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VECTOR BUNDLES AND THEIR APPLICATIONS



VECTOR BUNDLES AND THEIR APPLICATIONS

Glenys LUKE Oxford University
Oxford

Alexander MISHCHENKO Moscow State University
Moscow

KLUWER ACADEMIC PUBLISHERS Boston/London/Dordrecht



CONTENTS

v



PREFACE

In the last few years the use of geometric methods has permeated many more branches of mathematics and the sciences. Briefly its role may be characterized as follows. Whereas methods of mathematical analysis describe phenomena `in the small', geometric methods contribute to giving the picture `in the large'. A second no less important property of geometric methods is the convenience of using its language to describe and give qualitative explanations for diverse mathematical phenomena and patterns. From this point of view, the theory of vector bundles together with mathematical analysis on manifolds (global analysis and differential geometry) has provided a ma jor stimulus. Its language turned out to be extremely fruitful: connections on principal vector bundles (in terms of which various field theories are described), transformation groups including the various symmetry groups that arise in connection with physical problems, in asymptotic methods of partial differential equations with small parameter, in elliptic operator theory, in mathematical methods of classical mechanics and in mathematical methods in economics. There are other currently less significant applications in other fields. Over a similar period, university education has changed considerably with the appearance of new courses on differential geometry and topology. New textbooks have been published but `geometry and topology' has not, in our opinion, been well covered from a practical applications point of view. Existing monographs on vector bundles have been mainly of a purely theoretical nature, devoted to the internal geometric and topological problems of the sub ject. Students from related disciplines have found the texts difficult to use. It therefore seems expedient to have a simpler book containing numerous illustrations and applications to various problems in mathematics and the sciences. Part of this book is based on material contained in lectures of the author, A.Mishchenko, given to students of the Mathematics Department at Moscow State University and is a revised version of the Russian edition of 1984. Some of the less important theorems have been omitted and some proofs simplified and clarified. The focus of attention was towards explaining the most important notions and geometric constructions connected with the theory of vector bundles. vii


viii

Vector bundles and their applications

Theorems were not always formulated in maximal generality but rather in such a way that the geometric nature of the ob jects came to the fore. Whenever possible examples were given to illustrate the role of vector bundles. Thus the book contains sections on locally trivial bundles, and on the simplest properties and operations on vector bundles. Further properties of a homotopic nature, including characteristic classes, are also expounded. Considerable attention is devoted to natural geometric constructions and various ways of constructing vector bundles. Basic algebraic notions involved in describing and calculating K-theory are studied and the particularly interesting field of applications to the theory of elliptic pseudodifferential operators is included. The exposition finishes with further applications of vector bundles to topology. Certain aspects which are well covered in other sources have been omitted in order to prevent the book becoming too bulky.


1
INTRODUCTION TO THE LOCALLY TRIVIAL BUNDLES THEORY

1.1

LOCALLY TRIVIAL BUNDLES

The definition of a locally trivial bundle was coined to capture an idea which recurs in a number of different geometric situations. We commence by giving a number of examples. The surface of the cylinder can be seen as a disjoint union of a family of line segments continuously parametrized by points of a circle. The M¨ obius band can be presented in similar way. The two dimensional torus embedded in the three dimensional space can presented as a union of a family of circles (meridians) parametrized by points of another circle (a parallel). Now, let M be a smooth manifold embedded in the Euclidean space RN and T M the space embedded in RN â RN , the points of which are the tangent vectors of the manifold M . This new space T M can be also be presented as a union of subspaces Tx M , where each Tx M consists of all the tangent vectors to the manifold M at the single point x. The point x of M can be considered as a parameter which parametrizes the family of subspaces Tx M . In all these cases the space may be partitioned into fibers parametrized by points of the base. The examples considered above share two important properties: a) any two fibers are homeomorphic, b) despite the fact that the whole space cannot be presented as a Cartesian product of a fiber with the base (the parameter space), if we restrict our consideration to some small region of the base the part of the fiber space over this region is such a Cartesian product. The two properties above are the basis of the following definition.

1


2

Chapter 1

Definition 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 space F such that for any point x B there is a neighborhood U x for which the inverse image p-1 (U ) is homeomorphic to the Cartesian product U â F . Moreover, it is required that the homeomorphism : U â F -p-1 (U ) preserves fibers, it is a `fiberwise' map, that is, the fol lowing equality holds: p((x, f )) = x, x U, f F. The space E is cal led total space of the bund le or the fiberspace , the space B is cal led the base of the bund le , the space F is cal led the fiber of the bund le and the mapping p is cal led the projection . The requirement that the homeomorphism be fiberwise means in algebraic terms that the diagram U âF U where : U â F -U, (x, f ) = x is the projection onto the first factor is commutative. One problem in the theory of fiber spaces is to classify the family of all locally trivial bundles with fixed base B and fiber F . Two locally trivial bundles p : E -B and p : E -B are considered to be isomorphic if there is a homeomorphism : E -E such that the diagram E - E p p B= B is commutative. It is clear that the homeomorphism gives a homeomorphism of fibers F -F . To specify a locally trivial bundle it is not necessary to be given the total space E explicitly. It is sufficient to have a base B , a fiber F and a family of mappings such that the total space E is determined `uniquely' (up to isomorphisms of bundles). Then according to the definition of a locally trivial bundle, the base B can be covered by a family of open sets {U } such


- =



p-1 (U ) p U


Introduction to bund le theory

3

that each inverse image p-1 (U ) is fiberwise homeomorphic to U â F . This gives a system of homeomorphisms : U â F -p
-1

(U ).

Since the homeomorphisms preserve fibers it is clear that for any open subset V U the restriction of to V â F establishes the fiberwise homeomorphism of V â F onto p-1 (V ). Hence on U â U there are two fiberwise homeomorphisms : (U U ) â F -p-1 (U U ), : (U U ) â F -p-1 (U U ). Let denote the homeomorphism -1 which maps (U U ) â F onto itself. The locally trivial bundle is uniquely determined by the following collection: the base B , the fiber F , the covering U and the homeomorphisms


: (U U ) â F -(U U ) â F.

The total space E should be thought of as a union of the Cartesian products U â F with some identifications induced by the homeomorphisms . By analogy with the terminology for smooth manifolds, the open sets U are called charts , the family {U } is called the atlas of charts , the homeomorphisms are called the coordinate homeomorphisms and the are called the transition functions or the sewing functions . Sometimes the collection {U , } is called the atlas. Thus any atlas determines a locally trivial bundle. Different atlases may define isomorphic bundles but, beware, not any collection of homeomorphisms forms an atlas. For the classification of locally trivial bundles, families of homeomorphisms that actually determine bundles should be selected and then separated into classes which determine isomorphic bundles. For the homeomorphisms to be transition functions for some locally trivial bundle: = -1 . (1.1) Then for any three indices , , on the intersection (U U U ) â F the following relation holds: = Id, where Id is the identity homeomorphism and for each ,


= Id.

(1.2)


4

Chapter 1

In particular thus Hence for an atlas the


= Id,
1 = - .

(1.3)

should satisfy = Id,






= Id.

(1.4)

These conditions are sufficient for a locally trivial bundle to be reconstructed from the base B , fiber F , atlas {U } and homeomorphisms . To see this, let E = (U â F ) be the disjoint union of the spaces U â F . Introduce the following equivalence relation: the point (x, f ) U â F is related to the point (y , g ) U â F iff x = y U U and (y , g ) =


(x, f ).

The conditions (1.2), (1.3) guarantee that this is an equivalence relation, that is, the space E is partitioned into disjoint classes of equivalent points. Let E be the quotient space determined by this equivalence relation, that is, the set whose points are equivalence classes. Give E the quotient topology with respect to the pro jection : E -E which associates to a (x, f ) its the equivalence class. In other words, the subset G E is called open iff -1 (G) is open set. There is the natural mapping p from E to B : p (x, f ) = x. Clearly the mapping p is continuous and equivalent points maps to the same image. Hence the mapping p induces a map p : E -B which associates to an equivalence class the point assigned to it by p . The mapping p is continuous. It remains to construct the coordinate homeomorphisms. Put = |U âF : U â F -E . Each class z p-1 (U ) has a unique representative (x, f ) U â F . Hence is a one to one mapping onto p-1 (U ). By virtue of the quotient topology


Introduction to bund le theory

5

on E the mapping is a homeomorphism. It is easy to check that (compare with (1.1)) -1 = . So we have shown that locally trivial bundles may be defined by atlas of charts {U } and a family of homeomorphisms { } satisfying the conditions (1.2), (1.3). Let us now determine when two atlases define isomorphic bundles. First of all notice that if two bundles p : E -B and p : E -B with the same fiber F have the same transition functions { } then these two bundles are isomorphic. Indeed, let : U -p-1 (U ). : U -p = Then We construct a homeomorphism : E -E . Let x E . The atlas {U } covers the base B and hence there is an index such that x p -1 (U ). Set (x) =
-1 -1 -1

(U ). =
-1

be the corresponding coordinate homeomorphisms and assume that =
-1 -1

.

=

.

(x).

It is necessary to establish that the value of (x) is independent of the choice of index . If x p -1 (U ) also then = Hence the and other coordinate some = defined in
-1 - - (x) = -1 1 1 (x) = -1 -1





(x) =

(x).

definition of (x) is independent of the choice of chart. Continuity necessary properties are evident. Further, given an atlas {U } and homomorphisms { }, if {V } is a finer atlas (that is, V U for ( )) then for the atlas {V }, the coordinate homomorphisms are a natural way =
( ) |(V âF )

: V â F -p-1 (V ).

The transition functions tions
1

1 ,2

for the new atlas {V } are defined using restric: (V


,2

=

(1 ),(2 ) |(V

1

V2 )âF

1

V2 ) â F -(V

1

V 2 ) â F .


6

Chapter 1

Thus if there are two atlases and transition functions for a common refinement, that is, a finer atlas with transition restrictions, it can be assumed that the two bundles have , are two systems of the transition functions (for the isomorphic bundles then the transition functions ,

two bundles, with functions given by the same atlas. If same atlas), giving must be related.

Theorem 1 Two systems of the transition functions , and define isomorphic local ly trivial bund les iff there exist fiber preserving homeomorphisms h : U â F -U â F such that


=h

-1

h .

(1.5)

Pro of. Suppose that two bundles p : E -B andp : E -B with the coordinate homeomorphisms and are isomorphic. Then there is a homeomorphism : E -E . Let - h = 1 -1 . Then h
-1

h = -1
-1



- 1

-1

=

=



- 1



-1



-1

= .

Conversely, if the relation (1.5) holds, put = h
-1



-1

.

(1.6)

The definition (1.6) is valid for the subspaces p -1 (U ) covering E . To prove that the right hand sides of (1.6) coincide on the intersection p -1 (U U ) the relations (1.5) are used: h
-1 - 1 = -1 h - 1 -1 -1 = -1 -1 -1 = h h h-1

= =

- -1 h-1 1

- 1

-1

=

h-1

- 1

.


Introduction to bund le theory

7

Examples
1. Let E = B â F and p : E -B be pro jections onto the first factors. Then the atlas consists of one chart U = B and only one the transition function = Id and the bundle is said to be trivial . 2. Let E be the M¨ obius band. One can think of this bundle as a square in the plane, {(x, y ) : 0 x 1, 0 y 1} with the points (0, y ) and (1, 1 - y ) identified for each y [0, 1]. The pro jection p maps the space E onto the segment Ix = {0 x 1} with the endpoints x = 0 and x = 1 identified, that is, onto the circle S 1 . Let us show that the map p defines a locally trivial bundle. The atlas consists of two intervals (recall 0 and 1 are identified) U = {0 < x < 1}, U = {0 x < 1 1 } { < x 1}. 2 2

The coordinate homeomorphisms may be defined as following: : U â Iy -E , (x, y ) = (x, y ), : U â Iy -E (x, y ) = (x, y ) for 0 x < 1 , 2

(x, y ) = (x, 1 - y ) for

1 < x 1. 2

The intersection of two charts U U consists of union of two intervals U 1 U = (0, 2 ) ( 1 , 1). The transition function have the following form 2


= (x, y ) for 0 < x < = (x, 1 - y ) for

1 < x < 1. 2

1 , 2

The M¨ obius band is not isomorphic to a trivial bundle. Indeed, for a trivial bundle all transition functions can be chosen equal to the identity. Then by Theorem 1 there exist fiber preserving homeomorphisms h : U â Iy -U â Iy , h : U â Iy -U â Iy , such that = h
-1 h

in its domain of definition (U U ) â Iy . Then h , h are fiberwise homeomorphisms for fixed value of the first argument x giving homeomorphisms of


8

Chapter 1

interval Iy to itself. Each homeomorphism of the interval to itself maps end points to end points. So the functions h (x, 0), h (x, 1), h (x, 0), h (x, 1) are constant functions, with values the functions h-1 h (x, 0). On the constant because it equals zero for 1 2 < x < 1.This contradiction shows a trivial bundle. equal to zero or one. The same is true for other hand the function (x, 0) is not each 0 < x < 1 and equals one for each 2 that the M¨ obius band is not isomorphic to

3. Let E be the space of tangent vectors to two dimensional sphere S2 embedded in three dimensional Euclidean space R3 . Let p : E -S2 be the map associating each vector to its initial point. Let us show that p is a locally trivial bundle with fiber R2 . Fix a point s0 S2 . Choose a Cartesian system of coordinates in R3 such that the point s0 is the North Pole on the sphere (that is, the coordinates of s0 equal (0, 0, 1)). Let U be the open subset of the sphere S2 defined by inequality z > 0. If s U, s = (x, y , z ), then x2 + y 2 + z 2 = 1, z > 0. Let e = ( , , ) be a tangent vector to the sphere at the point s. Then x + y + z = 0, that is, = -(x + y )/z . Define the map by the formula (x, y , z , , ) = (x, y , z , , , -(x + y )/z ) giving the coordinate homomorphism for the chart U containing the point s0 S2 . Thus the map p gives a locally trivial bundle. This bundle is called the tangent bund le of the sphere S2 . : U â R2 -p
-1

(U )


Introduction to bund le theory

9

1.2

THE STRUCTURE GROUPS OF THE LOCALLY TRIVIAL BUNDLES

The relations (1.4,1.5) obtained in the previous section for the transition functions of a locally trivial bundle are similar to those involved in the calculation of one dimensional cohomology with coefficients in some algebraic sheaf. This analogy can be explain after a slight change of terminology and notation and the change will be useful for us for investigating the classification problem of locally trivial bundles. Notice that a fiberwise homeomorphism of the Cartesian product of the base U and the fiber F onto itself : U â F -U â F, (1.7)

can be represented as a family of homeomorphisms of the fiber F onto itself, parametrized by points of the base B . In other words, each fiberwise homeomorphism defines a map : U -Homeo (F), ¯ (1.8)

where Homeo (F) is the group of all homeomorphisms of the fiber F . Furthermore, if we choose the right topology on the group Homeo (F) the map ¯ becomes continuous. Sometimes the opposite is true: the map (1.8) generates the fiberwise homeomorphism (1.7)with respect to the formula (x, f ) = (x, (x)f ). ¯ So instead of a family of functions : U U -Homeo (F), ¯ can be defined on the intersection U U and having values in the group Homeo (F). In homological algebra the family of functions is called a ¯ one dimensional cochain with values in the sheaf of germs of functions with values in the group Homeo (F). The condition (1.4) from the section 1.1 means that (x) = Id, ¯ (x) (x) (x) = Id. ¯ ¯ x U U U . and we say that the cochain { } is a cocycle. The condition (1.5) means ¯ that there is a zero dimensional cochain h : U -Homeo (F) such that (x) = h ¯
-1

¯

(x) (x)h (x), x U U . ¯


10

Chapter 1

Using the language of homological algebra the condition (1.5) means that cocycles { } and { } are cohomologous. Thus the family of locally trivial ¯ ¯ bundles with fiber F and base B is in one to one correspondence with the one dimensional cohomology of the space B with coefficients in the sheaf of the germs of continuous Homeo (F)­valued functions for given open covering {U }. Despite obtaining a simple description of the family of locally trivial bundles in terms of homological algebra, it is ineffective since there is no simple method of calculating cohomologies of this kind. Nevertheless, this representation of the transition functions as a cocycle turns out very useful because of the situation described below. First of all notice that using the new interpretation a locally trivial bundle is determined by the base B , the atlas {U } and the functions { } taking the value in the group G = Homeo (F). The fiber F itself does not directly take part in the description of the bundle. Hence, one can at first describe a locally trivial bundle as a family of functions { } with values in some topological group G, and after that 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 locally trivial bundle can be generalized and the structure of bundle made richer by requiring that both the transition functions and the functions h are not arbitrary but take values ¯ in some subgroup of the homeomorphism group Homeo (F).Thirdly, sometimes information about locally 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 locally trivial bundle with additional structure -- the group where the transition functions take their values. Definition 2 Let E ,B ,F be topological spaces and G be a topological group which acts continuously on the space F . A continuous map p : E -B is said to be a local ly trivial bund le with fiber F and the structure group G if there is an atlas {U } and the coordinate homeomorphisms : U â F -p-1 (U ) such that the transition functions have the form


= -1 : (U U ) â F -(U U ) â F (x, f ) = (x, (x)f ), ¯


Introduction to bund le theory

11

where : (U U )-G are continuous functions satisfying the conditions ¯ (x) 1, x U , ¯ (x) (x) (x) 1, x U U U . ¯ ¯ ¯ The functions are also cal led the transition functions ¯ Let : E -E be p: the an isomorphism of locally trivial bundles with the structure group G. Let and be the coordinate homeomorphisms of the bundles p : E -B and E -B , respectively. One says that the isomorphism is compatible with structure group G if the homomorphisms
-1

(1.9)

: U â F -U â F

are determined by continuous functions h : U -G, defined by relation -1 (x, f ) = (x, h (x)f ). (1.10)

Thus two bundles with the structure group G and transition functions ¯ and are isomorphic, the isomorphism being compatible with the structure group G, if (x) = h (x) (x)h (x) ¯ ¯ (1.11) for some continuous functions h : U -G. So two bundles whose the transition functions satisfy the condition (1.11) are called equivalent bund les . It is sometimes useful to increase or decrease the structure group G. Two bundles which are not equivalent with respect of the structure group G may become equivalent with respect to a larger structure group G , G G . When a bundle with the structure group G admits transition functions with values in a subgroup H , it is said that the structure group G is reduced to subgroup H . It is clear that if the structure group of the bundle p : E -B consists of only one element then the bundle is trivial. So to prove that the bundle is trivial, it is sufficient to show that its the structure group G may be reduced to the trivial subgroup. More generally, if : G-G is a continuous homomorphism of topological groups and we are given a locally trivial bundle with the structure group G and the transition functions


: U U -G


12

Chapter 1

then a new locally trivial bundle may be constructed with structure group G for which the transition functions are defined by (x) = (


(x)). (with respect to the

This operation is called a change of the structure group homomorphism ).

Remark
Note that the fiberwise homeomorphism : U â F -U â F in general is not induced by continuous map : U -Homeo (F). ¯ (1.12)

Because of lack of space we will not analyze the problem and note only that later on in all our applications the fiberwise homeomorphisms will be induced by the continuous maps (1.12) into the structure group G. Now we can return to the third situation, that is, to the possibility to choosing a space as a fiber of a locally trivial bundle with the structure group G. Let us consider the fiber F =G with the action of G on F being that of left translation, that is, the element g G acts on the F by the homeomorphism g (f ) = g f , f F = G. Definition 3 A local ly trivial bund le with the structure group G is cal led principal G­bund le if F = G and action of the group G on F is defined by the left translations. An important property of principal G­bundles is the consistency of the homeomorphisms with the structure group G and it can be described not only in terms of the transition functions (the choice of which is not unique) but also in terms of equivariant properties of bundles.


Introduction to bund le theory

13

Theorem 2 Let p : E -B be a principal G­bund le, : U â G-p-1 (U ) be the coordinate homeomorphisms. 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, that is, p(x) = p(xg ), x E , g G. 2. the homeomorphism -1 transforms the right action of the group G on the total space into right translations on the second factor, that is, (x, f )g = f (x, f g ), x U , f , g G. (1.13)

Pro of. According to the definitions 2 and 3, the transition functions have the following form where : U U -G ¯ are continuous functions satisfying the conditions (1.9). Since an arbitrary point z E can be represented in the form z = (x, f ) for some index , the formula (1.13) determines the continuous right action of the group G provided that this definition is independent of the choice of index . So suppose that z = (x, f ) = (x, f ). We need to show that the element z g does not depend on the choice of index, that is, (x, f g ) = (x, f g ).


= -1

(x, f ) = (x, (x)f ), ¯


14

Chapter 1

or (x, f g ) = -1 (x, f g ) = (x, f g ) or f g = (x)f g . ¯ However, Hence f = (x)f . ¯ Thus multiplying (1.15) by g on the right gives (1.14). Theorem 2 allows us to consider principal G­bundles as having a right action on the total space. Theorem 3 Let : E -E (1.16) be a fiberwise map of principal G­bund les. The map (1.16) is the isomorphism of local ly trivial bund les with the structure group G, that is, compatible with the structure group G iff this map is equivariant (with respect to right actions of the group G on the total spaces). Pro of. Let p : E -B , p : E -B be locally trivial principal bundles both with the structure group G and let , be coordinate homeomorphisms. Then by the definition (1.10), the map is an isomorphism of locally trivial bundles with structure group G when -1 (x, g ) = (x, h (x)g ). for some continuous functions h : U -G. It is clear that the maps defined by (1.17) are equivariant since [-1 (x, g )]g1 = (x, h (x)g )g1 = = (x, h (x)g g1 ) = -1 (x, g g1 ). (1.17) (1.15) (x, f ) = -1 (x, f ) =


(1.14)

(x, f ) = (x, f ), ¯


Introduction to bund le theory

15

Hence the map is equivariant with respect to the right actions of the group G on the total spaces E and E . Conversely, let the map be equivariant with respect to the right actions of the group G on the total spaces E and E . By Theorem 2, the map -1 is equivariant with respect to right translations of the second coordinate of the space U â G. Since the map -1 is fiberwise, it has the following form -1 (x, g ) = (x, A (x, g )). The equivariance of the map (1.18) implies that A (x, g g1 ) = A (x, g )g1 for any x U , g , g1 G. In particular, putting g = e that A (x, g1 ) = A (x, e)g1 So putting h (x) = A (x, e), it follows that h (x)g = A (x, g ). and -1 (x, g ) = (x, h (x)g ). (1.18)

The last identity means that is compatible with the structure group G. Thus by Theorem 3, to show that two locally trivial bundles with the structure group G (and the same base B ) are isomorphic it necessary and sufficient to show that there exists an equivariant map of corresponding principal G­ bundles (inducing the identity map on the base B ). In particular, if one of the bundles is trivial, for instance, E = B â G, then to construct an equivariant map : E -E it is sufficient to define a continuous map on the subspace {(x, e) : x B , } E = B â G into E . Then using equivariance, the map is extended by formula (x, g ) = (x, e)g . The map{(x, e) : x B , }-E can be considered as a map s : B -E satisfying the property ps(x) = x, x B . (1.20) The map (1.19) with the property (1.20) is called a cross­section of the bundle. So each trivial principal bundle has cross­sections. For instance, the (1.19)



16

Chapter 1

map B -B â G defined by x-(x, e) is a cross­section. Conversely, if a principal bundle has a cross­section s then this bundle is isomorphic to the trivial principal bundle. The corresponding isomorphism : B â G-E is defined by formula (x, g ) = s(x)g , x B , g G. Let us relax our restrictions on equivariant mappings of principal bundles with the structure group G. Consider arbitrary equivariant mappings of total spaces of principal G­bundles with arbitrary bases. Each fiber of a principal G­bundle is an orbit of the right action of the group G on the total space and hence for each equivariant mapping : E -E of total spaces, each fiber of the bundle p : E -B maps to a fiber of the bundle p : E -B . In other words, the mapping induces a mapping of bases : B -B and the following diagram is commutative E - E p p . B - B Let U B be a chart in the base B and let U be a chart such that (U ) U .
- The mapping 1 makes the following diagram commutative
-1

(1.21)

(1.22)



(1.23)

U â G p U



-








U â G p . U



(1.24)

-

In diagram (1.24), the mappings p and p are pro jections onto the first factors. So one has
-1

(x , g ) = ((x ), h (x )g ).


Introduction to bund le theory

17

Hence the mapping (1.22) is continuous. Compare the transition functions of these two bundles. First we have (x , ¯ Then ((x ), h1 (x ) ¯ = that is, h1 (x ) ¯ or h1 (x )1 2 (x )h ¯
1 1 1

2

(x )g ) =

1

2

- (x , g ) = 1 1 2 (x , g ).

2

(x )g ) =

- 1 -1 1 1 1 1 1 -1 1 -2

2 2

(x , g ) = (x , g ) =

-1 1

2 (x , g ) =

2

((x ), 1 ¯

((x ))h2 (x )g ),



2

(x ) = ¯
-1 2

1



2

((x ))h2 (x ),
1

(x ) = ¯



2

((x )).

