

d

d

n

d n

n-1

- · · · - C

dn-

1

d 1

0

with the following properties:

d k Dk +

k -1 d k = 0, (-1) Dk-1 dn-k+1 = 0, Dk = (-1)k(n-k) Dn-k . k +1

d

(6 3 )

Assume that the Poincare duality homomorphism induces an isomorphism of homology groups. Then the triple (C, d, D) is called a algebraic Poincare complex. This definition permits the construction of the algebraic Poincare complex (X ) for each triangulation of the combinatorial manifold X : (X ) = (C, d, D), where C = C ( X ; C ) is the graded chain complex of the manifold X with lo cal system of co efficients induced by the natural inclusion of the fundamental group = 1 (X ) in the group ring C , d is the boundary homomorphism, D = Dk , Dk = 1 [X ] + (-1) 2
k (n-k )

([X ])



,

where [X ] is the intersection with the open fundamental cycle of the manifold X . Put Fk = ik(k-1) Dk . (6 4 )


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Then the diagram
n C0 1 C1 2 · · · - Cn - - F0 F1 Fn

d

d

d

(6 5 )

n Cn - C

d

n-1

- · · · 1 C -

d - n

1

d

0

possesses more natural conditions of commutativity and conjugacy dk Fk + Fk-1 d Fk = (-1) Let F = n=0 Fk , F, k deg F = n.
n -k +1
n(n-1) 2

= 0, (6 6 )

Fn-k .

Over completion to a regular C ­algebra C [ ], one can define an element of hermitian K ­theory using the non-degenerate self-adjoint operator G = d + d + F : C -C. Then
h sign [C, G] = sign (C, d, D) K0 (C [ ]).

(6 7 )

6

Additional historical remarks

The only candidates which are homotopy invariant characteristic numbers are the higher signatures. Moreover, any homotopy invariant higher signature can be expressed from a universal symmetric equivariant signature of the non-simply connected manifold. Therefore, to lo ok for homotopy invariant higher signatures, one can search through different geometric homomorphisms : L (C ) Q-Q. On of them is the so-called Fredholm representation of the fundamental group. Application of the representation theory in the finite-dimensional case leads to Hirzebruchtype formulas for signatures with lo cal system of co efficients. But the collection of characteristic numbers which can be represented by means of finite-dimensional representations is not very large and in many cases reduces to the classical signature. The most significant here is the contribution by Lusztig ([28]) where the class of representations with indefinite metric is considered. The crucial step was to find a class of infinite-dimensional representations which preserve natural properties of the finite-dimensional representations. This infinite-dimensional analogue consists of a new functional-analytic construction as a pair of unitary infinite dimensional representations (T1 , T2 ) of the fundamental group in the Hilbert space H and a Fredholm operator F which braids the representations T1 and T2 upto compact operators. The triple = (T1 , F, T2 ) is called the Fredholm representation of the group .


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From the categorical point of view, the Fredholm representation is a relative representation of the group C ­algebra C [ ] in the pair of Banach algebras (B (H ), Q(H ) where B (H ) is the algebra of bounded operators on the Hilbert space H and Q(H ) is the Calkin algebra Q(H ) = B (H )/K(H ). Then, for different classes of manifolds one can construct a sufficiently rich resource of Fredholm representations. For one class of examples from amongst many others, there are the manifolds with Riemannian metric of nonpositive sectional curvature, the so-called hyperbolic fundamental groups. For the most complete description of the state of these problems, one may consult ([29]) and the bo ok ([2]).

References
[1] M.A. Naimark: Normed algebras, Nauka, Moscow, 1968 (in Russian), English translation: Wolters­No ordHoff, 1972. [2] A. Connes: Noncommutative Geometry, Academic Press, 1994. [3] L.S. Pontryagin: "Classification of some fiber bundles", Dokl. Akad. Nauk SSSR, Vol. 47(5), (1945), pp. 327­330 (in Russian). [4] L.S. Pontryagin: "Characteristic cycles of differntiable manifolds", Mathem. Sbornik, Vol. 21(2), (1947), pp. 233­284 (in Russian). [5] W. Browder: Homotopy type of differential manifolds, Collo quium on Algebraic Topology, Aarhus Universitet, Aarhus, Matematisk Institut, 1962, pp. 42­46. [6] S.P. Novikov: "Homotopy equivalent smoth manifolds, I", Izv.Akad.Nauk SSSR, Ser.Mat., Vol. 28, (1964), pp, 365­474. [7] F. Hirzebruch: "On Steenro d's reduced powers, the index of interia, and the To dd genus", Proc.Nat,Acad.Sci. U.S.A., Vol. 39, (1953), pp. 951­956. [8] V.A. Rokhlin: "New results in the theory of 4-dimensional manifolds", Dokl. Akad. Nauk SSSR, Vol. 84, (1952), pp. 221­224. [9] A. Poincare: "Analysis situs", Journal de l'Ecole Polytechniques, Vol. 1, (1895), pp. 1­121. [10] A.S. Mishchenko: "Homotopy invariant of non simply connected manifolds. Rational invariants. I", Izv. Akad. Nauk. SSSR, Vol. 34(3), (1970), pp. 501­514. [11] A.S. Mishchenko: "Controlled Fredholm representations", In: S.C. Ferry, A. Ranicki and J. Rosenberg (Eds.): Novikov Conjectures, Index Theorems and Rigidity, Vol. 1, London Math. So c. Lect. Notes Series, Vol. 226, Cambridge Univ. Press, 1995, pp. 174­200. [12] A.S. Mishchenko: "C ­algebras and K -theory", Algebraic topology, Lecture Notes in Mathematics, Vol. 763, Aarhus, 1979, pp. 262­274. [13] C.T.C. Wall: Surgery of compact manifolds, Academic Press, New York, 1971. [14] R. Thom: "Quelques properietes globales des varietes differentiables", Comm. Math. Helv., Vol. 28, (1954), pp. 17­86. [15] G. Luke and A.S. Mishchenko: Vector Bund les and Their Applications, Kluwer Academic Publishers, Dordrrecht, Boston, London, 1998. [16] D. Husemoller: Fiber Bund les, McGraw­Hill, N.Y., 1966.


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