(1.25)

By Theorem 1 the left part of (1.25) are the transition functions of a bundle isomorphic to the bundle p : E -B . (1.26) Thus any equivariant mapping of total spaces induces a mapping of bases : B -B . Moreover, under a proper choice of the coordinate homeomorphisms the transition functions of the bundle (1.26) are inverse images of the transition functions of the bundle (1.21). The inverse is true as well: if : B -B is a continuous mapping and p : E -B is a principal G­bundle then one can put U = -1 (U ), (x ) = ((x)). ¯ ¯ Then the transition functions (1.27) define a principal G­bundle p : -B , for which there exists an equivariant mapping : E -E (1.27)


18

Chapter 1

with commutative diagram (1.23). The bundle defined by the transition functions (1.27) is called the inverse image of the bund le p : E -B with respect to the mapping . In the special case when the mapping : B -B is an inclusion then we say that the inverse image of the bundle with respect to the mapping is the restriction of the bund le to the subspace B = (B ). In this case the total space of the restriction of the bundle to the subspace B coincides with the inverse image E = p-1 (B ) E . Thus if E -E is an equivariant mapping of total spaces of principal G­bundles then the bundle p : E -B is an inverse image of the bundle p : -B with respect to the mapping : B -B . Constructing of the inverse image is an important way of construction new locally trivial bundles. The following theorem shows that inverse images with respect to homotopic mappings are isomorphic bundles. Theorem 4 Let p : E -B â I be a principal G­bund le, where the base is a Cartesian product of the compact space B and the unit interval I = [0, 1], and let G be a Lie group. Then restrictions of the bund le p to the subspaces B â {0} and B â {1} are isomorphic. Pro of. Without of the loss of generality we can assume that the atlas {U } consists of an atlas {V } on the space B and a finite system of intervals [ak , ak+1 ] which cover the segment I , that is, U = V â [ak , a
k+1

].
k+1

Further, it suffices to assume that there is only one interval [ak , a equals I , so U = V â I . Then the transition functions depend on the two arguments x t I . Further, we can assume that the transition functions ¯ ¯ and continuous on the closures V V and thus are uniformly Hence, we can assume that there exists an open neighborhood O of element of the group G homeomorphic to a disc and such that


] which

V V and are defined continuous. the neutral

(x, t1 )-1 (x, t2 ) O, x V V , t1 , t2 I ,


Introduction to bund le theory

19

and the power O the functions

N

lies in a disk, where N is the number of charts. We construct h : V -G, 1 N ,

by induction. Let h1 (x) 1. ¯ ¯ Then the function h2 (x) is defined on the set V1 V2 by the formula h2 (x) =
-1 12

(x, 1)12 (x, 0) O.

(1.28)

Then the function (1.28) can be extended on the whole chart V2 , h2 : V2 -O. Further, notice that when < the function h (x) should satisfy the following condition h (x) = -1 (x, 1)h (x) (x, 0). (1.29) ¯ ¯ on the set V V . Assume that the functions h (x) O are defined for all < and satisfy the condition (1.29). Then the function h (x) is well defined on the set ¯ ¯ (V V )
<

and takes values in the set O . Hence, the function h can be extended on the ¯ chart V taking values in O . Corollary 1 If the transition functions (x) and (x) are homotopic within the class of the transition functions then corresponding bund les are isomorphic.

Examples
1. In section 1 we considered the M¨ obius band. The transition functions take two values in the homeomorphism group of the fiber: the identity homeomorphism e(y ) y , y I and homeomorphism j (y ) 1 - y , y I . The group generated by the two elements e and j has the order two since j 2 = e. So instead of the M¨ obius band we can consider corresponding principal bundle with the structure group G = Z2 . As a topological space the group G consists of two isolated points. So the fiber of the principal bundle is the discrete twopoint space. This fiber space can be thought of as two segments with ends which are identified crosswise. Hence the total space is also a circle and the pro jection p : S1 -S1 is a two-sheeted covering. This bundle is nontrivial since the total space of a trivial bundle would have two connected components.


20

Chapter 1

2. In example 3 of section 1 the tangent bundle of two dimensional sphere was considered. The coordinate homeomorphisms : U â R2 -R3 â R3 were defined by formulas that were linear with respect to the second argument. Hence the transition functions also have values in the group of linear transformations of the fiber F = R2 , that is, G = GL (2, R). It can be shown that the structure group G can be reduced to the subgroup O(n) of orthonormal transformations, induced by rotations and reflections of the plane. Let us explain these statements about the example of the tangent bundle of the two dimensional sphere S2 . To define a coordinate homeomorphism means to define a basis of tangent vectors e1 (x), e2 (x) at each point x U such that functions e1 (x) and e2 (x) are continuous. Let us choose two charts U = {(x, y , z ) : z = 1}, U = {(x, y , z ) : z = -1}. The south pole P0 = (0, 0, -1) belongs to the chart U . The north pole P1 = (0, 0, +1) belongs to the chart U . Consider the meridians. Choose an orthonormal basis for the tangent space of the point P0 and continue it along the meridians by parallel transfer with respect to the Riemannian metric of the sphere S2 to all of the chart U . Thus we obtain a continuous family of orthonormal bases e1 (x), e2 (x) defined at each point of U . In a similar way we construct a continuous family of orthonormal bases e1 (x), e2 (x) defined over U . Then the coordinate homeomorphisms are defined by the following formulas (x, , ) = e1 (x) + e2 (x) (x, , ) = e1 (x) + e2 (x). The transition function f = -1 expresses the coordinates of a tangent vector at a point x U U in terms of the basis e1 (x), e2 (x) by the coordinates of the same vector with respect to the basis e1 (x), e2 (x). As both bases are orthonormal, the change of coordinates ( , ) into coordinates ( , ) is realized by multiplication by an orthogonal matrix. Thus the structure group GL (2, R) of the tangent bundle of the sphere §2 is reducing to the subgroup O(2) GL (2, R). 3. Any trivial bundle with the base B can be constructed as the inverse image of the mapping of the base B into a one-point space {pt} which is the base of a trivial bundle.


Introduction to bund le theory

21

1.3

VECTOR BUNDLES

The most important special class of locally trivial bundles with given structure group is the class of bundles where the fiber is a vector space and the structure group is a group of linear automorphisms of the vector space. Such bundles are called vector bund les . So, for example, the tangent bundle of two-dimensional sphere S2 is a vector bundle. One can also consider locally trivial bundles where fiber is a infinite dimensional Banach space and the structure group is the group of invertible bounded operators of the Banach space. In the case when the fiber is Rn , the vector bundle is said to be finite dimensional and the dimension of the vector bundle is equal to n (dim = n). When the fiber is an infinite dimensional Banach space, the bundle is said to be infinite dimensional . Vector bundles possess some special features. First of all notice that each fiber p-1 (x), x B has the structure of vector space which does not depend on the choice of coordinate homeomorphism. In other words, the operations of addition and multiplication by scalars is independent of the choice of coordinate homeomorphism. Indeed, since the structure group G is GL (n, R) the transition functions


: (U U ) â Rn -(U U ) â Rn

are linear mappings with respect to the second factor. Hence a linear combination of vectors goes to the linear combination of images with the same coefficients. Denote by ( ) the set of all sections of the vector bundle . Then the set ( ) becomes an (infinite dimensional) vector space. To define the structure of vector space on the ( ) consider two sections s1 , s2 : s1 , s2 : B -E . Put (s1 + s2 )(x) = s1 (x) + s2 (x), x B , (s1 )(x) = (s1 (x)), R, x B . (1.30) (1.31)

The formulas (1.30) and (1.31) define on the set ( ) the structure of vector space. Notice that an arbitrary section s : B -E can be described in local terms. Let {U } be an atlas, : U â Rn -p-1 (U ) be coordinate homeomorphisms, = -1 . Then the compositions
-1

s : U -U â Rn


22

Chapter 1

are sections of trivial bundles over U and determine vector valued functions s : U -Rn by the formula (
-1

)(x) = (x, s ), x U .

On the intersection of two charts U U the functions s (x) satisfy the following compatibility condition s (x) =


(x)(s (x)).

(1.32)

Conversely, if one has a family of vector valued functions s : U -Rn which satisfy the compatibility condition (1.32) then the formula s(x) = (x, s (x)) determines the mapping s : B -E uniquely (that is, independent of the choice of chart U ). The map s is a section of the bundle .

1.3.1

Op erations of direct sum and tensor pro duct

There are natural operations induced by the direct sum and tensor product of vector spaces on the family of vector bundles over a common base B . Firstly, consider the operation of direct sum of vector bundles. Let 1 and 2 be two vector bundles with fibers V1 and V2 , respectively. Denote the transition functions of these bundles in a common atlas of charts by 1 (x) and 2 (x). Notice that values of the transition function 1 (x) lie in the group GL (V1 ) whereas the values of the transition function 2 (x) lie in the group GL (V2 ). Hence the transition functions 1 (x) and 2 (x) can be considered as matrix­values functions of orders n1 = dim V1 and n2 = dim V2 , respectively. Both of them should satisfy the conditions (1.9) from the section 2. We form a new space V = V1 V2 . The linear transformation group GL (V ) is the group of matrices of order n = n1 + n2 which can be decomposed into blocks with respect to decomposition of the space V into the direct sum V1 V2 . Then the group GL (V ) has the subgroup GL (V1 ) GL (V2 ) of matrices which have the following form: A= A1 0 0 A2 = A1 A2 , A1 = GL (V1 ), A2 = GL (V2 ).


Introduction to bund le theory

23

Then we can construct new the transition functions (x) =
1

(x)

2

(x) =

1 (x) 0 0 2 (x)

.

(1.33)

The transition functions (1.33) satisfy the conditions (1.9) from section 2, that is, they define a vector bundle with fiber V = V1 V2 . The bundle constructed above is called the direct sum of vector bund les 1 and 2 and is denoted by = 1 2 . The direct sum operation can be constructed in a geometric way. Namely, let p1 : E1 -B be a vector bundle 1 and let p2 : E2 -B be a vector bundle 2 . Consider the Cartesian product of total spaces E1 â E2 and the pro jection p3 = p1 â p2 : E1 â E2 -B â B . It is clear that p is vector bundle with the fiber V = V1 V2 . Consider the The diagonal of the bundle bundle is the that diagonal B â B , that is, the subset = {(x, x) : x B }. is canonically homeomorphic to the space B . The restriction p3 to B is a vector bundle over B . The total space E of this subspace E E1 â E2 that consists of the vectors (y1 , y2 ) such p1 (y1 ) = p2 (y2 ). It is easy to check that {U1 â U2 } gives an atlas of charts for the bundle p3 . The transition functions (1 2 )(1 2 ) (x, y ) on the intersection of two charts (U1 â U2 ) (U1 â U2 ) have the following form: (
1 2 )(1 2 )

(x, y ) =

1

1

1

(x) 2
2

0

2

0

(y )

.

Hence on the diagonal B the atlas consists of sets U (U â U ). Then the transition functions for the restriction of the bundle p3 on the diagonal have the following form: (
)()

(x, x) =

0 1 (x) 0 2 (x)

.

(1.34)

So the transition functions (1.34) coincide with the transition functions defined for the direct sum of the bundles 1 and 2 . Now let us proceed to the definition of tensor product of vector bundles. As before, let 1 and 2 be two vector bundles with fibers V1 and V2 and let 1 (x) and 2 (x) be the transition functions of the vector bundles 1 and 2 , 1 (x) GL (V1 ),
2

(x) GL (V2 ), x V V .


24

Chapter 1

Let V = V1 V2 . Then form the tensor product A1 A2 GL (V1 V2 ) of the two matrices A1 GL (V1 ), A2 GL (V2 ). Put (x) = Now we have obtained a satisfy the conditions (1.9) with fiber V = V1 V2 tensor product of bundles
1

(x) 2 (x).

family of the matrix value functions (x) which from the section 2. The corresponding vector bundle and transition functions (x) will be called the 1 and 2 and denoted by = 1 2 .

What is common in the construction of the operations of direct sum and operation of tensor product? Both operations can be described as the result of applying the following sequence of operations to the pair of vector bundles 1 and 2 : 1. Pass to the principal GL (V1 )­ and GL (V2 )­ bundles; 2. Construct the principal ( GL (V1 ) â GL (V2 ))­ bundle over the Cartesian square B â B ; 3. Restrict to the diagonal , homeomorphic to the space B . 4. Finally, form a new principal bundle by means of the relevant representations of the structure group GL (V1 ) â GL (V2 ) in the groups GL (V1 V2 ) and GL (V1 V2 ), respectively. The difference between the operations of direct sum and tensor product lies in choice of the representation of the group GL (V1 ) â GL (V2 ). By using different representations of the structure groups, further operations of vector bundles can be constructed, and algebraic relations holding for representations induce corresponding algebraic relations vector bundles. In particular, for the operations of direct sum and tensor product the following well known relations hold: 1. Associativity of the direct sum (1 2 ) 3 = 1 (2 3 ).


Introduction to bund le theory

25

This relation is a consequence of the commutative diagram GL (V1 V2 ) â GL (V3 ) 1 2 GL (V1 ) â GL (V2 ) â GL (V3 ) GL (V1 V2 V3 ) 3 4 GL (V1 ) â GL (V2 V3 ) where 1 (A1 , 2 3 (A1 , 4 Then 2 1 (A1 , A2 , A3 ) = (A1 A2 ) A3 , 4 3 (A1 , A2 , A3 ) = A1 (A2 A3 ). It is clear that 2 1 = 4 since the relation (A1 A2 ) A3 = A1 (A2 A3 ) is true for matrices. 2. Associativity for tensor products (1 2 ) 3 = 1 (2 3 ). This relation is a consequence of the commutative diagram GL (V1 V2 ) â GL (V3 ) 1 2 GL (V1 ) â GL (V2 ) â GL (V3 ) GL (V1 V2 V3 ) 3 4 GL (V1 ) â GL (V2 V3 ) The commutativity is implied from the relation (A1 A2 ) A3 = A1 (A2 A3 ) for matrices.
3

A2 (B A2 (A

, A3 ) = (A1 A2 , A3 ), , A3 ) = B A3 , , A3 ) = (A1 , A2 A3 ), 1 , C ) = A1 C.


26

Chapter 1

3. Distributivity: (1 2 ) 3 = (1 3 ) (2 3 ). This property is implied by the corresponding relation (A1 A2 ) A3 = (A1 A3 ) (A2 A3 ). for matrices. 4. Denote the trivial vector bundle with the fiber Rn by n. The total space ¯ of trivial bundle is homeomorphic to the Cartesian product B â Rn and it follows that n = ¯ ¯ . . . ¯ n times). ¯11 1( and ¯ = , 1 n = . . . (n times). ¯

1.3.2

Other op erations with vector bundles

Let V = Hom (V1 , V2 ) be the vector space of all linear mappings from the space V1 to the space V2 . For infinite dimensional Banach spaces we will assume that all linear mappings considered are bounded. Then there is a natural representation of the group GL (V1 ) â GL (V2 ) into the group GL (V ) which to any pair A1 GL (V1 ), A2 GL (V2 ) associates the mapping (A1 , A2 ) : Hom (V1 , V2 )-Hom (V1 , V2 ) by the formula (A1 , A2 )(f ) = A2 f A
-1 1

.

(1.35)

Then following the general method of constructing operations for vector bundles one obtains for each pair of vector bundles 1 and 2 with fibers V1 and V2 and transition functions 1 (x) and 2 (x) a new vector bundle with fiber V and transition functions (x) = (1 (x), 2 (x)) (1.36) This bundle is denoted by HOM (1 , 2 ). When V2 = R1 , the space Hom (V1 , R1 ) is denoted by V1 . Correspondingly, when 2 = ¯ the bundle HOM ( , ¯ will be denoted by and called the dual 1 1)


Introduction to bund le theory

27

bund le . It is easy to check that the bundle can be constructed from by means of the representation of the group GL (V ) to itself by the formula A-(At ) There is a bilinear mapping V â V -R1 , which to each pair (x, h) associates the value h(x). Consider the representation of the group GL (V ) on the space V â V defined by matrix A 0 A- 0 (A, 1) (see (1.35)). Then the structure group GL (V â V ) of the bundle is reduced to the subgroup GL (V ). The action of the group GL (V ) on the V â V has the property that the mapping is equivariant with respect to trivial action of the group GL (V ) on R1 . This fact means that the value of the form h on the vector x does not depend on the choice of the coordinate system in the space V . Hence there exists a continuous mapping ¯ : -¯, 1 which coincides with on each fiber. Let k (V ) be the k-th exterior power of the vector space V . Then to each transformation A : V -V is associated the corresponding exterior power of the transformation k (A) : k (V )-k (V ), that is, the representation k : GL (V )- GL (k (V )). The corresponding operation for vector bundles will be called the operation of the k-th exterior power and the result denoted by k ( ). Similar to vector spaces, for vector bundles one has 1 ( ) = , k ( ) = 0 for k > dim , k (1 2 ) = k
=0 -1

, A GL (V ).

( 1 )

k-

(2 ),

(1.37)


28

Chapter 1

where by definition

0 ( ) = ¯. 1

It is convenient to write the relation (1.37) using the partition function. Let us introduce the polynomial t ( ) = 0 ( ) + 1 ( )t + 2 ( )t2 + . . . + n ( )tn . Then t (1 2 ) = t (1 ) t (2 ). (1.38) and the formula (1.38) should be interpreted as follows: the degrees of the formal variable are added and the coefficients are vector bundles formed using the operations of tensor product and direct sum.

1.3.3

Mappings of vector bundles
i = {pi : Ei -B , Vi is fiber}.

Consider two vector bundles 1 and 2 where

Consider a fiberwise continuous mapping f : E1 -E2 . The map f will be called a linear map of vector bund les or homomorphism of bund les if f is linear on each fiber. The family of all such linear mappings will be denoted by Hom (1 , 2 ). Then the following relation holds: Hom (1 , 2 ) = (HOM (1 , 2 )). (1.39)

By intuition, the relation (1.39) is evident since elements from both the lefthand and right-hand sides are families of linear transformations from the fiber V1 to the fiber V2 , parametrized by points of the base B . To prove the relation (1.39), let us express and right-hand sides of (1.39) in terms of lo {U } and coordinate homeomorphisms 1 , of the mapping f : E1 -E2 we construct a elements from both the left-hand cal coordinates. Consider an atlas 2 for bundles 1 , 2 . By means family of mappings:

(2 )-1 f 1 : U â V1 -U â V2 , defined by the formula: [(2 )-1 f 1 ](x, h) = (x, f (x)h),


Introduction to bund le theory

29

for the continuous family of linear mappings f (x) : V1 -V2 . On the intersection of two charts U U two functions f (x) and f (x) satisfy the following condition 2 (x)f (x) = f (x)1 (x), or f (x) =
2

(x)f (x)1 (x).

Taking into account the relations (1.35), (1.36) we have f (x) =


(x)(f (x)).

(1.40)

In other words, the family of functions f (x) V = Hom (V1 , V2 ), x U satisfies the condition (1.40), that is, determines a section of the bundle HOM (1 , 2 ). Conversely, given a section of the bundle HOM (1 , 2 ), that is, a family of functions f (x) satisfying condition (1.40) defines a linear mapping from the bundle 1 to the bundle 2 . In particular, if 1 = ¯, V1 = R1 1 then Hom (V1 , V2 ) = V2 . Hence Hence ¯ HOM (1, 2 ) = 2 . (2 ) = Hom (¯, 2 ), 1

that is, the space of all sections of vector bundle 2 is identified with the space of all linear mappings from the one dimensional trivial bundle ¯ to the bundle 1 2 . The second example of mappings of vector bundles gives an analogue of bilinear form on vector bundle. A bilinear form is a mapping V â V -R1 ,


30

Chapter 1

which is linear with respect to each argument. Consider a continuous family of bilinear forms parametrized by points of base. This gives us a definition of bilinear form on vector bundle, namely, a fiberwise continuous mapping f : -¯ 1 (1.41)

which is bilinear in each fiber and is called a bilinear form on the bund le . Just as on a linear space, a bilinear form on a vector bundle (1.41) induces a linear mapping from the vector bundle to its dual bundle ¯ f : - , such that f decomposes into the composition ¯ f Id - -¯, 1 where Id : - is the identity mapping and
¯ f Id -

¯ is the direct sum of mappings f and Id on each fiber. When the bilinear form f is symmetric, positive and nondegenerate we say that f is a scalar product on the bund le . Theorem 5 Let be a finite dimensional vector bund le over a compact base space B . Then there exists a scalar product on the bund le , that is, a nondegenerate, positive, symmetric bilinear form on the . Pro of. We must construct a fiberwise mapping (1.41) which is bilinear, symmetric, positive, nondegenerate form in each fiber. This means that if x B , v1 , v2 p-1 (x) then the value f (v1 , v2 ) can be identified with a real number such that f (v1 , v2 ) = f (v2 , v1 ) and f (v , v ) > 0 for any v p-1 (x), v = 0.

Consider the weaker condition f (v , v ) 0.


Introduction to bund le theory

31

Then we obtain a nonnegative bilinear form on the bundle . If f1 , f2 are two nonnegative bilinear forms on the bundle then the sum f1 + f2 and a linear combination 1 f1 + 2 f2 for any two nonnegative continuous functions 1 and 2 on the base B gives a nonnegative bilinear form as well. Let {U } be an atlas for the bundle . The restriction |U is a trivial bundle and is therefore isomorphic to a Cartesian product U â V where V is fiber of . Therefore the bundle |U has a nondegenerate positive definite bilinear form f : |U |U -¯. 1 In particular, if v p-1 (x), x U and v = 0 then f (v , v ) > 0. Consider a partition of unity {g } subordinate to the atlas {U }. Then 0 g (x) 1, g (x) 1,


Supp g U . We extend the form f by formula ¯ f (v1 , v2 ) = g (x)f 0
(v1 ,v2 )

v1 , v2 p-1 (x) x U , v1 , v2 p-1 (x) x U .

(1.42)

It is clear that the form (1.42) defines a continuous nonnegative form on the bundle . Put f (v1 , v2 ) = f (v1 , v2 ). (1.43)


The form (1.43) is then positive definite. Actually, let 0 = v p-1 (x). Then there is an index such that g (x) > 0. This means that x U and f (v , v ) > 0. Hence and f (v , v ) > 0. ¯ f (v , v ) > 0


32

Chapter 1

Theorem 6 For any vector bund le over a compact base space B with dim = n, the structure group GL (n, R) reduces to subgroup O(n). Pro of. Let us give another geometric interpretation of the property that the bundle is locally trivial. Let U be a chart and let : U â V -p
-1

(U )

be a trivializing coordinate homeomorphism. Then any vector v V defines a section of the bundle over the chart U : U -p
-1

(U ), (x)

= (x, v ) p-1 (U ).

If v1 , . . . , vn is a basis for the space V then corresponding sections
k (x) = (x, vk )

form a system of sections such that for each point x U the family of vectors 1 (x), . . . , n (x) p-1 (x) is a basis in the fiber p-1 (x). Conversely, if the system of sections
1 , . . . , n : U -p-1 (U )

forms basis in each fiber then we can recover a trivializing coordinate homeomorphism (x, i v i ) = i i (x) p-1 (U ).
i i

From this point of view, the transition function = -1 has an inter pretation as a change of basis matrix from the basis {1 (x), . . . , n (x)} to {1 (x), . . . , n (x)} in the fiber p-1 (x), x U U . Thus Theorem 6 will be proved if we construct in each chart U a system of sections {1 , . . . , n } which form an orthonormal basis in each fiber with respect to a inner product in the bundle . Then the transition matrices from one basis {1 (x), . . . , n (x)} to another basis {1 (x), . . . , n (x)} will be orthonormal, that is, (x) O(n). The proof of Theorem 6 will be completed by the following lemma. Lemma 1 Let be a vector bund le, f a scalar product in the bund le and {U } an atlas for the bund le . Then for any chart U there is a system of sections {1 , . . . , n } orthonormal in each fiber p-1 (x), x U .


Introduction to bund le theory

33

Pro of. The proof of the lemma simply repeats the Gramm-Schmidt method of construction of orthonormal basis. Let 1 , . . . , n : U -p-1 (U ) be an arbitrary system of sections forming a basis in each fiber p-1 (x), x U . Since for any x U , 1 (x) = 0 one has f (1 (x), 1 (x)) > 0. Put 1 (x) = 1 (x) f (1 (x), 1 (x)) .

The new system of sections 1 , 2 , . . . , n forms a basis in each fiber. Put 2 (x) = 2 (x) - f (2 (x), 1 (x))1 (x). The new system of sections 1 , 2 , 3 (x), . . . , n forms a basis in each fiber. The vectors 1 (x) have unit length and are orthogonal to the vectors 2 (x) at each point x U . Put 2 (x) 2 (x) = . f (2 (x), 2 (x)) Again, the system of sections 1 , 2 , 3 (x), . . . , n forms a basis in each fiber and, moreover, the vectors 1 , 2 are orthonormal. Then we rebuild the system of sections by induction. Let the sections 1 , . . . , k , k+1 (x), . . . , n form a basis in each fiber and suppose that the sections 1 , . . . , k be are orthonormal in each fiber. Put
k

k

+1

(x) = k

+1

(x) -
i=1

f ( k

k+1

(x), i (x))i (x),

k

+1

(x) = f (

+1

(x)
k+1

. (x))

k+1

(x),

It is easy to check that the system 1 , . . . , k+1 , k+2 (x), . . . , n forms a basis in each fiber and the sections 1 , . . . , k+1 are orthonormal. The lemma is proved by induction. Thus the proof ofh Theorem 6 is finished.


34

Chapter 1

Remarks
1. In Lemma 1 we proved a stronger statement: if {1 , . . . , n } is a system of sections of the bundle in the chart U which is a basis in each fiber p-1 (x) and if in addition vectors {1 , . . . , k } are orthonormal then there are sections {k+1 , . . . , n } such that the system {1 , . . . , k ,
k+1

, . . . , n }

is orthonormal in each fiber. In other words, if a system of orthonormal sections can be extended to basis then it can be extended to orthonormal basis. 2. In theorems 5 and 6 the condition of compactness of the base B can be replaced by the condition of paracompactness. In the latter case we should first choose a locally finite atlas of charts.

1.4

LINEAR TRANSFORMATIONS OF BUNDLES

Many properties of linear mappings between vector spaces can be extended to linear mappings or homomorphisms between vector bundles. We shall consider some of them in this section. Fix a vector bundle with the base B equipped with a scalar product. The value of the scalar product on a pair of vectors v1 , v2 p-1 (x) will be denoted by v1 , v2 . Consider a homomorphism f : - of the vector bundle to itself. The space of all such homomorphisms Hom ( , ) has the natural operations of summation and multiplication by continuous functions. Thus the space Hom ( , ) is a module over the algebra C (B ) of continuous functions on the base B . Further, the operation of composition equips the space Hom ( , ) with the structure of algebra. Using a scalar product in the bundle one can introduce a norm in the algebra Hom ( , ) and hence equip it with a structure of a Banach algebra: for each homomorphism f : - put f = sup
v =0

f (v ) , v v, v .

(1.44)

where v=


Introduction to bund le theory

35

The space Hom ( , ) is complete with respect to the norm (1.44). Indeed, if a sequence of homomorphisms fn : - is a Cauchy sequence with respect to this norm, that is, lim fn - fm = 0
n,m-

then for any fixed vector v p-1 (x) the sequence fn (v ) p-1 (x) is a Cauchy sequence as well since fn (v ) - fm (v ) leq fn - fm · v . Hence there exists a limit f (v ) = lim fn (v ).
n-

The mapping f : - is evidently linear. To show its continuity one should consider the mappings h
n,

= -1 fn : U â V -U â V ,

which are defined by the matrix valued functions on the U . The coefficients of these matrices give Cauchy sequences in the uniform norm and therefore the limit values are continuous functions. This means that f is continuous. The scalar product in the vector bundle defines an adjoint linear mapping f by the formula f (v1 ), v
2

= v1 , f (v2 ) , v1 , v2 p-1 (x).

The proof of existence and continuity of the homomorphism f is left to reader as an exercise.

1.4.1

Complex bundles

The identity homomorphism of the bundle to itself will be denoted by 1. Consider a homomorphism I : - which satisfies the condition The restriction Ix = I
|p-1 (x)

I 2 = -1.


36

Chapter 1

is an automorphism of the fiber p

-1

(x) with the property

2 Ix = -1.

Hence the transformation Ix defines a structure of a complex vector space on p-1 (x). In particular, one has dim V = 2n. Let us show that in this case the structure group GL (2n, R) is reduced to the subgroup of complex transformations GL (n, C). First of all notice that if the system of vectors {v1 , . . . , vn } has the property that the system {v1 , . . . , vn , I v1 , . . . , I vn } is a real basis in the space V , dim V = 2n, then the system {v1 , . . . , vn } is a complex basis of V . Fix a point x0 . The space p-1 (x) is a complex vector space with respect to the operator Ix0 , and hence there is a complex basis {v1 , . . . , vn } p-1 (x). Let U x0 be a chart for the bundle . There are sections {1 , . . . , n } in the chart U such that k (x0 ) = vk , 1 k n. Then the system of sections {1 , . . . , n , I 1 , . . . , I n } forms a basis in the fiber p-1 (x0 ) and therefore forms basis in each fiber p-1 (x) in sufficiently small neighborhood U x0 . Hence the system {1 (x), . . . , n (x)} forms a complex basis in each fiber p-1 (x) in the neighborhood U x0 . This means that there is a sufficient fine atlas {U } and a system of sections {1 (x), . . . , n (x)} on -1 each chart U giving a complex basis in each fiber p (x) in the neighborhood U x0 . Fix a complex basis {e1 , . . . , en } in the complex vector space Cn . Put : U â Cn
n

- p-1 (U ),
n n uk k (x) + k=1 n k=1 zk k (x), k=1 vk Ix (k (x))

(x,
k=1

zk ek )

= =

zk

=

uk + ivk , 1 k n.

Then the transition functions = -1 are determined by the transi tion matrices from the complex basis {1 (x), . . . , n (x)} to the complex basis {1 (x), . . . , n (x)}. These matrices are complex, that is, belong to the group


Introduction to bund le theory

37

GL (n, C). A vector bundle with the structure group GL (n, C) is called a complex vector bund le . Let be a real vector bundle. In the vector bundle , introduce the structure of a complex vector bundle by means the homomorphism I : - given by I (v1 , v2 ) = (-v2 , v1 ), v1 , v2 p-1 (x). (1.45)

The complex vector bundle defined by (1.45) is called the complexification of the bund le and is denoted by c . Conversely, forgetting of the structure of complex bundle on the complex vector bundle turns it into a real vector bundle. This operation is called the realification of a complex vector bund le and is denoted by r . It is clear that rc = . The operations described above correspond to natural representations of groups: c : GL (n, R)- GL (n, C), r : GL (n, C)- GL (2n, R). Let us clarify the structure of the bundle cr . If is a complex bundle, that is, is a real vector bundle with a homomorphism I : - giving the structure of complex bundle on it. By definition the vector bundle cr is a new real vector bundle = with the structure of complex vector bundle defined by a homomorphism I1 : I1 (v1 , v2 ) - , = (-v2 , v1 ). (1.46)

The mapping (1.46) defines a new complex structure in the vector bundle which is, in general, different from the complex structure defined by the mapping I . Let us split the bundle in another way: f : - , f (v1 , v2 ) = (I (v1 + v2 ), v1 - v2 ). and define in the inverse image a new homomorphism I2 (v1 , v2 ) = (I v1 , -I v2 ).


38

Chapter 1

Then f I 2 = I1 f because f I2 (v1 , v2 ) = f (I v1 , -I v2 ) = (v2 - v1 , I (v1 + v2 )), I1 f (v1 , v2 ) = I1 (I (v1 + v2 ), v1 - v2 ) = (v2 - v1 , I (v1 + v2 )). (1.48) (1.49) (1.47)

Comparing (1.49) and (1.48) we obtain (1.47). Thus the mapping f gives an isomorphism of the bundle cr (in the image) with the sum of two complex vector bundles: the first is and the second summand is homeomorphic to but with different complex structure defined by the mapping I , I (v ) = -I (v ). ¯ This new complex structure on the bundle is denoted by . The vector bundle ¯ is called the complex conjugate of the complex bund le . Note that the vector ¯ bundles and are isomorphic as real vector bundles, that is, the isomorphism is compatible with respect to the large structure group GL (2n, R) but not isomorphic with respect to the structure group GL (n, C). Thus we have the formula ¯ cr = .

The next proposition gives a description of the complex conjugate vector bundle in term of the transition functions. Prop osition 1 Let the : U - GL (n, C) be transition functions of a complex vector bund le . Then the complex conju¯ gate vector bund le is defined by the complex conjugate matrices . ¯ Pro of. Let V be a complex vector space of dimension n with the operator of multiplication by and imaginary unit I . Denote by V the same space V with a new complex structure defined by the operator I = -I . The complex basis {e1 , . . . , en } of the space V is simultaneously the complex basis of V . But if the vector v V has complex coordinates (z1 , . . . , zn ) then the same vector v in V has coordinates (z1 , . . . , zn ). ¯ ¯


Introduction to bund le theory

39

Hence if {f1 , . . . , fn } is another complex basis then the complex matrix of change from the first coordinate system to another is defined by expanding vectors {f1 , . . . , fn } in terms of the basis {e1 , . . . , en }. For the space V these expansions are fk =
j

zj k ej ,

(1.50)

and in the space V these expansions are fk =
j

zj k ej . ¯

(1.51)

Relations (1.50) and (1.51) prove the proposition (1). Prop osition 2 A complex vector bund le has the form = c if and only if there is a real linear mapping : - (1.52) such that 2 = 1, I = -I , (1.53)

where I is the multiplication by the imaginary unit. Pro of. First we prove the necessity. As a real vector bundle the bundle has the form = . Then define a mapping : - by formula (v1 , v2 ) = (v1 , -v2 ). The mapping is called the operation of complex conjugation . If the vector v has the property (v ) = v then v is called a real vector. Similarly, the section is called real if ( (x)) = (x) for any x B . A homomorphism f : - is called real if f = f . If f is real and v is real then f (v ) is real as well. The sufficiency we shall prove a little bit later.


40

Chapter 1

1.4.2

Subbundles

Let f : 1 -2 be a homomorphism of vector bundles with a common base B and assume that the fiberwise mappings fx : (1 )x -(2 )x have constant rank. Let p1 : E1 -B p2 : E2 -B be the pro jections of the vector bundles 1 and 2 . Put E0 E Theorem 7 = = {y E1 : f (y ) = 0 p-1 (x), x = p1 (y )} 2 f (E1 ).

1. The mapping p0 = p1 |E0 : E0 -B

is a local ly trivial bund le which admits a unique vector bund le structure 0 such that natural inclusion E0 E1 is a homomorphism of vector bund les. 2. The mapping p = p2 |E : E -B (1.54) is a local ly trivial bund le which admits a unique vector bund le structure such that the inclusion E E2 and the mapping f : E1 -E are homomorphisms of vector bund les. 3. There exist isomorphisms : : such that composition f -1 : 0 - has the matrix form f
-1 1 2

- 0 , - ,

=

0 Id 00

.


Introduction to bund le theory

41

The bundle 0 is called the kernel of the mapping f and denoted Ker f , the bundle is called the image of the mapping f and denoted Im f . So we have dim 1 = dim Ker f + dim Im f . Pro of. Assume that vector bundles 1 and 2 are equipped with scalar products. Firstly, let us show that the statement 2 holds, that is, the mapping (1.54) gives a vector bundle. It is sufficient to prove that for any point x0 B there is a chart U x0 and a system of continuous sections 1 , . . . , k : U -E which form a basis in each subspace fx (p-1 (x)) p-1 (x). For this fix a basis 1 2 {e1 , . . . , ek } in the subspace fx0 (p-1 (x0 )). The mapping 1 fx0 : p-1 (x0 )-fx0 (p 1
-1 1

(x0 ))

is an epimorphism and hence we may choose vectors g1 , . . . , gk p-1 (x0 ) such 1 that fx0 (gj ) = ej , 1 j k . Since 1 is a locally trivial bundle for each x0 there is a chart U x0 such that over U the bundle 1 is isomorphic to a Cartesian product U â p-1 (x0 ). 1 Therefore there are continuous sections 1 , . . . , k : U -E such that j (x0 ) = gj , 1 j k . Consider the sections j = f (j ), 1 j k . Then j (x0 ) = f (j (x0 )) = f (gj ) = ej , 1 j k . The sections j are continuous and hence there is a smaller neighborhood U x0 such that for any point x U the system {1 (x), . . . , k (x)} forms a basis of the subspace fx (p-1 (x)). Thus we have shown that E is the total space of 1 a vector bundle and the inclusion E E2 is a homomorphism. Uniqueness of the vector bundle structure on the E is clear. Now we pass to the statement 1. Denote by P : 1 -1 the fiberwise mapping which in each fiber p-1 (x) is the 1 orthonormal pro jection onto the kernel Ker fx p-1 (x). It sufficient to check 1 that P is continuous over each separate chart U . Notice that the kernel Ker fx is the orthogonal complement of the subspace Im fx . The rank of fx equals the rank of fx . Hence we can apply part 2 of Theorem 7 to the mapping f .
1


42

Chapter 1

Thus the image Im f is a locally trivial vector bundle. Now let {1 , . . . , k } be a system of sections, j : U -Im f E1 ,
forming a basis of the subspace Im fx p-1 (x) for any x U . Then applying 1 the pro jection P to a vector y p-1 (x) gives a decomposition of the vector y 1 into a linear combination k

y = P (y ) +
j =1

j j (x)

satisfying P (y ), j (x) = 0, 1 j k or y-
j =1 k

j j (x), l (x) = 0, 1 l k .

The coefficients j satisfy the following system of linear equations
k

j j (x), l (x) = y , l (x) , 1 l k .
j =1

The matrix j (x), l (x) depends continuously on x. Hence the inverse matrix also depends continuously and, as a consequence, the numbers j depend continuously on x. It follows that the pro jection P is continuous. The rank of P does not depend on x and so we can apply statement 2. Thus statement 1 is proved. The last statement is implied by the following decompositions 1 2 = = Ker f Im f , Ker f Im f .

(1.55)

Now we can complete the proof of Proposition 2. Consider a vector bundle with mapping (1.52) satisfying the condition (1.53). Consider a mapping P= 1 (1 + ) : - . 2 (1.56)

The mapping P is a pro jection in each fiber and has a constant rank. Applying Theorem 7 we get = Im P Ker P = 0 1


Introduction to bund le theory

43

and I (0 ) = 1 , I (1 ) = 0 . Hence = c0 . The proof of Proposition 2 is now completed. Theorem 8 Let be a vector bund le over a compact base B . Then there is a vector bund le over B such that ¯ = N = a trivial bundle. Pro of. Let us use Theorem 7. It is sufficient to construct a homomorphism ¯ f : -N , where the rank of f equals dim in each fiber. Notice that if is trivial then such an f exists. Hence for any chart U there is a homomorphism ¯ f : |U -N , rank f = dim . Let { } be a partition of unity subordinate to the atlas {U }. Then each mapping f can be extended by the zero trivial mapping to a mapping ¯ g : -N , g |U = f . The mapping g has the following property: if (x) = 0 then rank g |x = dim . Let g: g g (y ) ¯ ¯ - N = N , = g that is = (g1 (y ), . . . , g (y ), . . .).

It is clear that the rank of g at each point satisfies the relation rank g rank g dim . Further, for each point x B there is such an that (x) = 0. Hence, rank g dim . Therefore we can apply Theorem 7. By (1.57) we get Ker g = 0. ¯ Hence the bundle is isomorphic to Im g and N = Im g . (1.57)


44

Chapter 1

Theorem 9 Let 1 , 2 be two vector bund les over a base B and let B0 B a closed subspace. Let f0 : 1 |B0 -2 |B0 be a homomorphism of the restrictions of the bund les to the subspace B0 . Then the mapping f0 can be extended to a homomorphism f : 1 -2 , f |B0 = f0 . Pro of. In the case where the bundles are trivial the problem is reduced to constructing extensions of matrix valued functions. This problem is solved using Urysohn's lemma concerning the extension of continuous functions. For the general case, we apply Theorem 8. Let 1 1 and 2 2 be trivial bundles, let P : 2 2 -2 be a natural pro jection, and let Q : 1 -1 1 be the natural inclusion. Finally, let h0 = f0 0 : (1 1 )|B0 -(2 2 )|B
0

be the direct sum of f0 and the trivial mapping. Then there is an extension of h0 to a homomorphism h : 1 1 -2 2 . Let f = P hQ : 1 -2 . It is clear that f |B0 = f0 . (1.58)

Remark
Theorem 9 is true not only for compact bases but for more general spaces (for example, paracompact spaces). The proof then becomes a little bit more complicated.

Exercises
1. Using the notation of Theorem 9 and assuming that f0 : 1 |B0 -2 |B0 is a fiberwise monomorphism, prove that f0 can be extended to a fiberwise monomorphism f : 1 |U -2 |U in some neighborhood U B0 .


Introduction to bund le theory

45

¯ 2. Under the conditions of previous exercise prove there is a trivial bundle N and a fiberwise monomorphism ¯ g : 1 -2 N , extending f0 , that is, g |B0 = (f0 , 0).

1.5

VECTOR BUNDLES RELATED TO MANIFOLDS

The most natural vector bundles arise from the theory of smooth manifolds. Recall that by an n­dimensional manifold one means a metrizable space X such that for each point x X there is an open neighborhood U x homeomorphic to an open subset V of n­dimensional linear space Rn . A homeomorphism : U -V Rn is called a coordinate homeomorphism . The coordinate functions on the linear space Rn pulled back to points of the neighborhood U , that is, the compositions xj = xj : U -R1 are called coordinate functions on the manifold X in the neighborhood U . This system of functions {x1 , . . . , xn } defined on the neighborhood U is called a local system of coordinates of the manifold X . The open set U equipped with the local system of coordinates {x1 , . . . , xn } is called a chart . The system of charts {U , {x1 , . . . , xn }} is called an atlas if {U } covers the manifold X , that is, X = U . So each n­dimensional manifold has an atlas. If a point x X belongs to two charts, x U U , then in a neighborhood of x there are two local systems of coordinates. In this case, the local coordinates xj can expressed as functions of values of the local k coordinates {x1 , . . . , xn }, that is, there are functions f such that xk =
k

(x1 , . . . , xn ).

(1.59)


46

Chapter 1

The system of functions (1.59) are called a change of coordinates or transition functions from one local coordinate system to another. For brevity (1.59) will be written as xk = xk (x1 , . . . , xn ). If an atlas {U , {x1 , . . . , xn }} is taken such that all are differentiable functions of the class C k , 1 k the structure of differentiable manifold of the class functions are analytic functions then one says that X analytic manifold . In the case n = 2m, uk = xk , 1 k m,
k v k z

the transition functions then one says X has C k . If all the transition has the structure of an

= =

xm+k , 1 k m, k uk + iv , 1 k m,

and the functions
k z = k 1 m (z , . . . , z ) + i m+k 1 m (z , . . . , z )

are complex analytic functions in their domain of definition then X has the structure of a complex analytic manifold . Usually we shall consider infinitely smooth manifolds, that is, differentiable manifolds of the class C . A mapping f : X -Y of differentiable manifolds is called differentiable of class C k if in any neighborhood of the point x X the functions which express the coordinates of the image f (x) in terms of coordinates of the point x are differentiable functions of the class C k . It is clear that the class of differentiability k makes sense provided that k does not exceed the differentiability classes of the manifolds X and Y . Similarly, one can define analytic and complex analytic mappings. Let X be a n­dimensional manifold, {U , {x1 , . . . , xn }} be an atlas. Fix a point x0 X . A tangent vector to the manifold X at the point x0 is a 1 n system of numbers ( , . . . , ) satisfying the relations:
n k = j =1



j

x

k xj

(x0 ).

(1.60)

1 n The numbers ( , . . . , ) are called the coordinates or components of the vector with respect to the chart {U , {x1 , . . . , xn }}. The formula (1.60) gives the transformation law of the components of the tangent vector under the transition from one chart to another. In differential geometry such a law is called a tensor law of transformation of components of a tensor of the valency (1, 0). So in terms of differential geometry, a tangent vector is a tensor of the


Introduction to bund le theory

47

valency (1, 0). Consider a smooth curve passing through a point x0 , that is, a smooth mapping of the interval : (-1, 1)-X, (0) = x0 . In terms of a local coordinate system {x1 , . . . , xn } the curve is determined by a family of smooth functions xk (t) = xk ( (t)), t (-1, 1). t Clearly the numbers (1.61) satisfy a tangent vector at the point x0 to tangent vector to the curve and is
k =

Let

(xk (t))|

t=0

.

(1.61)

tensor law (1.60), that is, they define a manifold X . This vector is called the denoted by d (0), that is, dt

d (0). dt The family of all tangent vectors to manifold X is a denoted by T X . The set T X is endowed with a natural topology. Namely, a neighborhood V of the vector 0 at the point x0 contains all vectors in points x such that x U for some chart U and for some , = (x, x0 ) < ,
n

(
k=1

k 0

k - )

2

< .

The verification that the system of the neighborhoods V forms a base of a topology is left to the reader. Let : T X -X (1.62) be the mapping which to any vector associates its point x of tangency. Clearly, the mapping is continuous. Moreover, the mapping (1.62) defines a locally trivial vector bundle with the base X , total space T X and fiber isomorphic to the linear space Rn . If U is a chart on the manifold X , we define a homeomorphism : U â Rn - -1 (U ) which to the system (x0 , 1 , . . . , n ) associates the tangent vector whose components are defined by the formula
n k = j =1



j

x x

k j

(x0 ).

(1.63)


48

Chapter 1

It is easy to check that the definition (1.63) gives the components of a vector , that is, they satisfy the tensor law for the transformation of components of a tangent vector (1.60). The inverse mapping is defined by the following formula:
1 n -1 ( ) = ( ( ), , . . . , ), k where are components of the vector . Therefore the transition functions = -1 are determined by the formula

(x0 , 1 , . . . , n ) =

x0 ,



j

x x

1 j

(x0 ), . . . ,



j

x x

n j

(x0 ) .

(1.64)

Formula (1.64) shows that the transition functions are fiberwise linear mappings. Hence the mapping defines a vector bundle. The vector bundle : T X -X is called the tangent bund le of the manifold X . The fiber Tx X is called the tangent space at the point x to manifold X . The terminology described above is justified by the following. Let be an inclusion of the manifold X in the Euclidean space RN . In dinate system (x1 , . . . , xn ) in a neighborhood of the point x0X , f is determined as a vector valued function of the variables (x1 , . f (x) = f (x1 , . . . , xn ). f : X -RN a local coorthe inclusion . . , xn ) : (1.65)

If we expand the function (1.65) by a Taylor expansion at the point x0 = (x1 , . . . , xn ) : 0 0 f (x1 , . . . , xn ) = f (x1 , . . . , xn )+ 0 0
n

+
j =1

f x
j

(x1 , . . . , xn )xj + o(xk ). 0 0

Ignoring the remainder term o(xk ) we obtain a function g which is close to f: g (x1 , . . . , xn ) = f (x1 , . . . , xn )+ 0 0
n

+
j =1

f x
j

(x1 , . . . , xn )xj . 0 0

Then if the vectors f x
j

(x1 , . . . , xn )xj , 1 k n 0 0


Introduction to bund le theory

49

are linearly independent the function g defines a linear n­dimensional subspace in Rn . It is natural to call this space the tangent space to the manifold X . Any vector which lies in the tangent space to manifold X (having the initial point at x0 ) can be uniquely decomposed into a linear combination of the basis vectors: n f j = . (1.66) j j =1 x
k The coordinate s { } under a change of coordinate system change with respect to the law (1.60), that is, with respect to the tensor law. Thus the abstract k definition of tangent vector as a system of components { } determines by the formula (1.66) a tangent vector to the submanifold X in RN . Let

f : X -Y be a differentiable mapping of manifolds. Let us construct the corresponding homomorphism of tangent bundles, Df : T X -T Y . Let T X be a tangent vector at the point x0 and let be a smooth curve which goes through the point x0 , (0) = x0 , and which has tangent vector , that is, = d (0). dt

Then the curve f ( (t)) in the manifold Y goes through the point y0 = f (x0 ). Put d(f ( )) Df ( ) = (0). (1.67) dt This formula (1.67) defines a mapping of tangent spaces. It remains to prove that this mapping is fiberwise linear. For this, it is sufficient to describe the mapping Df in terms of coordinates of the spaces Tx0 X and Ty0 Y . Let 1 m {x1 , . . . , xn } and {y , . . . , y } be local systems of coordinates in neighborhoods of points the x0 and y0 , respectively . Then the mapping f is defined as a family of functions j j y = y (x1 , . . . , xn ). If xk = xk (t)


50

Chapter 1

are the functions defining the curve (t) then
k =

dxk (0). dt

Hence the curve f ( (t)) is defined by the functions
j j y = y (x1 (t), . . . , xn (t))

and the vector Df ( ) is defined by the components j = dy
n j n

dt

(0) =
k=1

y

j xk

(x0 )

dxk (0) = dt (1.68)

j y xk k=1

(x0 ) k .

Formula (1.68) definition does vector at the p mapping Df is

shows firstly that the mapping Df is well- defined since the not depend on the choice of curve but only on the tangent oint x0 . Secondly, the mapping Df is fiberwise linear. The called the differential of the mapping f .

Examples
1. Let us show that the definition of differential Df of the mapping f is a generalization of the notion of the classical differential of function. A differentiable function of one variable may be considered as a mapping of the space R1 into itself: f : R1 -R1 . The tangent bundle of the manifold R1 is isomorphic to the Cartesian product R1 â R1 = R2 . Hence the differential Df : R1 â R1 -R1 â R1 in the coordinates (x, ) is defined by the formula Df (x, ) = (x, f (x) ). The classical differential has the form df = f (x)dx. So Df (x, dx) = (x, df ).


Introduction to bund le theory

51

Exercises
Show that differential Df has the following properties: 1. D(f g ) = (Df ) (Dg ), 2. D(Id) = Id, 3. if f is a diffeomorphism the Df is isomorphism of bundles, 4. if f is immersion then Df is fiberwise a monomorphism. Consider a smooth manifold Y and a submanifold X Y . The inclusion j:XY is a smooth mapping of manifolds such that the differential Dj : T X -T Y is a fiberwise monomorphism. Then over the manifold X there are two vector bundles: the first is j (T Y ), the restriction of the tangent bundle of the manifold Y to the submanifold X , the second is its subbundle T X . According to Theorem 7, the bundle j (T Y ) splits into a direct sum of two summands: j (T Y ) = T X . The complement is called the normal bund le to the submanifold X of the manifold Y 1 . Each fiber of the bundle over the point x0 consists of those tangent vectors to the manifold Y which are orthogonal to the tangent space Tx0 (X ). The normal bundle will be denoted by (X Y ) or more briefly by (X ). It is clear that the notion of a normal bundle can be submanifolds but for any immersion j : X -Y of the manifold Y . It is known that any compact manifold X Euclidean space RN for some sufficiently large number N such an inclusion. Then j (T RN ) = T X (X RN ).
1 By construction the bundle dep ends on the choice of metric on T Y . However, different metrics yield isomorphic complementary summands

defined not only for manifold X into the has an inclusion in a . Let j : X -RN be


52

Chapter 1

The bundle T RN is trivial and so ¯ T X (X ) = N . (1.69)

In this case the bundle (X ) is called the normal bund le for manifold X (irrespective of the inclusion). Note, the normal bundle (X ) of the manifold X is not uniquely defined. It depends on inclusion into the space RN and on the dimension N . But the equation (1.69) shows that the bundle is not very far from being unique. Let 1 (X ) be another such normal bundle, that is, ¯ T X 1 (X ) = N1 . Then ¯ ¯ (X ) T (X ) 1 (X ) = (X ) N1 = N 1 (X ).

The last relation means that bundles (X ) and 1 (X ) became isomorphic after the addition of trivial summands. 2. Let us study the tangent bundle of the one dimensional manifold S1 , the circle. Define two charts on the S1 : U1 = {- < < }, U2 = {0 < < 2 }, where is angular parameter in the polar system of coordinates on the plane. On the U1 take the function x1 = , - < x1 < , as coordinate, whereas on the U2 take the function x2 = , 0 < x2 < 2 . The intersection U1 U2 consists of the two connected components V1 = {0 < < }, V2 = { < < 2 }. Then the transition function has the form x1 = x1 (x2 ) = x2 , 0 < x2 < , x2 - 2 , < x2 < 2 .


Introduction to bund le theory

53

Then by (1.64), the transition function

12

for the tangent bundle has the form

12 (x, ) =

x1 = . x2

This means that the transition function is the identity. Hence the tangent bundle T S1 is isomorphic to Cartesian product T S1 = S1 â R1 , in other words, it is the trivial one dimensional bundle. 3. Consider the two-dimensional sphere S2 . It is convenient to consider it as the extended complex plane S2 = C1 {}. We define two charts on the S2 U1 U2 = C1 = (C1 \{0}) {}.

1 Define the complex coordinate z1 = z on the chart U1 and z2 = z on the chart U2 extended by zero at the infinity . Then the transition function on the intersection U1 U2 has the form

z1

1 z2

and the tangent bundle has the corresponding transition function of the form 12 (z , ) = The real form of the matrix 12 (x, y ) =
12

z1 1 = - 2 = - z 2 . z2 z2 y2 - x2 2xy
i

is given by - z2 - z2 = -2xy y2 - x2 ,

- z2 x2

where z = x + iy . In polar coordinates z = e 12 (, ) =
2

this becomes .

cos 2 sin 2

- sin 2 cos 2

Let us show that the tangent bundle T S2 is not isomorphic to a trivial bundle. If the bundle T S2 were trivial then there would be matrix valued functions h1 : U1 h2 : U2 - - GL (2, R) GL (2, R), (1.70)


54

Chapter 1

such that 12 (, ) = h1 (, )h

-1 2

(, ).

The charts U1 , U2 are contractible and so the functions h1 , h2 are homotopic to constant mappings. Hence the transition function 12 (, ) must be homotopic to a constant function. On the other hand, for fixed the function 12 defines a mapping of the circle S1 with the parameter into the group SO(2) = S1 and this mapping has the degree 2. Therefore this mapping cannot be homotopic to a constant mapping. 4. Consider a vector bundle p : E -X where the base X is a smooth manifold. Assume that the transition functions


: U U - GL (n, R)

are smooth mappings. Then the total space E is a smooth manifold and also dim E = dim X + n. For if {U } is an atlas of charts for the manifold X then an atlas of the manifold E can be defined by V = p-1 (U ) = U â Rn . The local coordinates on V can be defined as the family of local coordinates on the chart U with Cartesian coordinates on the fiber. The smoothness of the functions implies smoothness of the change of coordinates. There is the natural question whether for any vector bundle over smooth manifold X there exists an atlas on the total space with smooth the transition functions . The answer lies in the following theorem. Theorem 10 Let p : E -X be an n­dimensional vector bund le and X a compact smooth manifold. Then there exists an atlas {U } on X and coordinate homeomorphisms : U â Rn -p-1 (U ) such that the transition functions are smooth.


: U U - GL (n, R)


Introduction to bund le theory

55

Pro of. Consider a sufficiently fine atlas for the bundle p and coordinate homeomorphisms : U â Rn -p-1 (U ). The transition functions


: U U - GL (n, R)

are in general only required to be continuous. The problem is to change the homeomorphisms to new coordinate homeomorphisms such that the new transition functions are smooth. In other words, one should find functions h : U - GL (n, R) such that the compositions h-1 (x) (x)h (x) are smooth. Let {U } be a new atlas such that ¯ U U U . We construct the functions h (x) by induction on the index , 1 N . Without loss of generality, we can assume that all the functions (x) are uniformly bounded, that is, Let


(x) C.

Choose smooth functions

1 , < 1. C (x), x U U such that 0<<




(x) -



(x) <

and which form a cocycle, that is,


1,







1.

Lemma 2 Let f (x) be a continuous function defined in a domain U of Euclidean space, let K be a compact subset such that K U . Then there is a neighborhood V K such that for each > 0 there is a smooth function g (x) defined on V such that |g (x) - f (x)| < , x V .


56

Chapter 1

Moreover, if the function f (x) is smooth in a neighborhood of a compact subset K then the function g can be chosen with additional property that g (x) f (x) in a neighborhood of K . Let us apply the lemma (2) for construction of functions . By the lemma ¯ ¯ there exists a smooth function 12 defined in a neighborhood of the set U1 U2 such that 12 (x) - 12 (x) < . Assume that we have chosen smooth functions (x) defined in neighborhoods ¯ ¯ of the subsets U U for all indices < 0 such that and (x) The function lemma (2).
1,0 +1

(x) -



(x) <

(x)



(x) 1, , , 0 .
12

(x) can be constructed in the same way as

using the

Assume that functions ,0 +1 (x) are constructed for all 0 0 and they satisfy the condition of cocycle:
,0 +1

(x) , (x)

,0 +1

(x), , a0 .

Then the required function 0 +1,0 +1 (x) can be defined in a neighborhood of ¯ ¯ ¯ the subset U U0 +1 U0 +1 for any 0 by formula
0 +1,0 +1 1 (x) = -,
0

+1

(x)

,0 +1

(x).

Hence the required function 0 +1,0 +1 (x) is defined in a neighborhood of the union ¯ ¯ ¯ V = 0 (U U0 +1 U0 +1 ) and satisfies the condition
0 +1,0 +1

(x) -

0

+1,0 +1

(x) < 2C .

(1.71)

By the lemma (2) the function can be extended 0 +1,0 +1 (x) from the closure ¯ ¯ of a neighborhood of the set V to a neighborhood of the set U0 +1 U0 +1 satisfying the same condition (1.71).


Introduction to bund le theory

57

By induction, a family of functions with property , (x) -
,

(x) < (2C )N = .

2

can be constructed. Now we pass to the construction of functions h satisfying the conditions (x) = h (x) (x)h-1 (x) or h (x) =


(x)h (x) (x).

(1.72)

We construct the functions h by induction. Put h1 (x) 1. Assume that the functions h (x), 1 0 have already been constructed satisfying the conditions (1.72) and 1 - h (x) < , < 0 . Then the function h of the set
0 +1

(x) is defined by the formula (1.72)in a neighborhood ¯ V = <0 U
0

+1

¯ U . (x) (1.73)

On the set V , the inequality 1 - h0 +1 (x) = 1 - 0 +1, (x)h (x) -1 C 2 h - ,0 +1 ,0 +1 2C 2
,0 +1

is satisfied. If the number is sufficient small then the function h0 +1 (x) can be extended from a neighborhood V on the whole chart U0 +1 satisfying the same condition (1.73). 5. Let us describe the structure of the tangent and normal bundles of a smooth manifold. Theorem 11 Let j : X Y be a smooth submanifold X of a manifold Y . Then there exists a neighborhood V X which is diffeomorphic to the total space of the normal bund le (X Y ). Pro of. Fix a Riemannian metric on the manifold Y (which exists by Theorem 5 and Remark 2 from the section 3). We construct a mapping f : (X )-Y .


58

Chapter 1

Consider a normal vector (X ) at the point x X Y . Notice that the vector is orthogonal to the subspace Tx (X ) Tx (Y ). Let (t) be the geodesic curve such that d (0) = . (0) = x, dt Put f ( ) = (1). The mapping f has nondegenerate Jacobian for each point of the zero section of the bundle (X ). Indeed, notice that 1. if = 0, Tx (Y ) then f ( ) = x, 2. f ( ) = (). Therefore the Jacobian matrix of the mapping f at a point of the zero section of (X ) that maps the tangent spaces Df : Tx ( (X )) = Tx (X ) x (X )-Tx (Y ) = Tx (X ) x (X ) is the identity. By the implicit function theorem there is a neighborhood V of zero section o f the bundle (X ) which is mapped by F diffeomorphically onto a neighborhood f (V ) of the submanifold X . Since there is a sufficiently small neighborhood V which is diffeomorphic to the total space of the bundle (X ) the proof of theorem is finished.

Exercise
Prove that the total space of the tangent bundle T X for the manifold X is diffeomorphic to a neighborhood V of the diagonal X â X . 6. There is a simple criterion describing when a smooth mapping of manifolds gives a locally trivial bundle. Theorem 12 Let f : X -Y be a smooth mapping of compact manifolds such that the differential Df is epimorphism at each point x X . Then f is a local ly trivial bund le with the fiber a smooth manifold. Pro of.


Introduction to bund le theory

59

Without loss of generality one can consider a chart U Y diffeomorphic to Rn and part of the manifold X , namely, f -1 (U ). Then the mapping f gives a vector valued function f : X -Rn . Assume firstly that n = 1. From the condition of the theorem we know that the gradient of the function f never vanishes. Consider the vector field grad f (with respect to some Riemannian metric on X ). The integral curves (x0 , t) are orthogonal to each hypersurface of the level of the function f . Choose a new Riemannian metric such that grad f is a unit vector field. Indeed, consider the new metric ( , )1 = ( , )(grad f , grad f ). Then (grad f , ) = (f ) = (grad 1 f , )1 = (grad 1 f , )(grad 1 f , grad 1 f ). Hence grad 1 f = Then (grad 1 f , grad 1 f )1 = (grad 1 f , grad 1 f )(grad f , grad f ) = (grad f , grad f ) = (grad f , grad f ) = 1. (grad f , grad f )2 Thus the integral curves d f ( (t)) = (grad f , grad f ) 1. dt Hence the function f ( (t)) is linear. This means that if f (x0 ) = f (x1 ), then f ( (x0 , t)) = f ( (x1 , t)) = f (x0 ) + t. Put g : Z â R1 -X, g (x, t) = (x, t). grad f . (grad f , grad f )

The mapping g is a fiberwise smooth homeomorphism. Hence the mapping f : -R1


60

Chapter 1

gives a locally trivial bundle. Further, the proof will follow by induction with respect to n. Consider a vector valued function f (x) = {f1 (x), . . . , fn (x)} which satisfies the condition of the theorem. Choose a Riemannian metric on the manifold X such that gradients grad f1 , . . . , grad f
n

are orthonormal. Such a metric exists. Indeed, consider firstly an arbitrary metric. The using the linear independence of the differentials {dfi } we know that the gradients are also independent. Hence the matrix aij = grad fi (x), grad fj (x) is nondegenerate in each point. Let bij (x) be the matrix inverse to the matrix aij (x) , that is, bi (x)aj (x) ij .


Put k =
i

bkj (x)grad fi (x).

Then k (fj ) =
i

bki (x)grad fi (fj ) = bki (x) grad fi , grad gj = bki (x)aij (x) kj .
i

=
i

Let U be a sufficiently small neighborhood of a point of the manifold X . The system of vector fields {1 , . . . , n } can be supplemented by vector fields n+1 , . . . , N to form a basis such that k (fi ) 0. Consider the new metric in the chart U given by i , k ,
j j



ij , 0.

Let be a partition of unity subordinate to the covering {U } and put ,
0

=


(x) , .


Introduction to bund le theory

61

Then i , j k ,
0 0



ij , 0

for any vector for which (fi ) = 0.Let grad 0 fi be the gradients of the functions fi with respect to the metric , 0 . This means that grad fi , for any vector . In particular one has grad fi , j grad fk ,
0 0 0

= (fi )



ij , 0

for any vector for which (fi ) = 0. Similar relations hold for the vector field i . Therefore i = grad fi , that is, grad fi , grad fj ij , the latter proves the existence of metric with the necessarily properties. Let us pass now to the proof of the theorem. Consider the vector function g (x) = {f1 (x), . . . , f
n-1

(x)}.

This function satisfies the conditions of the theorem and the inductive assumption. It follows that the manifold X is diffeomorphic to the Cartesian product X = Z â Rn-1 and the functions fi are the coordinate functions for the second factor. Then grad fn is tangent to the first factor Z and hence the function fn (x, t) does not depend on t Rn-1 . Therefore one can apply the first step of the induction to the manifold Z , that is, Z = Z1 â R1 . Thus X = Z1 â Rn .

7. An invariant formulation of the implicit function theorem Let f : X -Y be a smooth mapping of smooth manifolds. The point y0 Y is called a regular value of the mapping f if for any point x f -1 (y0 ) the differential


62

Chapter 1

Df : Tx (X )-Ty0 (Y ) is surjective. The implicit function theorem says that the inverse image Z = f -1 (y0 ) is a smooth manifold and that TX
|Z

= T Z Rm , m = dim Y .

In general, if W Y is a submanifold then the mapping f is said to be transversal along submanifold W when for any point x f -1 (W ) one has Tf
(x)

Y = Tf

(x)

W Df (Tx X ).

In particular, the mapping f is transversal 'along' each regular point y0 Y . The implicit function theorem says that the inverse image Z=f
-1

(W )

is a submanifold, the normal bundle (Z X ) is isomorphic to the bundle f ( (W Y )) and the differential Df is the fiberwise isomorphism (Z X )- (W Y ). 8. Morse functions on manifolds Consider a smooth function f on a manifold X . A point p0 is called a critical point if df (x0 ) = 0. A critical point x0 is said to be nondegenerate if the matrix of second derivatives is nondegenerate. This property does not depend on a choice of local coordinates. Let T X be the total space of the cotangent bundle of manifold X (that is, the vector bundle which is dual to the tangent bundle). Then for each function f : X -R1 there is a mapping df : X -T X (1.74)

adjoint to Df which to each point x X associates the linear form on Tx X given by the differential of the function f at the point x. Then in the manifold T X there are two submanifolds: the zero section X0 of the bundle T X and the image df (X ). The common points of these submanifolds correspond to critical points of the function f . Further, a critical point is nondegenerate if and only if the intersection of submanifolds X0 and df (X ) at that point is transversal. If all the critical points of the function f are nondegenerate then


Introduction to bund le theory

63

f is called a Morse function. Thus the function f is a Morse function if and only if the mapping (1.74) is transversal along the zero section X0 T X . 9. Oriented manifolds. A manifold X is said to be oriented if there is an atlas {U } such that all the transition functions have positive Jacobians at each point. The choice of a such atlas is called an orientation of the manifold X . If the manifold X is orientable then the structure group GL (n, R) of the tangent bundle T X reduces to the subgroup GL + (n, R) of matrices with positive determinant. Conversely, if the structure group GL (n, R) of the tangent bundle T X can be reduced to the subgroup GL + (n, R), then the manifold X is orientable. Indeed, let {U } be an atlas, = x
k xj

be the transition functions of the tangent bundle and


= h

-1 h

be the new transition functions such that det


> 0.

Of course, if the matrix valued functions h : U - GL (n, R) had the form of the Jacobian matrix, that is, if h(x) = then the functions y
k xj

k k y = y (x1 , . . . , xn )

could serve as changes of coordinates giving an orientation on the manifold X . But in general this is not true and we need to find new functions h . Notice that if we have a suitable system of functions then one can change this system in any way that preserves the signs of det h . Hence, select new functions as follows: ±1 0 . . . 0 0 1 ... 0 ¯ h (x) = . . .. ., . . .. . . . 0 0 ... 1


64

Chapter 1

where the sign of the first entry should coincide with the sign of det h (x) on each connected component of the chart U . Then the new transition functions ¯ ¯ (x) = h (x) satisfy the same condition det


¯ (x)h-1 (x)

(x) > 0.

¯ On the other hand, the functions h (x) are the Jacobians of a change of the coordinates: y
1 2

=

±x1 , x2 , ... xn .

y = ... ... n y =

and an atlas which gives an orientation of the manifold X has been found.

1.6

LINEAR GROUPS AND RELATED BUNDLES

This section consists of examples of vector bundles which arise naturally in connection with linear groups and their homogeneous spaces.

1.6.1

The Hopf bundle

The set of all one dimensional subspaces or lines of Rn+1 is called the real (n dimensional) projective space and denoted by RPn . It has a natural topology given by the metric which measures the smaller angle between two lines. The pro jective space RPn has the structure of smooth (and even real analytical) manifold. To construct the smooth manifold structure on RPn one should first notice that each line l in Rn+1 is uniquely determined by any nonzero vector x belonging to the line. Let {x0 , . . . , xn } be the Cartesian coordinates of such a vector x, not all vanishing. Then the line l is defined by the coordinates {x0 , . . . , xn } and any {x0 , . . . , xn }, = 0. Thus the point of RPn is given by a class [x0 : x1 : . . . : xn ] of coordinates {x0 , . . . , xn } (not all vanishing) determined up to multiplication by a nonzero real number . The class [x0 : x1 : . . . : xn ] gives the projective coordinates of the point of RPn .


Introduction to bund le theory

65

We define an atlas {Uk }

n k=0

on RPn as follows. Put

Uk = {[x0 : x1 : . . . : xn ] : xk = 0}, and define coordinates on Uk by the following functions:
yk =

x , 0 n, = k . xk

where in the numbering of the coordinates by index there is a gap when = k . The change of variables on the intersection Uk Uj , (k = j ) of two charts has the following form: yj when = j, k yj yk = (1.75) 1 k when = j. y
j

The formula (1.75) is well defined because k = j and on Uk Uj
k yj =

xk = 0. xj

All the functions in (1.75) are smooth functions making RPn into a smooth manifold of dimension n. Consider now the space E in which points have the form (l, x), where l is a one dimensional subspace of Rn+1 and x is a point on l. The space E differs from the space Rn+1 in that instead of zero vector of Rn+1 in the space E there are many points of the type (l, 0). The mapping p : E -RPn , (1.76)

which associates with each pair (l, x) its the first component, gives a locally trivial vector bundle. Indeed, the space E can be represented as a subset of the Cartesian product RPn â Rn+1 defined by the following system of equations in each local coordinate system: rank x0 y0 x1 y1 . . . xn = 1, . . . yn

where (x0 , x1 , . . . , xn ) are pro jective coordinates of a point of RPn , and (y0 , y1 , . . . , yn ) are coordinates of the point of Rn+1 . For example, in the case when U0 = {x0 = 0} we put x0 = 1. Then rank 1 y0 x1 y1 . . . xn = 1, . . . yn


66

Chapter 1

that is,

det 1 xk = 0, y0 yk det xk xj = 0, yk yj
+1

Hence the set E is defined in the RPn â Rn equations:

by the following system of n

fk (x1 , . . . , xn , y0 , . . . , yn ) = yk - xk y0 = 0, k = 1, . . . , n. The Jacobian matrix of the functions fk is -y0 0 ... 0 0 -y0 ... 0 . . . . . . . . . . . . 0 0 ... -y0 -x1 -x2 ... -xn 1 0 .. 0 1 .. ... ... .. 0 0 .. .0 .0 . ... .1

(1.77)

Clearly the rank of this matrix (1.77) is maximal. By the implicit function theorem the space E is a submanifold of dimension n + 1. As coordinates one can take (y0 , x1 , . . . , xn ). The pro jection (1.76) consists of forgetting the first coordinate y0 . So the inverse image p-1 (U0 ) is homeomorphic to the Cartesian product U0 â R1 . Changing to another chart Uk , one takes another coordinate yk as the coordinate in the fiber which depends linearly on y0 . Thus the mapping (1.76) gives a one dimensional vector bundle.

1.6.2

The complex Hopf bundle

This bundle is constructed similarly to the previous example as a one dimensional complex vector bundle over the base CPn . In the both cases, real and complex, the corresponding principal bundles with the structure groups O(1) = Z2 and U(1) = S1 can be identified with subbundles of the Hopf bundle. The point is that the structure group O(1) can be included in the fiber R1 , O(1) = {-1, 1} R1 , in a such way that the linear action of O(1) on R1 coincides on the subset {-1, 1} with the left multiplication. Similarly, the group U(1) = S1 can be included in C1 : S1 = {z : |z | = 1} C1 , such that linear action of U(1) on C1 coincides on S1 with left multiplication. Hence the Hopf bundle has the principal subbundle consisting of vectors of unit length. Let p : ES -RPn


Introduction to bund le theory

67

be principal bundle associated with the Hopf vector bundle. The points of the total space ES are pairs (l, x), where l is a line in Rn+1 and x l, |x| = 1. Since x = 0, the pair (l, x) is uniquely determined by the vector x. Hence the total space ES is homeomorphic to the sphere Sn of unit radius a nd the principal bundle p : Sn -RPn is the two-sheeted covering. In the case of the complex Hopf bundle, the associated principal bundle p : ES -CPn is ES = S and the fiber is a circle S1 . Both these two principal bundles are also called Hopf bund les. When n = 1, we have S3 -P1 , that is, the classical Hopf bundle S3 -S2 . (1.78)
2n+1

In this last case it is useful to describe the transition functions for the intersection of charts. Let us consider the sphere S3 as defined by the equation |z0 |2 + |z1 |2 = 1, in the two dimensional complex vector space C2 . The map (1.78) associates the point (z0 , z1 ) to the point in the CP1 with the pro jective coordinates [z0 : z1 ]. So over the base CP1 we have the atlas consisting of two charts: U0 U1 = = {[z0 , z1 ] : z0 = 0}, {[z0 , z1 ] : z1 = 0},

The points of the chart U0 are parametrized by the complex parameter w0 = z1 C z0
1

whereas points of the chart U1 are parametrized by the complex parameter w1 = The homeomorphisms 0 : p-1 (U0 ) - S1 â C1 = S1 â U0 , 1 : p-1 (U1 ) - S1 â C1 = S1 â U1 , z0 C1 . z1


68

Chapter 1

have the form 0 (z0 , z1 ) = 1 (z0 , z1 ) = z0 |z0 z1 |z1 z |z z , |z ,
1 0 0 1

= =

z |z z |z

0 0 1 1

| |

, [z0 : z1 ] , , [z0 : z1 ] .

The mappings 0 and 1 clearly are invertible:
-1 0

(, w0 ) = (, w1 ) =

1 + |w0 | 1 + |w1 |
2 2

, ,

w0 1 + |w0 | w1 1 + |w1 |
2 2

, .

-1 1

Hence the transition function 01 : S1 â (U0 U1 )-S1 â (U0 U1 ) is defined by the formula: 01 (, [z0:z1 ]) = that is, multiplication by the number z1 |z0 | w1 = . z0 |z1 | |w1 | z1 |z0 | , [z0 : z1 ], z0 |z1 | ,

1.6.3

The tangent bundle of the Hopf bundle

Let be the complex Hopf bundle over CPn and let T CPn be the tangent bundle. Then T CPn 1 = . . . = (n + 1) . When n = 1 this gives TCP1 1 = . (1.79)

Indeed, using the coordinate transition function w2 = 1 , w1


Introduction to bund le theory

69

the transition function for the tangent bundle TCP1 has the following form: 01 (w1 ) = - w1 ¯2 1 2 = - |w |4 . w1 1 (1.80)

On the other hand, the transition function for the Hopf bundle is 01 (w1 ) = w1 . |w1 | (1.81)

Using homotopies the formulas (1.80) and (1.81) can be simplified to 01 (w1 ) = w1 , ¯2 01 (w1 ) = w1 . This shows that the matrix functions w1 ¯2 0 0 1 and w1 ¯ 0 0 w1 ¯

are homotopic in the class of invertible matrices when w1 = 0, w1 C1 . In general, let us consider the manifold CPn as the quotient space of the unit sphere S2n+1 in Cn+1 by the action of the group S1 C1 of complex numbers with unit norm. The total space of the tangent bundle T CPn is the quotient space of the family of all tangent vectors of the sphere which are orthogonal to the orbits of the action of the group S1 . In other words, one has T CPn = {(u, v ) : u, v Cn+1 , |u| = 1, u, v = 0} , {(u, v ) (u, v ), S1 }

where u, v is the Hermitian inner product in Cn+1 Consider the quotient space {(u, s) : u Cn+1 , |u| = 1, s C1 } ¯ ¯ A= . {(u, s) (u, s), S1 } ¯ ¯ Associate to each pair (u, s) A the line l which passes through vector u ¯ ¯ Cn+1 and the vector su l. If (u, s) is an equivalent pair (which passes ¯ ¯ through vector u) then it corresponds the same line l and the same vector ¯ su = (s)(u). This means that the space A is homeomorphic to the total ¯ space of vector bundle . Hence the total space of the vector bundle (n + 1) is homeomorphic to the space B= {(u, s) : u, v Cn+1 , |u| = 1, } . {(u, v ) (u, v ), S1 }


70

Chapter 1

The space T CPn = is clearly a subspace of B. A complementary subbundle can be defined as a quotient space D= {(u, v ) : u, v Cn+1 , |u| = 1, v = su, s C1 } . {(u, v ) (u, v ), S1 }

The latter is homeomorphic to the space D= {(u, s) : u Cn+1 , |u| = 1, s C1 } , {u u, S1 }

which is homeomorphic to the Cartesian product T CPn â C1 . Thus one has the isomorphism (1.79).

1.6.4

Bundles of classical manifolds

Denote by Vn,k the space where the points are orthonormal sets of k vectors of n­dimensional euclidean space (Rn or Cn ). If we need we will write VR n,k or VC . Correspondingly, let us denote by Gn,k the space in which points are n,k k ­dimensional subspaces of n­dimensional euclidean space (Rn or Cn ). By expanding a k ­frame to a basis in Cn we obtain V
C n,k

= U(n)/U(n - k ),

where U(n - k ) U(n) is a natural inclusion of unitary matrices: A Similarly, the space G
n,k n-k

-

1k 0

0 A
n-k

.

is homeomorphic to the homogeneous space U(n)/ (U(k ) U(n - k )) ,

where U(k ) U(n - k ) U(n) is natural inclusion (Ak , B
n-k

)-

Ak 0

0 . Bn-k

Generally speaking, if G is a Lie group, and H G is a subgroup then the pro jection p : G-G/H


Introduction to bund le theory

71

is a locally trivial bundle (a principal H ­bundle) since the rank of Jacobian matrix is maximal and consequently constant. Hence the following mappings give locally trivial bundles Vn-k1 ,k-k1 Vn,k - Vn,k1 , V
C n,k

U(k) C - Gn,k .

(The fibers are shown over the arrows.) In particular, V
R n,n VC n,n VR 1 n, VC 1 n,

= O(n), = U(n), =S =S
n-1

, .

2n-1

Hence we have the following locally trivial bundles U(n-1) U(n) - S2
n-1 -1

,

O(n-1) O(n) - Sn

,

All the mappings above are defined by forgetting some of the vectors from the frame. The manifolds Vn,k are called the Stieffel manifolds, and the Gn,k called Grassmann manifolds.


72

Chapter 1


2
HOMOTOPY INVARIANTS OF VECTOR BUNDLES

In the section 1.2 of the chapter 1 (Theorem 3) it was shown that if the base of a locally trivial bundle with a Lie group as structure group has the form B â I then the restrictions of the bundle to B â {0} and B â {1} are isomorphic. This property of locally trivial bundles allows us to describe bundles in the terms of homotopy properties of topological spaces.

2.1

THE CLASSIFICATION THEOREMS
p : E -B

Assume that the base B of a locally trivial bundle

is a cellular space, that is, the space B is a direct limit of its compact subspaces [B ]n , B = limn- [B ]n , and each space [B ]n is constructed from [B ](n-1) by attaching to it a finite n family of discs Di by using continuous mappings S
n-1 i

-[B ]

(n-1)

.

The subspace [B ]n is called the n­dimensional skeleton of the space B . Let us consider a principal G­bundle, pG : EG -BG , (2.1)

73


74

Chapter 2

such that all homotopy groups of the total space EG are trivial: i (EG ) =), 0 i < . (2.2)

Theorem 13 Let B be a cel lular space. Then any principal G­bund le p : E -B (2.3)

is isomorphic to the inverse image of the bund le (2.1) with respect to a continuous mapping f : B -BG . (2.4) Two inverse images of the bund le (2.1) with respect to the mappings f , g : B -B
G

are isomorphic if and only if the mappings f and g are homotopic. Corollary 2 The family of al l isomorphism classes of principal G­bund les over the base B is in one to one correspondence with the family of homotopy classes of continuous mappings from B to BG . Corollary 3 If two cel lular spaces B and B are homotopy equivalent then the families of al l isomorphism classes of principal G­bund les over the bases B and B are in one to one correspondence. This correspondence is defined by inverse image with respect to a homotopy equivalence B -B . Corollary 4 If the space BG is a cel lular space then it is defined by the condition (2.2) up to homotopy equivalence. Pro of of Theorem 13. By Theorem 2 of section 1.2 of chapter 1 the bundle (2.3) is isomorphic to an inverse image with respect to a continuous mapping (2.4) if and only if there is an equivariant mapping of total spaces of these bundles. Let B be a cellular


Homotopy Invariants

75

space, B0 be a cellular subspace, (that is, B0 is union of some cells of the space B and is closed subspace). Consider a principal bundle (2.3) and let E0 = p-1 (B0 ). Let F0 : E0 -EG (2.5) be an equivariant continuous mapping. Let us show that F0 can be extended to a continuous equivariant mapping F : E -EG . (2.6)

Lemma 3 Any principal G­bund le p over a disc Dn is isomorphic to a trivial bund le. Pro of. In fact, put h : Dn â I h(x, t) Then h(x, 0) h(x, 1) x 0. - Dn , = tx, 0 x 1, x Dn .

Then by Theorem 3 of the section 1.2 of the chapter 1, inverse images of the bundle p with respect to the mappings h0 (x) h1 (x) = h(x, 0), = h(x, 1)

are isomorphic. On the other hand, the inverse image of the bundle p with respect to the identity mapping h1 is isomorphic to the bundle p and the inverse image with respect to the mapping h0 is trivial. Returning to the proof of Theorem 13, let the (n - 1)­dimensional skeleton n [B ](n-1) lie in the subspace B0 with a cell Di outside B0 . Let
n i : Di -B


76

Chapter 2

be the natural inclusion. Then i (Si
(n-1)

) [B ](

n-1)

B0 .

(2.7)

n Consider the restriction of the bundle p to the cell Di . By Lemma 3 this n restriction is a trivial bundle, that is, the total space is isomorphic to Di â G.By (2.7) the mapping (2.5) induces the equivariant mapping

(F0 )|(

Si

(n-1)

âG)

: (Si

(n-1)

â G) - EG x Si -EG .
(n-1)

F0 (x, g ) = F0 (x, 1)g , Consider the mapping F0 (x, 1) : Si Since
(n-1) (n-1)

, g G. (2.8)

(EG ) = 0,

the mapping (2.8) can be extended to a continuous mapping
n F (x, 1) : Di -EG .

Put

n F (x, g ) = F (x, 1)g , x Di , g G.

(2.9)

It is clear that the mapping (2.9) is equivariant and extends the mapping F0 n to the set p-1 (B0 Di ) E0 . Using induction on the number of cells we can establish the existence of an equivariant mapping (2.6) which extends the mapping (2.5). The first statement of the theorem follows by putting B0 = . The second statement of the theorem follows if we substitute B for B â I and put B0 = (B â {0}) (B â {1}). Corollaries 2 and 3 follow directly from Theorem 13. Pro of of Corollary 4. Let p : EG p : EG be two principal G­ bundles such that i (EG ) = i (EG ) = 0, 0 i , - BG , - B
G


Homotopy Invariants

77

and let BG and BG be cellular spaces. Then by Theorem 13 there are two continuous mappings f : BG g : BG such that f (p ) = p, g (p) = p . Hence (g f ) (p) = (g f ) (p ) = By the second part of Theorem 13, gf fg Id. Id. p = (Id) (p), p = (Id) (p ). - BG , - BG

This means that the spaces BG and BG are homotopy equivalent. The bundle (2.1) is called the universal principal G­bund le and the base B is called the classifying space for bundles with structure group G
G

2.2

EXACT HOMOTOPY SEQUENCE

It was shown in Theorem 13 of section 2.1 that the homotopy groups of the total space total space are needed for the construction of a universal principal G­bundle. Definition 4 A mapping p : X -Y is said to satisfy the lifting homotopy axiom if for any cel lular space P , any homotopy f : P â I -Y and any mapping g0 : P â {0}-Y


78

Chapter 2

satisfying the condition pg0 (x, 0) = f (x, 0), there is an extension g : P â I -Y satisfying a similar condition pg (x, t) = f (x, t), x P, t I . (2.10)

The mapping g is a lifting mapping (for mapping p if the two mappings f and g lifting homotopy axiom states that if the mapping then any homotopy f (x, t) has a Theorem 14 Let

the mapping f ) with respect to the satisfy the condition (2.10). So the initial mapping f (x, 0) has a lifting lifting homotopy.

p : E -B be a local ly trivial bund le. Then p satisfies the lifting homotopy axiom. Pro of. Assume firstly that (2.11) is a trivial bundle, that is, E = B â F. Then the lifting mapping g0 can be described as g0 (x, 0) = (f (x, 0), h(x)). Then let g (x, t) = (f (x, t), h(x)).

(2.11)

(2.12)

Clearly the mapping (2.12) is a lifting mapping for the mapping f . Further, let P0 P be a cellular subspace with a lifting homotopy g0 : P0 â I -E = B â F. In other words, the lifting mapping is defined on the subspace (P0 â I ) (P â {0}) P â I .


Homotopy Invariants

79

We need to extend the mapping g0 to a lifting mapping g : P â I -E . Notice that it is sufficient to be able to construct an extension of the mapping g0 on a single cell k Di â I P â I . The restriction of the mapping g0 defines the lifting mapping on the subset
k (Di â {0}) (Si (k-1) k â I ) Di â I .

(2.13)

k This subset has the form of a `glass' where the `bottom' corresponds to Di â {0} (k-1) k and the `sides' to Si â I . So the Cartesian product Di â I is homeomorphic to itself in such a way that the subset (2.13) lies on the `bottom'. Hence the problem of the construction of an extension of the mapping g0 from (2.13) to the whole Cartesian product is equivalent to the problem of an extension of k k the lifting mapping from the `bottom' Di â {0} to the Di â I and the latter problem is already solved. To complete the proof, consider a sufficiently fine cell decomposition of the space

P = D and the interval

k i

I = Il such that image of each Cartesian product of a cell with an interval lies in a k chart of the bundle p. Then the restriction of the mapping f onto Di â Il maps into a chart U and the lifting mapping g maps it into the total space of the bundle p|p-1 (U ) . The latter is a trivial bundle. Hence the problem is reduced to the case we have proved. Theorem 15 Let p : E -B be a local ly trivial bund le, x0 B , y0 p-1 (x0 ) = F and let j : F -E be a natural inclusion of the fiber in the total space. Then there is a homomorphism of homotopy groups : k (B , x0 )-
k -1

( F , y0 )


80

Chapter 2

such that the sequence of homomorphisms . . . -k (F, y0 )-k (E , y0 )-k (B , x0 )- j p - k-1 (F, y0 )-k-1 (E , y0 )-k-1 (B , x0 )- . . . j p . . . -1 (F, y0 )-1 (E , y0 )-1 (B , x0 ) is an exact sequence of groups. Pro of. The elements of homotopy group k (X, x0 ) can be represented as homotopy classes of continuous mappings : I k -X such that To construct the mapping let : I k -B be a representative of an element of the group k (B , x0 ) and note that Ik = I Hence (I
k -1 k-1 j


p





(2.14)

( I k ) = x0 .

â I. â {1}) = x0 .
k-1

â {0}) = (I

k -1

The mapping can be considered as a homotopy of the cube I (u, 0) y0 , u I
k -1

. Put

.

Then the mapping is a lifting of the mapping (u, 0). By the lifting homotopy axiom there is a lifting homotopy :I such that p (u, t) (u, t). In particular, p (u, 1) x0 . This means that (u, 1) maps the cube I ( I
k-1 k -1 k -1

â T -E ,

into the fiber F . Moreover, since

â I ) = x0


Homotopy Invariants

81

the lifting can be chosen such that ( I
k-1

â I ) y0 .

Hence the mapping (u, 1) maps I k-1 to the point y0 . This means that the mapping (u, 1) defines an element of the group k-1 (F, y0 ). This homomorphism is well defined. Indeed, if the mapping is homotopic to the mapping then the corresponding lifting mappings and are homotopic and a homotopy between and can be constructed as a lifting mapping for the homotopy between and . Hence the homomorphism is well defined. The proof of additivity of is left to the reader. Next we need to prove the exactness of the sequence (2.14). 1. Exactness at the term k (E , y0 ). If : Sk -F

represents an element of the group k (F, y0 ), then the mapping pj : Sk -B represents the image of this element with respect to homomorphism p j . Since p(F ) = x0 , we have Hence Im j Ker p . Conversely, let : Sk -E pj (Sk ) = x0 .

be a mapping such that p is homotopic to a constant, that is, [] Ker p . Let be a homotopy between p and a constant mapping. By the lifting homotopy axiom there is a lifting homotopy between and a mapping which is lifting of the constant mapping. Thus p (Sk ) = x0 , that is, (Sk ) F.


82

Chapter 2

Hence [ ] = [] Im j , that is, Ker p Im j . 2. Exactness in the term k (B , y0 ). Let : I k -E

represent an element of the group k (B , x0 ). The composition p : I k = I
k -1

â I -B

is lifted by . According to definition, the restriction (u, 1) : I represents the element [p]. Since (u, 1) y0 , we have p [] = 0. Now let : Ik = Ik
-1 k -1

-f E

â I -B

represent an element of the group k (B , x0 ) and :I
k-1

â I -E

be a lifting of the mapping . Then the restriction (u, 1) represents the element []. If the mapping (u, 1) : I
k -1

-F

is homotopic to the constant mapping then there is a homotopy :I
k -1

â I -F

between (u, 1) and constant mapping. Let us now construct a new mapping :I
k-1

â I -E ,


Homotopy Invariants

83

(u, 1) = Then

(u, 2t), (u, 2t - 1), (u, 2t), x0 ,

1 2

0t 1 2 t 1.

p (u, 1) =

1 2

0t 1 2 t 1.

Hence the mapping p is homotopic to the mapping , that is, p [ ] = [] or Ker Im p . 3. Exactness in the term k (F, y0 ). Let the mapping :I
k+1

= I k â I -B
+1

represent an element of the group k

(B , x0 ) and let

: I k â I -E lift the mapping . Then the restriction (u, 1) represents the element ([]) k (F, y0 ). Hence the element j ([]) is represented by the mapping (u, 1). The latter is homotopic to constant mapping (u, 0), that is, j ([]) = 0. Conversely, if : I k â I -E

is a homotopy between (u, 1) F and constant mapping (u, 0), then according to the definition the mapping (u, 1) represents the element ([p]). Hence Ker j Im .


84

Chapter 2

Remark
Let 0 (X, x0 ) denote the set of connected components of the space X where the fixed element [x0 ] is the component containing the point x0 . Then the exact sequence (2.14) can be extended as follows . . . -1 (F, y0 )-1 (E , y0 )-1 (B , x0 )- j p -0 (F, y0 )-0 (E , y0 )-0 (B , x0 ). This sequence is exact in the sense that the image coincides with the inverse image of the fixed element.
j


p



Examples
1. Consider the covering R1 -S1

in which the fiber Z is the discrete space of integers. The exact homotopy sequence . . . -k (Z)-k (R1 )-k (S1 )- . . . j p . . . -1 (Z)-1 (R1 )-1 (S1 )- j p -0 (Z)-0 (R1 )-0 (S1 )


j

p



has the following form . . . -0-0-k (S1 )-0- . . . j p . . . -0-0-1 (S1 )- j p -Z-0-0, since k (R1 ) = 0, k (Z) = 0, Hence 1 (S1 ) = k (S ) = 2. Consider the Hopf bundle p : S3 -S2 .
1 j p

k 0, k 1.

Z, 0 when k 2.


Homotopy Invariants

85

The corresponding exact homotopy sequence . . . -k (S1 )-k (S3 )-k (S2 )- . . .


j

p



... ... ... has the following form:

-3 (S1 )-3 (S3 )-3 (S2 )- j p -2 (S1 )-2 (S3 )-2 (S2 )- j p -1 (S1 )-1 (S3 )-1 (S2 )-




j

p



... . . . since

-0-k (S3 )-k (S2 )-0 dots j p . . -0-Z-3 (S2 )- j p . . -0-0-2 (S2 )- j p . . -Z-0-0,


j

p



3 (S1 ) = 2 (S1 ) = 2 (S3 ) = = 1 (S3 ) = 1 (S2 ) = 0. Hence 2 (S2 ) = 1 (S1 ) = Z, 3 (S2 ) = Z. The space S2 is the simplest example of a space where there are nontrivial homotopy groups in degrees greater than the dimension of the space. Notice also that the Hopf bundle pro jection p : S3 -S2 gives a generator of the group 3 (S2 ) = Z. Indeed, from the exact homotopy sequence we see that a generator of the group 3 (S3 ) is mapped to a generator of the group 3 (S2 ) by the homomorphism p . A generator of the group 3 (S3 ) is represented by the identity : S3 -S3 and hence p [] = [p] = [p]. 3. In the general case there are two bundles: the real Hopf bundle Sn -RPn with fiber Z2 , and complex Hopf bundle S
2n-1

-CPn


86

Chapter 2

with fiber S1 . In the first case, the exact homotopy sequence shows that 1 (Sn ) = 0-1 (RPn )-0 (Z2 ) = Z2 -0, k (Sn ) = 0-k (RPn )-k-1 (Z2 ) = 0, Hence k (RPn ) = 0,
1 1

1 (RPn ) = Z2 , 2 k n.

2 k n - 1.

When n = 1, we have RP = S . In the complex case, one has 0 = 2 (S2n-1 )-2 (CPn )-1 (S1 ) = Z -1 (S2n-1 ) = 0-1 (CPn )-0 (S1 ) = 0 k (S2n-1 ) = 0-k (CPn )-k-1 (S1 ) = 0 where 2n - 1 > k > 2. Hence 1 (CPn ) = 2 (CPn ) = k (CPn ) = 0, Z, 0 for 3 k < 2n - 1. -

4. Consider natural inclusion of matrix groups j : O(n)-0(n + 1). In previous chapter it was shown that j is a fiber of the bundle p : O(n + 1)-Sn . So we have the exact homotopy sequence . . . -k
+1

(Sn )-k (O(n))-k (O(n + 1))-k (Sn )- . . .


j

(2.15)

If k + 1 < n that is k n - 2 then k (O(n)) k (O(n + 1)). Hence k (O(k + 2)) k (O(k + 3)) k (O(N )) for N k + 2. This property is called the stability of homotopy groups of the series of orthogonal groups . In particular, when k = 1 one has 1 (O(1)) = 1 (O(2)) = 1 (O(3)) = 0, Z, 1 (O(4)) = . . . = 1 (O(N )).


Homotopy Invariants

87

Also SO(3) = RP3 , and so 1 (O(3) = 1 (RP3 ) = Z2 . Hence 1 (O(n)) = Z2 for n 3. Moreover from (2.15) one has the exact sequence Z = 1 (O(2))-1 (O(3)) = Z2 -0. This means that the inclusion O(2) O(3) induces epimorphism of fundamental groups. 5. Consider the inclusion in the complex case j : U(n)-U(n + 1) giving a fiber of the bundle p : U(n + 1)-S The exact homotopy sequence . . . -k
+1 2n+1

.

(S

2n+1

)-k (U(n))-k (U(n + 1))-k (S2


j

k+1

)- . . . (2.16)

shows that k (U(n)) = k (U(n + 1)) for k < 2n. and, in particular, 1 (U(1)) = . . . = 1 (U(n)). From U(1) = S it follows that 1 (U(n)) = Z, For k = 2, 2 (U(2)) = . . . = 2 (U(n)). From (2.16) it follows that . . . -p3 (S3 )-2 (U(1))-2 (U(2))-2 (S3 )- . . . , and hence 0-2 (U(2))-0, or 2 (U(2)) = 0. n 1.
1


88

Chapter 2

Hence 2 (U(n)) = 0 for any n. Let us now calculate 3 (U(n)). Firstly, for n 2, 3 (U(2)) = . . . = 3 (U(n)). and so it is sufficient to calculate the group 3 (U(2)). Consider the locally trivial bundle p : U(2)-S1 , (2.17) where p(A) = det A. The fiber of (2.17) is the group SU(2). Thus we have the exact homotopy sequence . . . -4 (S1 )-3 (SU(2))-3 (U(2))-3 (S 1 )- . . . . The end terms are trivial and hence 3 (SU(2)) = 3 (U(2)). Further, the group SU(2) is homeomorphic to the 3­dimensional sphere S3 C2 , and therefore 3 (SU(2)) = Z. Thus, 3 (U (1)) = 3 (U (n)) = 0, Z, n 2.

6. In the same spirit as the two previous examples consider locally trivial bundle p : Sp(n)-S4n-1 with a fiber Sp(n - 1) where Sp(n) is the group of orthogonal quaternionic transformations of the quaternionic space Kn = R4n . Each such transformation can be described as a matrix with entries which are quaternions. Hence each column is a vector in Kn = R4n . The mapping p assigns the first column to each matrix. The exact homotopy sequence . . . -k (Sp(n - 1))-k (Sp(n))-k (S4n-1 )- . . . . . . -1 (Sp(n - 1))-1 (Sp(n))-1 (S4n-1 )-0 gives the following table of isomorphisms 1 (Sp(n)) = 1 (Sp(1)) = 1 (S3 ) = 0, 2 (Sp(n)) = 2 (Sp(1)) = 2 (S3 ) = 0, pi3 (Sp(n)) = 3 (Sp(1)) = 3 (S3 ) = Z, k (Sp(n)) = k (Sp(n - 1)) for k < 4n - 2.


Homotopy Invariants

89

7. Let p : EG -B
G

be a universal principal G­bundle. By the exact homotopy sequence . . . -k (EG )- we have k (BG ) k since k (EG ) = 0 for all k 0. 8. Let G = S1 . Then the space BS1 can be constructed as a direct limit CP = lim CPn , and total space ES1 the direct limit S Therefore, 2 (CP ) k (CP ) =Z = 0 for k = 2.
-1 k -1

(BG )-

k -1

(G)-k

-1

(EG )- . . .

(G)

= lim S2

n+1

.

(2.18)

The table (2.18) means that the space CP is an Eilenberg­McLane complex K (Z, 2). Thus the same space CP classifies both the one­dimensional complex vector bundles and two­dimensional integer cohomology group. Hence the family Line (X ) of one dimensional complex vector bundles over the base X is isomorphic to the cohomology group H 2 (X, Z). Clearly, there is a one to one correspondence l : Line (X )-[X, CP ] defined by the formula l([f ]) = f ( ),

where f : X -CP is a continuous mapping, [f ] [X, CP ], and is universal one­dimensional vector bundle over CP . Then the isomorphism : Line (X )-H 2 (X, Z), mentioned above is defined by the formula ( ) = l
-1

( )



(a) H 2 (X, Z), Line (X ),

where a H 2 (CP , Z) is a generator.


90

Chapter 2

Theorem 16 The additive structure in the group H 2 (X, Z) corresponds with the tensor product in the family of one­dimensional complex vector bund les over X with respect to the isomorphism . Pro of. The structure group of a one­dimensional complex vector bundle is U(1) = S1 . The tensor product is induced by the homomorphism : S1 â S1 -S1 , which is defined by the formula (z1 , z2 ) = z1 z2 . The homomorphism (2.19) induces a mapping of the classifying spaces : CP â CP -CP such that universal bundle over CP


(2.19)

(2.20)

goes to tensor product

( ) = 1 2 . Hence the tensor product structure in the Line (X ) is induced by the composition f1 âf2 X -X â X - CP â CP -CP . (2.21) On the other hand the mapping (2.21) induces the additive structure on the two­dimensional cohomology group. Indeed, the group structure (2.19) has the property that (z , 1) = (1, z ) = z . Hence the same holds for (2.20): (x, x0 ) = (x0 , x) = x. Therefore, Thus ([f1 ] [f2 ]) = = ( (f1 â f2 )) (a) = () (f1 â f2 ) (a) = = () (f1 â f2 ) (a 1 + 1 a) =
= () (f1 (a) 1 + 1 f2 (a)) = f1 (a) + f2 (a) = = ([f1 ]) + ([f2 ]).

(a) = (a 1) + (1 a) H 2 (CP , Z).


Homotopy Invariants

91

2.3

CONSTRUCTIONS OF THE CLASSIFYING SPACES
pG : EG -BG .

In this section we shall construct a universal principal G­bundle

There are at least two different geometric constructions of a universal G­bundle. The problem is to construct a contractible space EG with a free action of the group G.

2.3.1

Lie groups
U(n) U(N + n)

Let us start with G = U(n). Consider the group U(N + n). The two subgroups:

and U(N ) U(N + n) are included as matrices X 0 0 , X U(n) 1 and 10 , Y U(N ) 0Y

These subgroups commute and hence the group U(n) acts on the quotient space U(N + n)/U(N ). This action is free and the corresponding quotient space is homeomorphic to U(N + n)/U(N ) U(n). Thus one has a principal U(n)­bundle U(N + n)/U(N )-U(N + n)/U(N ) U(n). It is easy to see that total space of this bundle is homeomorphic to the complex Stieffel manifold VN +n,n and the base is homeomorphic to complex Grassmann manifold GN +n,n . Now consider the inclusion j : U(N + n)-U(N + n + 1) defined by T - T 0 0 , T U(N + n). 1


92

Chapter 2

The subgroup U(N ) maps to the subgroup U(N + 1): 10 - 0Y 10 0Y 00 1 0 0=0 0 1 00 Y0 01 , Y U(N ),

and the subgroup U(n) maps to itself. Hence the inclusion j generates an inclusions of principal U(n)­bundles U(N + )/U(N ) n p U(N + n)/ (U(N ) U(n)) or -
j j

U(N + n + /U(N + 1) 1) p

- U(N + n + 1)/ (U(N + 1) U(n)) VN n+1 + p . j - GN +n+1,n = lim V , -
j

VN n,n + p
,n

(2.22)

Let V

GN +n,n denote the direct limit V
,n

N +n,n

and G

,n

the direct limit G
,n

= lim GN

+n,n

.

The commutative diagram (2.22) induces the mapping p:V
,n

-G,n .

(2.23) this it is sufficient the definition of a GN +n,n is open be constructed as

The mapping (2.23) gives a principal U(n)­bundle. To prove to show that p gives a locally trivial bundle. Notice that by direct limit, a set U G,n is open iff for any N the set U in the space GN +n,n . Hence charts for the bundle (2.23) can a union = N where N G is a open chart in G
N +n,n N +n,n

such that N N
+1

.

The existence of the sequence {N } follows from Theorem 3 chapter 1 and from the fact that any closed cellular subset has a sufficiently small neighborhood which contracts to the closed set. Let us show that k ( V
,n

) = 0.

(2.24)


Homotopy Invariants

93

Consider the bundle U(N + n)-VN with fiber U(N ). The inclusion j : U(N )-U(N + n) induces an isomorphism of the homotopy groups for k < 2N (see example 2.2 of the previous section). Hence from the exact homotopy sequence -k (U(N ))-k (U(N + n))-k (VN +n,n )- -k-1 (U(N ))-k-1 (U(N + n))- . . . it follows that k (V for k < 2N . Since k (V
,n N +n,n +n,n

)=0
N +n,n

) = lim k (V

)

the equality (2.24) holds for any k . Thus we have shown that (2.23) is a universal principal U(n)­bundle. Now let G U(n) be a closed subgroup. Since we already know that there is a universal principal U(n)­bundle (2.23), the group U(n) freely acts on the total space V,n . Hence the group G also acts freely on the same space V,n . Prop osition 3 The mapping pG : V
,n

-BG = V

,n

/G

gives a local ly trivial principal G­bund le. Pro of. Notice that the quotient mapping q : U(n)-U(n)/G gives a locally trivial G­bundle. Since the bundle (2.23) is locally trivial, and for any point x V,n there is an equivariant open neighborhood W x such that W U(n) â W0 , W0 G,n ,


94

Chapter 2

then the mapping pG |W : W = U(n) â W0 -U(n)/G â W0 gives a locally trivial bundle and the set U(n)/G â W0 is an open set. Hence the mapping pG gives a locally trivial bundle. Prop osition 4 For each finite dimensional matrix group G there is a universal principal G­bund le. Pro of. Consider a subgroup HG such that H is deformation retract of G, so k (H) k (G). Consider the commutative diagram EH - EG (2.26) (2.25)

BH - BG The corresponding homotopy sequences give the following diagram - k ) (H - k ( H ) E - k ( H ) B - k-(H) 1
k -1

- . - (2.27)

- k (G) - k (EG ) Since

- k (BG )

-

(G)

k (EH ) = k (EG ) = 0 for all k , and because of (2.25), k (BH ) k (BG ). On the other hand if there is universal principal H­bundle then we can define EG as the total space of the associated bundle over BH with a new fiber G. Then from (2.26) and (2.27) we have that k (EG ) = 0. This means that for the group G there is universal principal G­bundle. To finish the proof, it is sufficient to notice that any matrix group has a compact subgroup which is deformation retract of the group.


Homotopy Invariants

95

2.3.2

Milnor construction

There is another construction (due to J.Milnor) which can be used for any topological group. The idea consists of trying to kill homotopy groups of the total space starting from a simple G­bundle. Let us start from the bundle with one point as base G-pt. (2.28) The total space of the (2.28) consists of a single fiber G. In general, 0 (G) = 0. To kill 0 we attach a system of paths to G which connect pairs of points of the group in such a way that the left translation on the group can be extended to the paths. It is convenient to consider two copies of the group G and the family of intervals which connect each point of the first copy to each point of the second. This space is called the join of two copies of the group G and is denoted by G G. Each point of G G is defined by a family (0 , g0 , 1 , g1 ) where g0 , g1 G and 0 0 , 1 1, 0 + 1 = 1 are barycentric coordinates of a point on the unit interval, with the convention that (0, g0 , 1, g1 ) = (0, g0 , 1, g1 ), (1, g0 , 0, g1 ) = (1, g0 , 0, g1 ), The action of G on G G is defined by h (0 , g0 , 1 , g1 ) = (0 , hg0 , 1 , hg1 ), h G. Similarly, the join of (k + 1) copies of the group G is denoted by Gk = G G · · · G (k+1 times). A point of the join has the form x = (0 , g0 , 1 , g1 , . . . , k , gk ) where g0 , g1 , . . . gk G (2.29) g0 , g0 , g1 G g0 , g1 , g1 G


96

Chapter 2

are arbitrary elements and (0 , 1 , . . . k ) : i 0, 0 + 1 + · · · + k = 1 are barycentric coordinates of the point in a k ­dimensional simplex; this description (2.29) of the point x should be factorized by relation (0 , g0 , 1 , g1 , . . . , k , gk ) (0 , g0 , 1 , g1 , . . . , k , gk ) iff i = i for all i and gi = gi for each i when i = 0. The group G acts on the G
k

by formula

h(0 , g0 , 1 , g1 , . . . , k , gk ) = (0 , hg0 , 1 , hg1 , . . . , k , hgk ), h G. Prop osition 5 When k - 1, j (G Pro of. The join X Y is the union X Y = ((C X â Y )) ((X â C Y )) (2.30)
(k+1)

) = 0.

where C X, C Y are cones over X and Y . The union (2.30) can represented as the union X Y = ((C X â Y )) C Y ) ((X â C Y )) C X ) = P1 P2 where P1 P2 = ((C X â Y ) C Y ) , = ((X â C Y ) C X ) .

Both spaces P1 and P2 are contractible and intersection is P3 = P1 P2 = (X â Y ) C X C Y .


Homotopy Invariants

97

Hence both P1 and P2 are homotopy equivalent to the cone C P3 . So the union P1 P2 is homotopy equivalent to the suspension S P3 . By induction, the space G(k+1) is homotopy equivalent to the S k Z for some Z . It is easy to check that 0 (G
(k+1)

) = H0 (G

(k+1)

) = 0.

Hence, using the Hurewich theorem j (G Hence G has trivial homotopy groups.
(k+1)

) = Hj (G

(k+1)

) = 0 for j k .
k

= lim G

Examples
1. For any real vector bundle the structure group can be taken as O(n), for some n. The corresponding classifying space can be constructed as a limit of real Grassmann manifolds: BO(n) = lim GR N 2. Let EU(n)-BU(n) be the universal principal U(n)­bundle. Then BO(n) EU(n)/O(n), and hence there is the quotient mapping c : BO(n)-BU(n) which gives a locally trivial bundle with the fiber U(n)/O(n). The mapping c corresponds to the operation of complexification of real vector bundles which is also denoted by c . 3. The family of all real vector bundles over a cellular space X , for which the complexifications are trivial bundles, is given by the image of the homomorphism Im ([X, U(n)/O(n)]-[X, BO(n)]) . (2.31)
+n,n

=G

R ,n


98

Chapter 2

4. The family [X, U(n)/O(n)] is bigger than the image (2.31). It is easy to see that each continuous mapping f : X -U(n)/O(n) defines a real vector bundle for which the complexification is trivial and gives concrete trivialization of the bundle c . Prop osition 6 Each element of the family [X, U(n)/O(n)] defines a real vector bund le with a homotopy class of trivializations of the bund le c . Pro of. The mapping U(n)-U(n)/O(n) gives a principal O(n)­bundle. Hence each mapping f : X -U(n)/O(n) induces a principal O(n)­ bundle over X and an equivariant mapping of total spaces g O - Un) ( . f X - U(n)/O(n) Let U be the total space of the principal U(n)­bundle associated with the complexification of O . Then the space O lies in the space U and this inclusion O U is equivariant with respect to the actions of the group O(n) on O and the group U(n) on U . Therefore, the map g extends uniquely to an equivariant map g : U -U(n). ¯ The latter defines a trivialization of the bundle U . If f1 , f2 : X -U(n)/O(n) are homotopic then there is a bundle O over X â I with a trivialization of the complexification. Since the restrictions of the bundle O to the X â {t} are all isomorphic, the trivializations of U |X â{0} and U |X â{1} are homotopic. Conversely, let O be a principal O(n)­bundle and fix a trivialization of the


Homotopy Invariants

99

complexification U . This means that there is map g giving a commutative ¯ diagram g ¯ U - Un) EU(n) ( . X - {pt} BU(n) Hence there is a commutative diagram

O & b &

U

g ¯ E U(n)

g

E U(n) b & &

E EU(n)

E EU(n) b & &

c X & b c& X The mapping f

E b & c& U(n)/O(n) E

c pt

c E BU(n) b & c& BO(n) E

f : X -U(n)/O(n) is induced by the mapping g and generates the bundle O and the trivializa¯ tion g of the bundle cO . Hence the family of homotopy classes of mappings ¯ [X, U(n)/O(n)] can be interpreted as the family of the equivalence classes of pairs ( , ) where is real bundle over X and is an isomorphism of c with a trivial bundle. Two pairs (1 , 1 ) and (2 , 2 ) are considered equivalent if the bundles 1 and 2 are isomorphic and the isomorphisms 1 and 2 are homotopic in the class of isomorphisms.

2.4

CHARACTERISTIC CLASSES

In previous sections we showed that, generally speaking, any bundle can be obtained as an inverse image or pull back of a universal bundle by a continuous mapping of the base spaces. In particular, isomorphisms of vector bundles over X are characterized by homotopy classes of continuous mappings of the space X to the classifying space BO(n) (or BU(n) for complex bundles). But it


100

Chapter 2

is usually difficult to describe homotopy classes of maps from X into BO(n). Instead, it is usual to study certain invariants of vector bundles defined in terms of the homology or cohomology groups of the space X . Following this idea, we use the term characteristic class for a correspondence which associates to each n-dimensional vector bundle over X a cohomology class ( ) H (X ) with some fixed coefficient group for the cohomology groups. In addition, we require functoriality : if f : X -Y is a continuous mapping, an n­dimensional vector bundle over Y , and = f ( ) the pull-back vector bundle over X , then ( ) = f (( )), (2.32)

where in (2.32) f denotes the induced natural homomorphism of cohomology groups f : H (Y )-H (X ). If we know the cohomology groups of the space X and the values of all characteristic classes for given vector bundle , then might hope to identify the bundle , that is, to distinguish it from other vector bundles over X . In general, this hope is not justified. Nevertheless, the use of characteristic classes is a standard technique in topology and in many cases gives definitive results. Let us pass on to study properties of characteristic classes. Theorem 17 The family of al l characteristic classes of n­dimensional real (complex) vector bund les is in one­to­one correspondence with the cohomology ring H (BO(n)) (respectively, with H (BU(n))). Pro of. Let n be the universal bundle over the classifying space BO(n) and a characteristic class. Then (n ) H (BO(n)) is the associated cohomology class. Conversely, if x H (BO(n)) is arbitrary cohomology class then a characteristic class is defined by the following rule: if f : X -BO(n) is continuous map and = f (n ) put ( ) = f (x) H (X ). (2.33)


Homotopy Invariants

101

Let us check that the correspondence (2.33) gives a characteristic class. If g : X -Y is continuous map and h : Y -BO(n) is a map such that then ( ) = ((hg ) (n )) = (hg ) (x) = g (h (x)) = g ((h (n ))) = g (( )). If f : BO(n)-BO(n) is the identity mapping then (n ) = f (x) = x. Hence the class corresponds to the cohomology class x. We now understand how characteristic classes are defined on the family of vector bundles of a fixed dimension. The characteristic classes on the family of all vector bundles of any dimension should be as follows: a characteristic class is a sequence = {1 , 2 , . . . , n , . . .} (2.34) where each term n is a characteristic class defined on vector bundles of dimension n. Definition 5 A class of the form (2.34) is said to be stable if the fol lowing condition holds: n+1 ( 1) = n ( ), (2.35) for any n­dimensional vector bund le . In accordance with Theorem 17 one can think of n as a cohomology class n H (BO(n)) . Let : BO(n)-BO(n + 1) = g (n ), = g ( ),


102

Chapter 2

be the natural mapping for which (n
+1

) = n 1.

This mapping is induced by the natural inclusion of groups O(n) O(n + 1). Then the condition (2.35) is equivalent to: (n
+1

) = n .

(2.36)

Consider the sequence BO(1)-BO(2)- . . . -BO(n)-BO(n + 1)- . . . and the direct limit BO = lim BO(n).
-

Let

H (BO) = lim H (BO(n)) .
-

Condition 2.36 means that the family of stable characteristic classes is in oneto-one correspondence with the cohomology ring H (BO). Now we consider the case of cohomology with integer coefficients. Theorem 18 The ring H (BU(n); Z) of integer cohomology classes is isomorphic to the polynomial ring Z[c1 , c2 , . . . , cn ], where ck H 2k (BU(n); Z). The generators {c1 , c2 , . . . , cn } can be chosen such that 1. the natural mapping : BU(n)-BU(n + 1) satisfies (ck ) = (cn
+1

(2.37)

ck , k = 1, 2, . . . , n, (2.38)

) = 0;


Homotopy Invariants

103

2. for a direct sum of vector bund les we have the relations ck ( ) = ck ( ) + ck-1 ( )c1 ( )+ + ck-2 ( )c2 ( ) + . . . + c1 ( )ck-1 ( ) + ck ( ) = =
+ =k

c ( )c ( ),

(2.39)

where c0 ( ) = 1. The condition (2.38) means that the sequence {0, . . . , ck , ck , . . . , ck , . . .} is a stable characteristic class which will also be denoted by ck . This notation was used in (2.39). If dim < k then by (2.38) it follows that ck ( ) = 0. Formula (2.39) can be written in a simpler way. Put c = 1 + c1 + c2 + . . . + ck + . . . (2.40)

The formal series (2.40) has a well defined value on any vector bundle since in the infinite sum (2.40) only a finite number of the summands will be nonzero: c( ) = 1 + c1 ( ) + c2 ( ) + . . . + ck ( ), if dim = k . Hence from (2.39) we see that c( ) = c( )c( ). (2.41)

Conversely, the relations (2.39) may be obtained from (2.41) by considering the homogeneous components in (2.41). Pro of. Let us pass now to the proof of Theorem 18. The method we use for the calculation of cohomology groups of the space BU(n) involves spectral sequences for bundles. Firstly, using spectral sequences, we calculate the cohomology groups of unitary group U(n). Since H 0 (Sn ) H n (Sn ) H k (Sn ) = Z, = Z, = 0, when k = 0 and k = n,


104

Chapter 2

the cohomology ring H (Sn ) = = = k H k (Sn ) = H 0 (Sn ) H n (Sn )

is a free exterior algebra over the ring of integers Z with a generator an H n (Sn ). The choice of the generator an is not unique: one can change an for (-an ). We write H (Sn ) = (an ). Now consider the bundle U(2)-S3 with fibre S1 . The second term of spectral sequence for this bundle is
E2 ,

=
p,q

E

p,q 2

=

= H (S3 , H (S1 )) = H S3 ) H (S1 = = (a3 ) (a1 ) = (a1 , a3 ). The differential d2 vanishes except possibly on the generator
0 1 a1 E2 ,1 = H 0

S3 , H 1 (S1 ) . S3 , H 0 (S1 ) = 0.

But then Hence

2 d2 (1 a1 ) E2 ,0 = H

2

d2 (1 a1 ) = 0, d2 (a3 1) = 0, d2 (a3 a1 ) = d2 (a3 1)a1 - a3 d2 (1 a1 ) = 0. Hence d2 is trivial and therefore E Similarly d3 = 0 and Continuing, dn = 0 and
En, +1 , E p,q 3 p,q = E2 .

p,q p,q p,q E4 = E3 = E2 .

=E

, n

= ... = E

, 2

= (a1 , a3 ),

= (a1 , a3 ).


Homotopy Invariants

105

The cohomology ring H (U(2)) is associated to the ring (a1 , a3 ), that is, the ring H (U(2)) has a filtration for which the resulting factors are isomorphic to the homogeneous summands of the ring (a1 , a3 ). In each dimension, n = p + q , p,q the groups E vanish except for a single value of p, q . Hence H H H H Let
0 1 3 4

(U(2)) = (U(2)) = (U(2)) = (U(2)) =

Z, 1, E0 , 0, E3 , 1, E3 .

u1 H 1 (U(2)) , u3 H 3 (U(2))

be generators which correspond to a1 and a3 , respectively. As a1 a3 is a gener1, ator of the group E3 , the element u1 u3 is a generator of the group H 4 (U(2)). It is useful to illustrate our calculation as in figure 2.1, where the nonempty p,q cells show the positions of the generators the groups Es for each fixed s­level of the spectral sequence.

q6 6

a

1

1

a1 a @ @ @ @ a3 @ @ @ @ @ @ @ @
Figure 2.1

3

p

For brevity the tensor product sign is omitted. The arrow denotes the action of the differential ds for s = 2, 3. Empty cells denote trivial groups. Thus we have shown H (U(2)) = (u1 , u3 ).


106

Chapter 2

6 6

u2 q

n-3

u1 1

@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ u1 a3 @ @ @ @ @ a2n-1 @ @ @ @ @ @ @
Figure 2.2

p

Proceeding inductively, assume that H (U(n - 1)) = (u1 , u3 , . . . , u
2n-3

), u

2k - 1

H

2k-1

(U(n - 1)) , 1 k n - 1,

and consider the bundle

U(n)-S2

n-1

with fiber U(n - 1). The E2 member of the spectral sequence has the following form:
E2 , = H S2n-1 ; H (U(n - 1)) = = (a2n-1 ) (u1 , . . . , u2n-3 ) = (u1 , u3 , . . . , u

2n-3

,a

2n-1

).

(see fig.2.2).


Homotopy Invariants

107

The first possible nontrivial differential is d d2
n-1

2n-1

. But

(uk ) = 0, k = 1, 3, . . . , 2n - 3,

and each element x E 0,q decomposes into a product of the elements uk . Thus d2n-1 (x) = 0. Similarly, all subsequent differentials ds are trivial. Thus
p,q p,q p,q E = . . . = Es = . . . = E2 = = H p S2n-1 , H q (U(n - 1)) , = (u1 , u3 , . . . , u2n-3 , u2n-1 ).

, E

Let now show that the ring H (U(n)) is isomorphic to exterior algebra (u1 , u3 , . . . , u
2n-3

, u2

n-1

).

, Since the group E has no torsion, there are elements v1 , v3 , . . . , v2n-3 H (U(n)) which go to u1 , u3 , . . . , u2n-3 under the inclusion U(n - 1) U(n). 2 All the vk are odd dimensional. Hence elements of the form v11 v33 · · · v2nn-3 -3 where k = 0, 1 generate a subgroup in the group H (U(n)) mapping iso2n morphically onto the group H (U(n - 1)). The element a2n-1 E -1,0 has filtration 2n - 1. Hence the elements of the form v11 v33 · · · v 2n form a basis of the group E consisting of the elements v1 1 v 3 3 -1,0 2n-3 2n-3 a2n-1

. Thus the group H (U(n)) has a basis , k = 0 , 1.
2n-3

···v

2n-3 2n-1 2n-3 a2n-1

Thus

H (U(n)) = = (v1 , v3 , . . . , v

,v

2n-1

).

Now consider the bundle EU(1)-BU(1) with the fibre U(1) = S . From the exact homotopy sequence 1 (EU(1)) -1 (BU(1)) - it follows that 1 (BU(1)) = 0. At this stage we do not know the cohomology of the base, but we know the cohomology of the fibre H S1 = (u1 ),
0 1 1

S


108

Chapter 2

q6 6 u1 1 u1 c @ @ @ @ @ c u1 c2 @ @ @ @2 c u1 cn cn u1 cn+1 @ @ @ @ @ n+1 c

p

Figure 2.3

and cohomology of the total space H (EU(1)) = 0. This means that We know that E and hence
p,q p,q E = s Es = 0. p,q 2

= H p (BU(1)) , H q (S1 ),

p,q E2 = 0 when q 2.

In the figure 2.3 nontrivial groups can only occur in the two rows with q = 0 and q = 1. Moreover, p, p, p, E2 1 E2 0 u1 E2 0 . But and it follows that
p,q E2 = 0 for q 2, p,q Es = 0 for q 2.

Hence all differentials from d3 on are trivial and so E Also and thus the differential
p,q 3 p,q = . . . = E = 0.

p,q p,q E3 = H (E2 , d2 ) p, d2 : E2 1 -E p+2,0


Homotopy Invariants

109

is an isomorphism. Putting c = d2 (u1 ), we have d2 (u1 ck ) = d2 (u1 )ck = ck
+1

.

Hence the cohomology ring of the space BU(1) is isomorphic to the polynomial ring with a generator c of the dimension 2: H (BU(1)) = Z[c]. Now assume that H (BU(n - 1)) = Z[c1 , . . . , cn-1 ]

and consider the bundle BU(n - 1)-BU(n) with the fiber U(n)/U(n - 1) = S2n-1 . The exact homotopy sequence gives us that 1 (BU(n)) = 0. We know the cohomology of the fiber S2n-1 and the cohomology of the total space BU(n - 1). The cohomology of the latter is not trivial but equals the p, ring Z[c1 , . . . , cn-1 ]. Therefore, in the spectral sequence only the terms Es 0 p,2n-1 and Es may be nontrivial and
p, E2 2n-1 p, = E2 0 a2 n-1

= H p (BU(n)) = a2

n-1

.

(see figure 2.4). Hence the only possible nontrivial differential is d2n and therefore E
p,q 2 p,q = . . . = E2n ,

p,q p,q p,q H (E2n , d2n ) = E2n+1 = . . . = E p,q It is clear that if n = p + q is odd then E = 0. Hence the differential

d

2n

0, : E2n2

n-1

2n, -E2n

0

p,q p,q is a monomorphism. If p + q = k < 2n - 1 then E2 = E . Hence for odd k ,0 k 2n - 1, the groups E2n are trivial. Hence the differential k, d2n : E2n2 n-1 k+2 -E2n n, 0


110

Chapter 2

6 6

a2 q

n-1

1

@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @

cn

p

Figure 2.4

is a monomorphism for k 2n. This differential makes some changes in the k+2n, term E2n+1 0 only for even k 2n. Hence, for odd k 2n, we have E Thus the differential
k+2n,0 2n+1 n-1

= 0.
0

k, d2n : E2n2

k+2 -E2n n,

is a monomorphism for k 4n. By induction one can show that
k, E2n0 = 0

for arbitrary odd k , and the differential
k, d2n : E2n2 n-1 k+2 -E2n n, 0

is a monomorphism. Hence E
k,2n-1 2n+1 k+2n, E2n+1 0

= =

0
k2 E2n n,0 k, /E2n2 n-1 k+2 = E n,0 .

p,q The ring H (BU(n - 1)) has no torsion and in the term E only one row is k+2n,0 nontrivial (q = 0). Hence the groups E have no torsion. This means that image of the differential d2n is a direct summand.


Homotopy Invariants

111

Let cn = d2n (a2 and then d2n (xa It follows that the mapping H (BU(n)) -H (BU(n)) defined by the formula x-cn x is a monomorphism onto a direct summand and the quotient ring is isomorphic to the ring H (BU(n - 1)) = Z[c1 , . . . , cn-1 ]. Thus H (BU(n)) = Z[c1 , . . . , cn Consider now the subgroup Tn = U(1) â . . . â U(1) (n times ) U(n) (2.42)
-1 2n-1 n-1

).

) = xc2n .

, cn ].

of diagonal matrices. The natural inclusion Tn U(n) induces a mapping jn : BTn -BU(n). But and hence BTn = BU(1) â . . . â BU(1), H (BTn ) = Z[t1 , . . . , tn ]. (2.43)

Lemma 4 The homomorphism
jn : Z[c1 , . . . , cn ]-Z[t1 , . . . , tn ]

induced by the mapping (2.43) is a monomorphism onto the direct summand of al l symmetric polynomials in the variables (t1 , . . . , tn ). Pro of. Let : U(n)-U(n)


112

Chapter 2

be the inner automorphism of the group induced by permutation of the basis of the vector space on which the group U(n) acts. The automorphism acts on diagonal matrices by permutation of the diagonal elements. In other words, permutes the factors in the group (2.42). The same is true for the classifying spaces and the following diagram BTn BTn - BU(n) - BU(n)
j
n

j

n

is commutative. The inner automorphism is homotopic to the identity since the group U(n) is connected. Hence, on the level of cohomology, the following diagram jn H (BTn ) - H (BU(n)) jn n H (BT ) - H (BU(n)) or jn Z[c1 , . . . , cn ] - Z[t1 , . . . , tn ] . jn Z[c1 , . . . , cn ] - Z[t1 , . . . , tn ] is commutative. The left homomorphism is the identity, whereas the right permutes the variables (t1 , . . . , tn ). Hence, the image of jn consists of symmetric polynomials. Now let us prove that the image of j sufficient to show that the inclusion
n

is a direct summand. For this it is

U(k ) â U(1) U(k + 1) induces a monomorphism in cohomology onto a direct summand. Consider the corresponding bundle B (U(k ) â U(1)) -BU(k + 1) with fiber U(k + 1)/ (U(k ) â U(1)) = CPk .
E2 , = H

The E2 term of the spectral sequence is


BU(k + 1); H (CPk ) .

(2.44)

For us it is important here that the only terms of (2.44) which are nontrivial occur when p and q are even. Hence all differentials ds are trivial and E
p,q 2 p,q = E .


Homotopy Invariants

113

None of these groups have any torsion. Hence the group H p (BU(k + 1); Z) = E is a direct summand. It is very easy to check that the rank of the group H k (BU(n)) and the subgroup of symmetric polynomials of the degree k of variables (t1 , . . . , tn ) are the same.
p,0

H (BU(k ) â BU(1))

Using Lemma 4, choose generators c1 , . . . , cn H (BU(n)) as inverse images of the elementary symmetric polynomials in the variables t1 , . . . , tn H (BU(1) â . . . â BU(1)) . Then the condition (2.38) follows from the fact that the element cn+1 is mapped by jn+1 to the product t1 · . . . · tn+1 , which in turn is mapped by the inclusion (2.37) to zero. Condition 2.39 follows from the properties of the elementary symmetric polynomials: k (t1 , . . . , tn , t
n+1

,...,t

n+m

)=
+ =k

(t1 , . . . , tn ) (t

n+1

,...,t

n+m

).

The proof of Theorem 18 is finished. The generators c1 , . . . , cn are not unique but only defined up to a choice of signs for the generators t1 , . . . , tn . Usually the sign of the tk is chosen in such way that for the Hopf bundle over CP1 the value of c1 on the fundamental circle is equal to 1. The characteristic classes ck are called Chern classes . If X is complex analytic manifold then characteristic classes of the tangent bundle T X are simply called characteristic classes of manifold and one writes ck (X ) = ck (T X ).

Example
Consider the complex pro jective space CPn . It was shown in the section 6 of the chapter 1 that the tangent bundle T CPn satisfies the relation ¯ T CPn 1 = (n + 1)



114

Chapter 2

where is the Hopf bundle. Hence ¯ ck (T CPn ) = ck (T CPn 1) = ck ((n + 1) ) = n+1 k t. = k (-t1 , -t2 , . . . , -tn )|tk =t = (-1)k k From this we can see that the tangent bundle of CPn is nontrivial. We have studied in detail the algebraic structure of characteristic classes of complex bundles. The proof was based on simple geometric properties of the structure group U(n) and some algebraic properties of the cohomology theory. Similarly, we can describe the characteristic classes of real vector bundles using geometric properties of the structure group O(n). However, as we shall use real bundles very little in the following we shall omit some details and refer the reader to other books (Milnor and Stasheff, Husemoller) for further information. Unlike U(n), the nontrivial elements of finite order in O(n) all have order two. In studying H (BO(n)), it is convenient to describe two separate rings: H (BO(n); Q) and H (BO(n); Z2 ). Theorem 19 1. The ring H (BSO(2n); Q) is isomorphic to the ring of polynomials Q[p1 , p2 , . . . , pn-1 , ], where pk H
4k

(BSO(2n); Q) , H

2n

(BSO(2n); Q) .

2. The ring H (BS = O(2n + 1); Q) is isomorphic to the ring of polynomials Q[p1 , p2 , . . . , pn ], where pk H
4k

(BSO(2n + 1); Q) .

3. Generators {pk } can be chosen such that the natural mapping BSO(2n)-BSO(2n + 1) induces a homomorphism such that pk goes to pk if 1 k n - 1 and p goes to 2 . The natural mapping BSO(2n + 1)-BSO(2n + 2) induces a homomorphism such that pk goes to pk , 1 k n, and goes to zero.
n


Homotopy Invariants

115

4. For a direct sum of bund les: pk ( ) =
+ =k

p ( )p ( ),

where p0 = 1. Pro of. The group SO(2) is isomorphic to the circle S1 = U(1). Therefore, the cohomology ring of BSO(2) is known: H (BSO(2); Q) = Q[], H 2 (BSO(2); Q). Consider the bundle BSO(2k - 2)-BSO(2k ) with the fiber SO(2k )/SO(2k - 2) = V2k,2 . The Stieffel manifold V total space of a bundle V2k,2 -S2k-1
2k-2 2k , 2

(2.45) is the (2.46)

with fiber S . The manifold V2k,2 may be described as the family of pairs of orthogonal unit vectors (e1 , e2 ). We think of the first vector e1 as parametrising the sphere S2k-1 and the second vector e2 as tangent to the sphere S2k-1 at the point e1 . Since the sphere S2k-1 is odd dimensional, there is a nontrivial vector field, that is, a section of the bundle (2.46) inducing a monomorphism H (S2
k -1

)-H (V2

k ,2

).

In the spectral sequence for the bundle (2.46) the only one possible nontrivial differential d2k vanishes. Hence
E2 , = e ,

= H (V

2k , 2

) = (a

2k-1

,a

2k-2

).

Consider the spectral sequence for the bundle (2.45) (see 2.5). We take the inductive assumption that H (BSO(2k - 2); Q) = Q[p1 , . . . , p
k -2

, ],

(2.47)

where H 2k-2 (BSO(2k - 2); Q). In particular, this means that the odd dimensional cohomology of BSO(2k - 2) is trivial. If d
2k - 1

(a2

k -2

) = v = 0,


116

Chapter 2

q6 6 a2k-2 · ·a2k-1

a2 a2

k -1

k -2

1

@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @

w

p

Figure 2.5

then d2 Hence
k -1

(a

2k-2

v ) = v 2 = 0. = 0.

2k E

-1,2k-2

The latter contradicts the assumption that H Therefore, one has that d2 and hence d2
k -1 k -1 4k-3

(BSO(2k - 2); Q) = 0. (a2 ) = 0.

k-2

= 0.


Homotopy Invariants

117

The dimension of the element a

2k - 1

is odd and hence ) = w = 0.

d2k (a

2k - 1

Proceeding as in the proof of Theorem 18, one can check that the ring H (BSO(2k ); Q) has only even dimensional elements and the differential , , d2k maps the subgroup a2k-1 E2k monomorphically onto the subgroup wE2k . Therefore, , , , E = E0 a2k-2 E0 which is associated to the polynomial ring (2.47). This means that p1 , . . . , p [] = [2 ] , E0 = E2 ,0 =
k-2 , E0 , a2k-2 , , E0 , Q[p1 , . . . , pk Q[p1 , . . . , pk

-2 -2

, 2 ], , 2 , w ].

Put p

k-1

= 2 . Then H (BSO(2k ); Q) = Q[p1 , . . . , p
k -1

, w ].

Consider the bundle BSO(2k - 2)-BSO(2k - 1) (2.48)

with the fiber S2k-2 . In the spectral sequence for the bundle (2.48), the member E2 , has only two nontrivial rows: E2 ,0 and E2 ,2k-2 = a2k-2 E2 ,0 . If d then d2 Hence
k -1 2k - 1

(a2

k -2

) = v = 0,

(a

2k-2

v ) = v 2 = 0. = 0.

2k E

-1,2k-2

and the latter contradicts the previous calculation since the odd dimensional cohomology of the space BSO(2k - 2) is trivial. This means that d
2k - 1

(a2k-2 ) = 0, d2k-1 = 0,

, E2 , = E .


118

Chapter 2

Hence the odd dimensional cohomology of the space BSO(2k - 1) is trivial. Consider the bundle BSO(2k - 1)-BSO(2k )
with fiber S2k-1 . The member E2 (2.49) has the form ,

(2.49)

of the spectral sequence of the bundle ).

E

, 2

= H (BSO(2k )) (a

2k-1

The odd dimensional cohomology of the total space BSO(2k - 1) is trivial and so d2k (a2k-1 ) = v = 0. Using the calculation for the bundle (2.45), we see that v = d2k (a2k ) = w. Hence d2k (a Thus E
, 2k+1 2k-1

x) = wx. ] = H (BSO(2k - 1); Q) .

, = E = Q[q1 . . . . , p

k -1

To prove the last relation of Theorem 19 we study the bundle jn : BSO(2) â . . . â BSO(2)-BSO(2n).
We need to prove that the image of jn coincides with the ring of symmetric 2 22 polynomials of variables {t1 , p2 , . . . , pn } where

tk H (BSO(2) â . . . â BSO(2); Q) are generators. We leave this proof for reader. The characteristic classes pk H
4k

(BSO(n); Q)

constructed in Theorem 19 are called the rational Pontryagin classes . The class H 2n (BSO(2n); Q) is called the Euler class .


Homotopy Invariants

119

Theorem 20 The ring H (BO(n); Z2 ) is isomorphic to the polynomial ring Z[w1 , . . . , wn ], H (BO(n); Z2 ) = Z[w1 , . . . , wn ], wk H k (BO(n); Z2 ) . The generators wk can be chosen such that 1. wk ( ) = 0 for k > dim ; 2. wk ( 1) = wk ( ); 3. wk ( ) =
+ =k

w ( )w ( ).

The proof of this theorem is left to the reader. The class wk is called Stieffel­ Whitney characteristic class .

Exercise
Prove that the structure group O(n) of the vector bundle can bes reduced to the subgroup SO(n) if and only if w1 ( ) = 0. For this consider the bundle BSO(n)-BO(n), a two­sheeted covering. If w1 ( ) = 0 then the vector bundle is said to be an oriented bund le .

Example
Consider the space BO(1) = RP and the universal one­dimensional Hopf vector bundle over it. It is clear that w1 ( ) = w1 H 1 (RP ; Z2 ) is a generator and w1 ( ) = w1 ( ) + w1 ( ) = 2w1 = 0. Hence the bundle is orientable. The bundle is nontrivial since
2 w2 ( ) = w1 ( )w1 ( ) = w1 = 0.


120

Chapter 2

2.5 2.5.1

GEOMETRIC INTERPRETATION OF SOME CHARACTERISTIC CLASSES The Euler class

Let X be a smooth, closed, oriented manifold of dimension 2n. We put the question: Is there a nonvanishing vector field on the manifold X ? Let T X be the tangent bundle and S X the bundle consisting of all unit tangent vectors. Then the existence of a nonvanishing vector field is equivalent to the existence of a section of the bundle S X . The bundle S X -X with the fiber S2
n-1

(2.50)

is classified by a mapping X -BSO(2n) (2.51)

and the universal bundle SESO(2n)-BSO(2n) with the fiber S2
n-1

.

The classifying space BSO(2n) can be taken as the space of 2n­dimensional oriented linear subspaces of an infinite dimensional vector space. The points of the total space SESO(2n) can be represented by pairs (L, l) where L is 2n­dimensional subspace and l L is a unit vector. Let l L be the orthonormal supplement to the vector l. Then the correspondence (L, l)-l gives a locally trivial bundle SESO(2n)-BSO(2n - 1). (2.52)

The fiber of this bundle is an infinite dimensional sphere. Hence the mapping (2.52) is a homotopy equivalence and the bundle (2.50) is classified by the bundle BSO(2n - 1)-BSO(2n), (2.53) with the fiber S2
n-1

.


Homotopy Invariants

121

p,q p,q Let Es be the spectral sequence for the bundle (2.53) and Es (X ) be the spectral sequence for the bundle (2.50). Let p,q p,q f : Es -Es (X )

be the homomorphism induced by the mapping (2.51). Then the element f () H
2n 2 (X ) = E2 n,0

(X )

is the Euler class of the manifold X . In the section 2.4 (Theorem 19), it was shown that pro jection p : BO(2n - 1)-BO(2n) sends the Euler class to zero. Hence the same holds for the pro jection p : S X -X where: If : X -S X is a section, then 0 = p (f ( )) = f (). (2.54) p (f ()) = 0.

Thus the existence of nonvanishing vector field on the manifold X implies that the Euler class of the manifold X is trivial.

2.5.2

Obstruction to constructing a section

In section 2.4 the Euler class was defined as a cohomology class with rational coefficients. But it is possible to define it for integer cohomology. As a matter of fact, the triviality of the integer Euler class is sufficient for the existence of nonvanishing vector field on an oriented 2n­dimensional manifold. On the other hand, the triviality of the rational Euler class implies the triviality of the integer Euler class since 2n­dimensional cohomology group has no torsion. Generally speaking, there are homology type obstructions to the existence of a section of a locally trivial bundle. Some of them may be described in terms of characteristic classes.


122

Chapter 2

Let p : E -X be a locally trivial bundle with fiber F and let X be a cellular space. Assume that there exists a section s : [X ]n-1 -E over (n - 1)­dimensional skeleton of X . We can ask whether it is possible to extend the section s over the n­dimensional skeleton [X ]n . It is sufficient to extend the section s over each n­dimensional cell i X separately. Over a cell i the bundle is trivial, that is, p-1 (i ) i â F, the restriction of the section s to the boundary i defines a mapping to the fiber F : s : i : -F, and which defines an element of the homotopy group of the fiber: [s]i
n-1

(F ).

This element will be well defined, that is, it does not depend on the choice of the fiber and on the choice of fixed point of F if both the base and fiber are simply connected. Thus necessary and sufficient conditions for the existence of an extension of the section s to the skeleton [X ]n are given by the following relations: [s]i = 0 In other words, the cochain [s] C n (X,
n-1 n-1

(F ).

(F ))

defined by the section s may be considered as a obstruction to the extension of the section s to the n­dimensional skeleton [X ]n . The cochain [s] is not arbitrary. In fact, it is cocycle: [s] = 0. To prove this, consider an (n + 1)­dimensional cell . Let f : -[X ]
n


Homotopy Invariants

123

be the mapping which defines the gluing of the cell to the skeleton [X ]n . Consider the composition ~ f : -[X ]n -[X ]n /[X ]n
-1

= i / i .

~ Using a homotopy of the mapping f we can assume that f is smooth everywhere except at the fixed point x0 of i / i and the points xi i / i are nonsingular. This means that ~ f -1 (x0 ) = i,j Dij where the Di,j are n­dimensional disks which are pairwise disjoint and for which each ~ f|Di,j : Di,j -i / i (2.55) is a diffeomorphism on the interior of Di,j . Let ti,j = ±1 be the degree of the mapping (2.55). Then for the cell chain generated by single cell we have the boundary homomorphism =
i


j

ti,j i .

with corresponding adjoint boundary homomorphism on cochains ( x)() =
i,j

ti,j x(i ).

(2.56)

Now the section s is defined on the \ i,j Di,j . The restriction of the section s to Di,j gives an element [s]i,j of n-1 (F ) for which [s]i,j = ti,j [s]i Thus from (2.56), ( [s])() =
i,j n-1

(F ).

ti,j [s](i ) =
i,j

ti,j [s]i =
i,j

[s]i,j .

Lemma 5 Let Dk be a family of disjoint disks and let g : \ k Dk -F
n-1

be a continuous mapping which on each Dk gives an element [g ]k Then [g ]k = 0 k ( F ) .
k

(F ).


124

Chapter 2

Lemma 5 finishes proof that [s] is cocycle. The obstruction theory says that the cocycle [s] C n (X, n-1 (F )) gives a cohomology class [s] H n (X ; n-1 (F )) called the obstruction to extending the section [s]. Further, if this obstruction is trivial then there is another section [s ] on the skeleton [X ]n which coincides with [s] on the skeleton [X ]n-2 . If n is smallest number for which the group H n (X ; n-1 (F )) is nontrivial then the corresponding obstruction [s] is called the first obstruction to the construction a section. Theorem 21 Let p : E -X be local ly trivial bund le with the fiber F and let X be a simply connected cel lular space. Let 0 (F ) = . . .
n-2 n-1

(F ) = 0 (F ) = Z.

p,q Let a H n-1 (F ; Z) be a generator and let Es be the spectral sequence of the bund le p. Then the first nontrivial differential acting on the element a is dn , n, n, En 0 = E2 0 = H n (X ; Z), and the element w = dn (a) H n (X ; Z) is the first obstruction to constructing a section of the bund le p.

Theorem 22 Let be a complex n­dimensional vector bund le over a base X . Then the Chern class ck ( ) H 2k (X ; Z) coincides with the first obstruction to constructing of n - k + 1 linear independent sections of the bund le . Pro of. Constructing n - k + 1 linearly independent sections of is the same as representing the bundle as a direct sum = n - k + 1. Consider the bundle p : BU(k - 1)-BU(n) with the fiber U(n)/U(k - 1). Let f : X -BU(n) be a continuous mapping inducing the bundle , that is, = f (n ). (2.58) (2.57)


Homotopy Invariants

125

6 6

a q

1

@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @

w

p

Figure 2.6

Representing the bundle as in (2.57) means constructing a mapping g : X -BU(k - 1) such that pg f . In other words, the first obstruction to constructing n - k + 1 linearly independent sections of the bundle coincides with the inverse image of the first obstruction to constructing a section of the bundle (2.58). Let us use Theorem 21. Consider the spectral sequence for the bundle (2.58) or the figure (2.7). Since 2 p (ck ) = 0, the element ck E2 k,0 is the image of the differential d2k , d2k (u2 By Theorem 21, we have
k -1

) = ck .

f (ck ) = ck ( )

giving the first obstruction to constructing n-k +1 linearly independent sections of the bundle x. There are similar interpretations for the Stieffel­Whitney classes and the Pontryagin classes.


126

Chapter 2

6 6

u2 q

k -1

1

@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @

c

k

p

Figure 2.7

2.6 2.6.1

K ­THEORY AND THE CHERN CHARACTER Definitions and simple prop erties

The family of (isomorphism classes of ) vector bundles over a fixed base is not a group with respect to the operation of direct sum since subtraction does not in general exist for any two bundles. However this family may be extended to form a group by introducing a formal subtraction. If the base is not connected we will consider a vector bundle as a union vector bundles of arbitrary dimensions, one on each connected component. Definition 6 Let K (X ) denotes the abelian group where the generators are (isomorphism classes of ) vector bund les over the base X subject to the fol lowing relations: [ ] + [ ] - [ ] = 0 (2.59) for vector bund les and , and where [ ] denotes the element of the group K (X ) defined by the vector bund le .


Homotopy Invariants

127

The group defined in Definition 6 is called the category of all vector bundles over the base X .

Grothendieck group of the

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 ). The relations (2.59) show that in the group K (X ) the direct sum corresponds to the group operation. Prop osition 7 Each element x K (X ) can be represented as a difference of two vector bund les: x = [1 ] - [2 ]. Pro of. In the general case, the element x is a linear combination
p

x=
i=1

ni [i ]

with the integer coefficients ni . Let us split this sum into two parts
p
1

p

2

x=
i=1

ni [i ] -
j =1

mj [j ]

with ni > 0, mj > 0. Using the relations (2.59) we have x= = [
p1 i=1
2 (i . . . i )] - [p=1 (j . . . j )] = j

n

i

times

m

j

times

[1 ] - [2 ].

Note: In Proposition 7 the bundle 2 may be taken to be trivial. In fact, using Theorem 8 from the chapter 1 there is a vector bundle 3 such that ¯ 2 3 = N


128

Chapter 2

is a trivial bundle. Then x = [1 ] - [2 ] = ([1 ] - [2 ]) + ([3 ] - [= 3]) = ¯ = ([1 ] + [3 ]) - ([2 ] + [3 ]) = [2 3 ) - N .

Prop osition 8 Two vector bund les 1 and 2 define the same element in the ¯ group K (X ) if and only if there is a trivial vector bund le N such that ¯ ¯ 1 N = 2 N . (2.60)

Condition 2.60 is called stable isomorphism of vector bund les . Pro of. If the condition (2.60) holds then by (2.59) ¯ ¯ [1 ] + [N ] = [2 ] + [N ], that is, [1 ] = [2 ]. Conversely, assume that [1 ] = [2 ] or [1 ] - [2 ] = 0 . Then by definition, this relation can be represented as a linear combination of determining relations: [1 ] - [2 ] =
j

j (j + j - j j ).

(2.61)

Without loss of generality we can consider that j ± 1. We transform the identity (2.61) in the following way: all summands with negative coefficients carry over to the other side of the equality. Then we have 1 +
j

(j j ) +
k

(k + k ) = (k k )
k

= 2 +
j

(j + j ) +

(2.62)

where j runs through all indices for which j = 1 and k runs through all indices for which k = -1. Both parts of the equality (2.62) are formal sums of vector bundles. Hence each summand on the left hand side is isomorphic to a summand on the right hand side and vice versa. The direct sum of all


Homotopy Invariants

129

summands in the left hand side is equal to the direct sum of all summand in the right hand side. Thus 1
j

(j j )
k

(k k ) = (k k ).
k

= Put

2
j

(j j )

=
j

(j j )
k

(k k ).

Then we have 1 = 2 . ¯ Finally, let be a vector bundle such that N = then ¯ ¯ 1 N = 2 N .

Prop osition 9 The operation of tensor product of vector bund les induces a ring structure in the additive group K (X ). We shall leave the proof to the reader.

Examples
1. We describe the ring K (x0 ) where x0 is a one point space. Each vector bundle over a point is trivial and characterized by the dimension of the fiber. Therefore, the family of all vector bundles over a point, with two operations -- direct sum and tensor product, is isomorphic to the semiring of positive integers. Hence the ring K (x0 ) is isomorphic to the ring Z of integers. In addition, the difference [ ] - [ ] corresponds the number dim - dim Z. 2. If X = {x1 , x2 } is a two point space then any vector bundle is defined by two integers -- the dimension of the fiber over the point x1 and the dimension of the fiber over the point x2 . Hence K (x0 ) = Z Z.


130

Chapter 2

2.6.2

K ­theory as a generalized cohomology theory

The group K (X ) is a ring. It is clear that a continuous mapping f : X -Y induces a ring homomorphism f : K (Y )-K (X ) which to each vector bundle associates the inverse image f ( ). If f , g : X -Y are two homotopic mappings then f = g : K (Y )-K (X ). In fact, there is a continuous map F : X â I -Y such that F (x, 0) = f (x); F (x, 1) = g (x). Then by Theorem 3 from the chapter 1, restrictions of the bundle F ( ) to the subspaces X â {0} and X â {1} are isomorphic. Hence f ( ) = g ( ). Therefore, if two spaces X and Y are homotopy equivalent then the rings K (X ) and K (Y ) are isomorphic. This means that the correspondence X -K (X ) is a homotopy functor from the category of homotopy types to the category of rings. One can extend this functor to an analogue of a generalized cohomology theory. Let (X, x0 ) be a cellular space with a base point. The natural inclusion x0 -X induces a homomorphism of rings: K (X )-K (x0 ) = Z. This homomorphism is defined by a formula for a difference [ ] - [ ]: [ ] - [ ]- dim - dim Z (2.63)


Homotopy Invariants

131

Note that if X has many connected components then we take the dimension of the bundle x over the component containing the base point x0 . Therefore, the homomorphism (2.63) will also be denoted by dim : K (X )-K (x0 ). Let K 0 (X, x0 ) denote the kernel of the homomorphism (2.63): 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 h. 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 ]). 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 23 The pair (X, Y ) induces an exact sequence K 0 (Y , x0 )-K 0 (X, x0 )-K 0 (X, Y )- -K -1 (Y , x0 )-K -1 (X, x0 )-K -1 (X, Y )- -K -2 (Y , x0 )-K -2 (X, x0 )-K -2 (X, Y )- . ................................................. . . . -K -n (Y , x0 )-K -n (X, x0 )-K -n (X, Y )- ................................................. Pro of. The sequence (2.64) is based on the so called Puppe sequence of spaces and may be is constructed for any pair (X, Y ): Y -X -X/ Y - -S 1 Y -S 1 X -S 1 (X/ Y )-

. . . .

. . .. .

(2.64)


132

Chapter 2

...

-S 2 Y -S 2 X -S 2 (X/ Y )- . . . ................................. -S n Y -S n X -S n (X/ Y )- . . . .................................

In the Puppe sequence, all continuous mapping may be taken as inclusions by substituting the gluing of a cone for each quotient space or factorization. Then the next space in the sequence can be considered as a quotient space. Hence to prove exactness it is sufficient to check exactness of the sequence K 0 (Y , x0 )-K 0 (X, x0 )-K 0 (X, Y ) at the middle term. Let be a vector bundle over X for which the restriction to the subspace Y is trivial. This means that the bundle is trivial in some neighborhood U of the Y . Hence the neighborhood U can serve as a chart of an atlas for . In addition, other charts can be chosen so that they do not intersect Y . Hence all the transition functions are defined away from the subspace Y . This means that the same transition functions define transition functions on the quotient space X/ Y , that is, they define a vector bundle over X/ Y such that j ([ ]) = [ ].
i


j



Remark
We have not quite obtained a generalized cohomology theory but only the part graded by nonnegative integers.

2.6.3

The Chern character

In the section 2.4 we defined Chern classes for any complex vector bundle. By Theorem 18, an arbitrary characteristic class is a polynomial of the Chern classes c1 ( ), . . . , cn ( ) with integer coefficients. If we consider characteristic classes in cohomology with rational coefficients then they may all be expressed as polynomials in Chern classes with rational coefficients. Consider now the function n (t1 , . . . , tn ) = e
e
1

+ et2 + . . . + e

t

n


Homotopy Invariants

133

and its the Taylor series


(t1 , . . . , tn ) =
k=0

n

tk + . . . + tk n 1 . k!

(2.65)

Each term of the series (2.65) is a symmetric polynomial in the variables t1 , . . . , tn and hence may be expressed as a polynomial in the elementary symmetric polynomials tk + . . . 1 k! 1 = 1 (t1 , . . ............. n = n (t1 , . . In other words, n (t1 , . . . , tn ) =
k=0

+ tk n

=

n (1 , . . . , n ), k

. , tn ) = t1 + t2 + . . . + tn , ..... ... ............... . , t n ) = t1 t2 · · · tn ,


n (1 , . . . , n ). k

Definition 7 Let be a n­dimensional complex vector bund le over a space X . The characteristic class


ch =
k=0 k=0

n (c1 ( ), . . . , cn ( )) k of the bund le .

(2.66)

is cal led the Chern character

In the formula (2.66), each summand is an element of the cohomology group H 2k (X, Q). If the space X is a finite cellular space then the Chern character ch is a nonhomogeneous element of the group H 2 ( X ; Q ) = H 2k ( X ; Q ) since from some number k0 on all the cohomology groups of the space X are trivial. In general, we consider that the formula (2.66) defines an element of the group


H (X ; Q) =
k=0



H k (X ; Q).

Theorem 24 The Chern character satisfies the fol lowing conditions:


134

Chapter 2

1. ch ( ) = ch ( ) + ch ( ); 2. ch 0 ( ) = dim ; 3. ch ( ) = ch ( )ch ( ). Pro of. As in the section 2.4, we can represent the Chern classes by elementary symmetric polynomial in certain variables, that is, ck ( ) = ck ( ) = ck ( =) = Using (2.66), ch k ( ) = ch k ( ) = ch k ( ) = = = 1k t + · · · + tk , n k! 1 1k m (c1 ( ), . . . , cn ( )) = + · · · + tk + t k n k ! n+1 n+m (c1 ( =), . . . , cn+m ( )) = k 1k t + · · · + tk + tk +1 + · · · + tk +m = n n n k! 1 ch k ( ) + ch k ( ). n (c1 ( ), . . . , cn ( )) = k k (t1 , . . . , tn ), k (tn+1 , . . . , tn+m ), k (t1 , . . . , tn , tn+1 , . . . , t

n+m

).

(2.67)

m

,

(2.68)

and from (2.68) property 1 follows. Property 2 follows from ch 0 ( ) = n (c1 ( ), . . . , cn ( )) = n = dim . 0 Strictly speaking this proof is not quite correct because we did not make sense of the formulas (2.67). We can make sense of them in the following way. For all three identities in Theorem 24 it is sufficient to check them over the special spaces X = BU(n) â BU(m), where = n is the universal vector bundle over the factor BU(n) and = is the universal vector bundle over the factor BU(m). Let f : BU(n) â BU(m)-BU(n + m) be a mapping such that f (n ) = n m .
m

+m


Homotopy Invariants

135

Then we must prove that f (ch
n+m

) = ch n + ch m .

(2.69)

Consider a commutative diagram f BU(n) â BU(m) u e e e E BU(n + m) ¡ ! ¡

ep ¡q e ¡ e ¡ e ¡ e ¡ BU(1) â . . . â BU(1) â BU(1) â . . . â BU(1)
n

e

¡

¡

times

m

times

where the mappings p, q correspond to the inclusions of the group of diagonal matrices into the unitary groups. In the notation of section 2.4, p = jn â jm , q = jn
+m

.

By the lemma 4 both homomorphisms p and q are monomorphisms on cohomology. Hence the identity (2.69) follows from the following identity p f (ch n that is, from or from q (ch ch q (
+m

) = p ch n + p ch m ,

n+m

) = p ch n + p ch m

n+m

) = ch p n + ch p m .

Let the i be one dimensional vector bundles, 1 i n + m, which are universal vector bundles over the various factors with corresponding number i. Then q
n+m p n p m

= 1 . . .

n+m

, .

= 1 . . . n , = n+1 . . . n

+m


136

Chapter 2

Therefore, we can put ti = c1 (i ) and then the formulas (2.67) make sense. We now pass to the proof of the property 3. As in the previous case it is sufficient to provide the proof for bundles Then =
i,j

= =

1 . . . n n+1 . . . n (i n

+m

.

+j

).

So we need to calculate the first Chern class c1 ( ) for the product of two one dimensional vector bundles which might be considered universal bundles over factors the BU(1). Consider a mapping f : BU(1) â BU(1)-BU(1) such that f ( ) = . Then for some integer and c1 ( ) = c1 ( ) + c1 ( ). When = 1 we have = and hence c1 ( ) = c1 (1) + c1 ( ) = c1 ( ), that is, = 1. Similarly, = 1. Finally, ch ( ) =
i,j

e e
i,j

c1 (i

n+j

)

= eti et
i,j
n +j

=

ti +t

n+j

=

=

e
i ti


n +j

=


j

e

t

= ch ch .

Theorem 24 shows that the Chern character can be extended to a homomorphism of rings ch : K (X )-H 2 (X ; Q).


3
GEOMETRIC CONSTRUCTIONS OF BUNDLES

To extend the study of the properties of the vector bundles we need further geometric ideas and constructions. This chapter is devoted to the most frequently used constructions which lead to deeper properties of vector bundles. They are Bott periodicity ­ the main instrument of the calculation of K ­theory, linear representations and cohomology operations in K ­theory and the Atiyah­Singer formula for calculating of the indices of elliptic operators on compact manifolds. The last of these will be considered in the next chapter.

3.1

THE DIFFERENCE CONSTRUCTION

Consider a vector bundle over the base X . Assume that the bundle is trivial over a closed subspace Y X . This means that there is a neighborhood U Y such that the restriction of the bundle to U is isomorphic to a Cartesian product: |U U â Rn . It is clear that the set U can serve as chart and, without loss of generality, we can consider that the other charts do not intersect with Y . Hence all the transition functions are defined at points of the complement X \Y . Therefore, the same transition functions can be used for the definition of a bundle over the new base -- the quotient space X/ Y . The bundle can be considered as a bundle obtained from by identification of fibers over Y with respect to the isomorphism |Y Y â Rn . This important construction can be generalized in the following way. Consider a pair (X, Y ) of cellular spaces and a triple (1 , d, 2 ) where 1 and 2 are vector 137


138

Chapter 3

bundles over X and d is an isomorphism d : 1 |Y -2 |Y . In the case when 2 is a trivial bundle we have the situation described above. Consider the family of all such triples. The triple (1 , d, 2 ) is said to be trivial if d can be extended to an isomorphism over the whole base X . If two isomorphisms d1 , d2 : 1 |Y -2 |Y are homotopic in the class of isomorphisms then the two triples (1 , d1 , 2 ) and (1 , d2 , 2 ) are said to be equivalent. If the triple (1 , h, 2 ) is trivial then the triples (1 , d, 2 ) and (1 1 , d h, 2 2 ) are also said to be equivalent. We denote by K2 (X, Y ) the family of classes of equivalent triples. Theorem 25 The family K2 (X, Y ) is an Abelian group with respect to the direct sum of triples. The group K2 (X, Y ) is isomorphic to K 0 (X, Y ) by the correspondence ¯ ¯ K 0 (X, Y ) ([ ] - [N ])-(p , d, N ) ¯ where p : X -X/ Y is natural projection and d : p |Y -N is natural isomorphism generated by trivialization on a chart which contains the point [Y ] X/ Y , (dim = N ). Pro of. The operation of summation is defined by (1 , d1 , 2 ) + (1 , d2 , 2 ) = (1 1 , d1 d2 , 2 2 ). Hence associativity is clear. It is clear that a trivial triple gives a neutral element. If (1 , d1 , 2 ) is trivial triple and d is homotopic to d in the class of isomorphisms then the triple (1 , d1 , 2 ) also is trivial. Indeed, assume firstly that both vector bundles 1 and 2 are trivial. Then triviality means that the ¯ mapping d : Y -GL(n) extends to the mapping d : X -GL(n). Hence there is a continuous mapping D : (Y â I ) (X â {0})-GL(n). The space (Y â I ) (X â {0}) is a retract in the space X â I . Hence D extends to a continuous mapping ¯ D : X â I -GL(n).


Geometric constructions of bund les

139

For nontrivial vector bundles the theorem can be proved by induction with respect to the number of charts. Hence the triple (1 , d1 , 2 ) is equivalent to a trivial triple if and only there is a trivial triple (1 , h, 2 ) such that the direct sum (1 1 , d h, 2 2 ) is trivial. What are the inverse elements? We show that the triple (1 2 , d d is equivalent to trivial triple. For this consider the homotopy Dt = When t = 0, D0 = and when t = 1, D1 = 01 . -1 0 d 0 0 d-1 d sin t cos : (1 2 )|Y -(1 2 )|Y - cos t d-1 sin t
-1

, 2 1 )

The latter matrix defines an isomorphism which can be extended to an isomorphism over the whole of X . Hence the family K2 (X, Y ) is a group. Consider a mapping : K 0 (X, Y )-K0 (X, Y )

¯ defined as follows. Let x K 0 (X, Y ) be represented as a difference x = [ ]-[N ], dim = N , let p : X -X/ Y be the pro jection, = p ( ), and let f : - be the canonical mapping. Then let d : |Y d(h) ¯ - N |Y = Y â RN = (y , f (h)), h y , y Y .

¯ The triple ( , d, N ), which we denote by (x), defines an element of the group 0 K (X, Y ). This element is well defined, that is, it does not depend on the representation of the element x. ¯ We now prove that is a monomorphism. Assume that the triple ( , d, N ) is equivalent to a trivial triple. This means that there is a trivial triple (1 , h, 2 ) ¯ such that direct sum ( 1 , d h, N 2 ) also is trivial. In particular, the bundles 1 and 2 are isomorphic over X . Without loss of generality we can ¯ assume that both 1 and 2 are trivial and h 1. Hence the triple ( N , d ¯ ¯ ¯ 1, N N ) is trivial and so the bundle N is isomorphic to trivial bundle ¯ ¯ N N . Hence x = 0.


140

Chapter 3

Next we a bundle (1 , d is trivial Hence

prove that is an epimorphism. Consider a triple (1 , d, 2 ). ¯ such that 2 = N . The triple ( , 1, ) is trivial and the ¯ 1, N ) is equivalent to the triple (1 , d, 2 ). Hence the bundle over Y . This means that 1 = p ( ) for a bundle over ¯ ([ ] - [N ]) = (1 , d, 2 ).

If is triple 1 X/ Y .

The definition of the groups K0 (X, Y ) has the following generalization. If (1 , d, 2 ) is a triple, it can be represented as a complex of vector bundles ¯-1 -d 2 -¯ 0 0
¯

(3.1)

which is exact over Y . For this we need to extend the isomorphism d (over Y ) ¯ to a morphism d of vector bundles over the whole of X . It is clear that such extension exists. Hence a natural generalization of short bundle complexes of type (3.1) is a bundle complex of arbitrary length
d1 d2 -1 ¯-1 -2 -3 - · · · dn n -¯, 0 - 0

(3.2)

which is exact over each point x Y . There is a natural operation of direct sum of bundle complexes of the type (3.2). The bundle complex (3.2) is said to be trivial if it is exact over each point of X . Two bundle complexes of type (3.2) are said to be equivalent if after addition of trivial complexes the corresponding vector bundles become isomorphic and corresponding morphisms become homotopic in the class of morphisms which preserve exactness over Y . The family of classes of equivalent bundle complexes of type (3.2) will be denoted by Kn (X, Y ). There is a natural mapping : K2 (X, Y )-Kn (X, Y ). A bundle complex of the form
dk ¯-¯- · · · -k -k 0 0 +1

- · · · -¯ 0

is said to be an elementary complex. Lemma 6 Every trivial bund le complex has a splitting into a sum of elementary trivial complexes. Pro of. follows from Theorem 7 of the chapter 1.


Geometric constructions of bund les

141

Lemma 7 Every bund le complex of type (3.2) is equivalent to an elementary complex, that is, the mapping is epimorphism. Pro of. Consider a bundle complex (3.2) The first step is to change the given complex (3.2) by adding a trivial elementary complex of a sufficiently high dimension in terms n - 2 and n - 1: ¯-1 - · · · -n 0
Dn Dn -3
-1

-

-3

-

-2



n-1

- n -¯, 0

Dn

n-2

-

Dn

-2

(3.3)

where Dk = dk , 1 k n - 4, d d Dn-3 = n-3 , Dn-2 = n- 0 0 0 ,D 1

2

n-1

= dn-

1

0.

The morphism dn-1 is an epimorphism over Y . Hence there is a neighborhood U Y such that dn-1 is epimorphism over U . Let be a function which equals 1 on X \U and 0 on Y , and be a function which equals 1 on Y and is such that 0. Consider the new bundle complex
n- n- ¯-1 - · · · -n-3 -3 n-2 -2 0 Dn-2 Dn-1 - n-1 - n -¯, 0

D

D

(3.4)

where Dk = dk , 1 k n - 4, d d Dn-3 = n-3 , Dn-2 = n- 0 0 0 ,D

2

n-1

= dn

-1

f .

and f : -n is an epimorphism over X . It is clear that the two complexes (3.3) and (3.4) are homotopic. So we have succeeded in constructing a new equivalent bundle complex (3.4) where the last morphism Dn-1 is an epimorphism over X . Hence we can split the bundle complex (3.4) into two direct summands: ¯-1 - · · · -n 0
Dn -3

-

-3

n-2

- Ker (Dn

Dn

-2

-1

)-¯, 0


142

Chapter 3

and

¯- Ker (D 0

n-2

)

Dn

- n -¯, 0
-1

the latter being a trivial elementary complex. Further, we can apply induction on the length of the bundle complex. Lemma 8 The mapping is a monomorphism. The proof is similar to the proof of Lemma 7. Thus we have shown Theorem 26 The natural mapping : K2 (X, Y )-Kn (X, Y ). is isomorphism. Using theorems 25 and 26 we can give different interpretations to important examples of K ­groups. The representation of elements of the group K 0 (X, Y ) as triples or bundle complexes is called the difference construction. It is justified by the fact that the restriction of the triple (1 , d, 2 ) to X equals [1 ] - [2 ] and the restriction of the bundle complex (3.2) to the space X equals the alternating sum [1 ] - [2 ] + . . . + (-1)n+1 [n ].

Examples
1. Let us describe the group K 0 (Sn ). We represent the sphere Sn as a quotient space of the disc Dn by its boundary Dn = Sn-1 , Sn = Dn /Sn-1 . Therefore, K 0 (Sn ) = K2 (Dn , Dn ). Since every bundle over Dn is trivial, any element of the group K2 (Dn , Dn ) may be defined as the homotopy class of a continuous mapping d : Dn = S Hence
n-1

-GL(N , C).

K 0 (S2 ) = 1 (U(N )) = 1 (U(1)) = 1 (S1 ) = Z.


Geometric constructions of bund les

143

Then

K (S2 ) = Z Z.

Similarly, Hence

K 0 (S3 ) = 2 (U(N )) = 2 (U(1)) = 0. K (S3 ) = Z.

2. Let S X denote the suspension of the space X , that is, the quotient space of the cone CX by its base, S X = CX/X . Then K 0 (S X ) = K2 (CX, X ) = [X, U(N )] where [X, U(N )] is the group of homotopy classes of mappings of X to unitary group U(N ), for sufficiently large N . 3. Consider a generalization of the difference construction. Let us substitute a triple (1 , d, 2 ) for two complexes of type (3.2) and a homomorphism giving a commutative diagram
11 12 1 - 2 - . . . d1 d2 d21 d22 0 - 1 - 2 - . . .

0 -

d

d

d1

-

,n-1

d2

-

,n-1

n - 0 dn n - 0

(3.5)

Assume that horizontal homomorphisms dij are exact over Y and vertical homomorphisms dk are exact over Y . It is natural to expect that the diagram (3.5) defines an element of the group K 0 (X/(Y Y )). Indeed, there is a new complex
1 2 0-1 -(2 1 )-(3 2 )- . . . -(n n

D

D

-1

)-n -0,

(3.6)

in which the horizontal homomorphisms are defined by the matrices Dk = d1k dk 0 d
2,k-1

.

This complex (3.6) is exact over the subspace (Y Y ).


144

Chapter 3

More generally, consider a diagram 0 - 0 -
11 - 11 d11 d21 - 21

d

12 - . . . 12 d12 d22 - . . . 22

d

d1

- -

,n-1

d2

,n-1

n - 0 1 d1n n - 0 2 . . .

. . . 0 - m 0 with anticommutative squares: dk
1 d

. . .
m1 - m

2

m2 - . . .

d

d

m,n-1

-

mn 0

- 0

0

,l+1 dkl

+ dk

+1,l dkl

= 0.

Assume that the horizontal rows are exact over a subset Y and the vertical columns are exact over a subset Y . Then this diagram defines an element of the group K 0 (X, Y Y ). 4. The previous construction allows us to construct a natural tensor product of K ­groups as a bilinear function K 0 (X, Y ) â K 0 (X , Y )-K 0 (X â X , X â Y Y â Y ). Consider a complex on X ,
1 2 0-1 -2 - . . . - n -0

(3.7)

d

d

dn

-1

(3.8)

which is exact over Y and a complex on X
1 2 0-1 -2 - . . . - n -0

d

d

d

n-1

(3.9)


Geometric constructions of bund les

145

which is exact over Y . Consider the tensor product of the two complexes (3.8) and (3.9) 0 0 - 0 -
1 1 1 - -1 d 1 d1 1 1 2 -

0
d 1
2 2 1 - 1 d1 d2 1 2 2 -

0
d 1

... ...

d

n-1

-

1

d

n-1

-

1

n 1 - 0 (-1)n d1 n 2 - 0 . . .

. . . 0 - 1 m 0
d1 1

. . . - 2 m 0
d2 1

-

...

d

n-1

-

1

n m 0

- 0

(3.10)

It is easy to check that this diagram has rows exact over X â Y and columns exact over Y â X . Hence the diagram (3.10) defines a tensor product as a bilinear operation of type (3.7). If X = X then using diagonal inclusion X X â X we have the tensor product as a bilinear operation K (X, Y ) â K (X, Y )-K (X, Y Y ). 5. 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 1 2 0-0 -1 -2 - . . . - n -0,

d

d

d

d

n-1

(3.11)

where = p is inverse image of the bundle and the homomorphism dk : k -k
+1



is defined as the exterior multiplication by the vector y E , y x , x = p(y ). It is known that if the vector y x is nonzero, y = 0, then the complex (3.11) is exact. Consider the subspace D( ) E which consists of all vectors y E such that |y | 1. Then the subspace S ( ) D( ) of all unit vectors gives the pair


146

Chapter 3

(D ), S ( )) for which the complex (3.11) is exact on S ( ). Denote the element defined by (3.11) by ( ) K 0 (D( ), S ( )). Then one has homomorphism given by multiplication by the element ( ) : K (X )-K 0 (D( ), S ( )) . (3.12)

The homomorphism (3.12) is called the Bott homomorphism. In next section we shall prove that the Bott homomorphism is an isomorphism. 6. There is a simple way to shorten the bundle complex (3.2). Without loss of generality we may assume that n = 2m. Put
m

0 = 1 3 . . . =
k=1



2k-1

,

m

1 = 2 4 . . . =
k=1

2k ,

D=

d1 0 0 0 0

d 2 d3 0 0 0

0 d 4 d5 ··· 0 0

··· ··· ··· ··· ···

0 0 0 dm 2 d2m
-2 -1

.

First of all notice that there is a sequence
0 1 0 · · · -0 -1 -0 - · · · ,

D

D

D

where d1 0 0 0 d3 0 · · · · · · · · · · · · 0 0 d
2m-1

D0 =

, D1 =

0 d2 0

0 ·· 0 ·· ·· 0 ··

· · · ·

0 0 d
2m-2

0 0 0

.

This sequence is exact over Y . On the other hand, D = D0 + D1 . Hence D is isomorphism over Y .


Geometric constructions of bund les

147

7. Using the previous construction we can define the tensor product of two triples d d 0-1 -2 -0, 0-1 -2 -0 as a triple 0-(1 1 ) (2 2 )-(1 2 ) (2 1 )-0, where D= d1 1d -1 d d 1
D

.

3.2

BOTT PERIODICITY

Consider the Bott element = (1) K 0 S2 which was defined in the section 3.1. The two natural pro jections from X â S2 to the factors X and S2 generate a natural homomorphisms from K (X ) and K (S ) to the ring K (X â S ). Then since K S2 Z[ ]/{ 2 = 0}, there is a homomorphism h : K (X )[t]/{t2 = 0}-K X â S2 , defined by the formula h(xt + y ) = x + y . Theorem 27 (Bott p erio dicity) The homomorphism (3.13) is an isomorphism. Pro of. We need to construct an inverse homomorphism to (3.13). Consider the pair X â S2 , X â {s0 } , where s0 is a fixed point. From the exact sequence in K ­theory: K S X â S2 -K (S X )-K X â S2 , X â {s0 } -K X â S
Si


(3.13)



j



2

-K (X )

i



we see that is trivial since i and S i are epimorphisms. Hence the homomorphism h sends the element ty K (X )[t]/{t2 = 0} to K X â S2 , X â {s0 } . ~ Denote the restriction of this homomorphism by h: ~ h : K (X )-K X â S2 , X â {s0 } . (3.14)


148

Chapter 3

~ h(y ) = y , y K (X ). The pair X â S2 , X â {s0 } is equivalent to the pair X â D2 , X â S1 . Hence each element of the group K 0 X â D2 , X â S1 is represented by a triple (1 , d, 2 ) where d is an isomorphism over X â S1 . Moreover, we can assume that both vector bundles 1 and 2 are trivial. Thus the isomorphism d is a continuous function d : X â S1 -U(n) for sufficiently large n. Equivalent triples can be obtained by using the two operations: stabilization of dimension generated by inclusion U(n) U(n + n ) and homotopies of d in the class of continuous mappings X â S1 -GL(C, n). Thus we have a continuous mapping d(x, z ) U(n), x X, z S1 C, |z | = 1.

The first step
We produce a smooth ­approximation of the function d(x, z ), for example, by the formula q d1 (x, z ) = 2
+

d(x, ei )d.
-

The second step
Consider the Fourier series:


d1 (x, z ) =
k=-

ak (x)z k ,

(3.15)

where 1 ak (x) = 2

2

d1 (x, e
0

-ik

)d.

The series (3.15) converges uniformly with respect to the variables x and z and equipotentially in the class of continuous functions d1 (x, z ) U(n).


Geometric constructions of bund les

149

The third step
Restrict to a finite part of the Fourier series:
N

fN (x, z ) =
k=-N

ak (x)z k .

The number N can be chosen sufficiently large so that fN (x, z ) GL(C, n).

The fourth step
Put pN (x, z ) = z N fN (x, z ).

The fifth step
Let pN (x, z ) =
k=0 2N

bk (x)z k GL(C, n).

Put b0 (x) -z 0 LN (p)(x, z ) = . . . 0 b1 (x) . . . bN 1 ... -z ... . . . 0 ...
-1

0 0 . . .

(x) bN (x) 0 0 GL(C, nN ). (3.16) . . . 1

-z

The sixth step
The function (3.16) is linear with respect to the variable z , LN (p)(x, z ) = A(x)z + B (x) GL(C, nN ). Hence there is a pro jection Q(x) : CnN -CnN


150

Chapter 3

which commutes with LN (p)(x, z ) and satisfies the following condition: the space CnN is split into two summands CnN = V+ (x) V- (x) such that the operator LN (p)(x, z ) is an isomorphism on V+ (x) when |z | 1 and is an isomorphism on V- (x) when |z | 1.

The seventh and final step
The family of V+ (x) induces a vector subbundle of the trivial bundle X â CnN . Put N (d) = [V+ (x)] - nN . (3.17)

We need to prove that the definition (3.17) gives the homomorphism inverse to (3.14).

3.3

PERIODIC K ­THEORY

Using the Bott periodicity we can define the groups K n (X, Y ) for any integer K n (X, Y ) n, negative or positive. First of all notice that the direct sum
n0

can be given a ring structure. In fact, K
-n

(X , Y ) = K

0

X â Dn , (Y â Dn ) (X â S

n-1

)

.

Hence the operation of tensor product considered in the example 3.1 of the section 3.1 gives the following pairing K
-n

(X, Y ) â K K
0

-m

(X , Y ) =
-1

=

X â Dn , (Y â Dn ) (X â Sn âK
0 m n m

)
m n

â
-1

X â D , (Y â D ) (X â Sm
m

)

-

- K (X â X â D â D , (X â Y â D â D ) (X â X â Dn â Sm-1 ) (Y â X â Dn â Dm ) (X â X â Sn-1 â Dm )) = K 0 (X â X â Dn â Dm , (((X â Y ) (Y â X )) â Dn â Dm )

0

=


Geometric constructions of bund les

151

=

(X â X â Dn â Sm K 0 (X â X â Dn+m ,

-1

) (X â X â Sn

-1

â Dm )) =
n+m-1

((Y â X ) (X â Y ) â Dn+m ) (X â X â S =K
-(n+m)

)) =

(X â X , (X â Y ) (Y â X )).

In particular, when X = X and Y = Y it gives the usual inner multiplication K
-n

(X , Y ) â K

-m

(X, Y )-K

-(n+m)

(X, Y ).

When X = S0 , Y = {s0 }, where S0 is a 0­dimensional sphere, we have the pairing K
-n

(X, Y ) â K -m (S0 , s0 )- -K -(n+m) (X â S0 , (X â s0 ) (Y â S0) ) = = K -(n+m) (X, Y ).

The Bott element K 0 (S2 , s0 ) = K -2 (S0 , s0 ) = Z generates a homomor~ phism h: K -n (X, Y )-K -(n+2) (X, Y ) (3.18) by the Bott periodicity isomorphism. This formula (3.18) justifies the term `Bott periodicity'. Thus, via the isomorphism (3.18), the groups K n (X, Y ) can be indexed by the integers modulo 2. Then the exact sequence for the pair (X, Y ) has the following form : ... -K 0 (X, Y )-K 0 (X, x0 )-K 0 (Y , x0 )- -K 1 (X, Y )-K 1 (X, x0 )-K 1 (Y , x0 )- -K 0 (X, Y )- ...

(3.19)

Let

K (X, Y ) = K 0 (X, Y ) K 1 (X, Y ). ... -K (X, Y )-K (X, x0 )-K (Y , x0 )- j i -K (X, Y )-K (X, x0 )-K (Y , x0 )- -K (X, Y )- ...
j


Then the sequence (3.19) can be written as
i


152

Chapter 3

where i and j are ring homomorphisms and is an operator of degree 1. Consider the Chern character defined in the section 2.6. It can be represented as a ring homomorphism of graded rings ch : K (X, Y )-H (X, Y ; Q), where H (X, Y ; Q) is considered as a Z2 ­graded ring by decomposition into a direct sum of the even and odd dimensional cohomology. Theorem 28 The homomorphism ch : K (X, Y ) Q-H (X, Y ; Q) is an isomorphism for any finite cel lular pair (X, Y ). Pro of. The proof is based on the Five Lemma and uses induction on the number of cells. The initial step consists of checking that the homomorphism ch : K (Dn , Sn-1 ) Q-H (Dn , Sn-1 ; Q) is an isomorphism. Let n = 2. Then the with generator = [ ] - 1, where is the Hopf bundle. Then ch = ch - 1 = ec1 - 1, where c1 H 2 (S2 , s0 ; Q) ch : K 0 (D2 , S1 ) Q-H 2 (D2 , S1 ; Q) is an isomorphism. For the odd case, one has K 1 (D2 , S1 ) = H odd (D2 , S1 ) = 0. Thus we have proved initial step. This finishes the proof of theorem. K 0 (D2 , S1 ) Z

is the integer generator. Hence the homomorphism


Geometric constructions of bund les

153

Corollary 5 For any element x H 2k (X ) there is a bund le and a number such that ch = dim + x + terms of higher dimension.

3.4

LINEAR REPRESENTATIONS AND BUNDLES

Consider a linear representation of a group G on a (complex) n­dimensional vector space : G-GL(n, C). If is a principal G­bundle over a base space X with transition functions : U -G then using the new transition functions =


: U -GL(n, R)

we obtains a new vector bundle . We apply this construction to the universal principal G­bundle G over the classifying space BG . Denote by R(G) the group of virtual finite dimensional representations of the group G. It is clear that the group R(G) is a ring with respect to tensor product of representations. By the previous construction there is a ring homomorphism b : R(G)-K (B G). (3.20)

Interest in the homomorphism (3.20) comes from the fact that it has applica tions to a number of different problems. The virtual bundle b() = [G ] is an invariant of the virtual representation . This bundle can be used to interpret algebraic properties of the representation in homotopy terms. Let G = U(n). Then elements of the group K (BU(n)) correspond to cohomology operations in K ­theory. If an element belongs to the image of (3.20) then the corresponding cohomology operation is determined by a linear representation of the structure group U(n). In particular, it could be interest to know if the homomorphism (3.20) is an epimorphism or to describe the image of (3.20).


154

Chapter 3

Consider some simple examples.

The group G = S1 .
Each linear complex representation of the group S1 = U(1) can be split into a direct sum of one­dimensional representations which in turn are described by the following formula k (z ) = z k , where z G = S1 is a complex number with unit modulus. So the irreducible representations are parametrized by the integers. It is clear that k s =
1 k +s

.
-1

Hence the ring R(S ) is generated by two elements 1 and relation 1 -1 = 1. Thus, R(S1 ) = Z[1 ,
-1

with the unique

]/{1

-1

= 1},

the so called ring of Laurent polynomials. On the other hand, the classifying space B G is homeomorphic to the direct limit B G = lim CPn = CP .
n-

The representation 1 is the identity representation. Hence b(1 ) = and b(k ) = 1 . . .
1 k times for k 0. Notice that the representation -1 is the complex conjugate of since z -1 - z for |z | = 1. Hence ¯ 1

1

b( Thus

-k

¯ ¯ ) = 1 . . . 1
k

times

Theorem 29 The image of the homomorphism b : R(S1 ) Q-K (BS1 ) Q, where K (BS1 ) Q is the completion with respect to the topology generated by the ideal of zero dimensional virtual bund les.


Geometric constructions of bund les

155

Pro of. Consider the homomorphism
b ch R(S1 )-K (BS1 )-H (BS1 ; Q)

After tensoring with Q we have
b ch R(S1 ) Q-K (BS1 ) Q-H (BS1 ; Q)

(3.21)

The group K (BS1 ) is defined to be the inverse limit K (BS1 ) = lim K (CPn ).
-

Since

H (BS1 ; Q) = lim H (CPn ; Q),
-

using theorem 28 we have that the homomorphism ch in (3.21) is a topological isomorphism. On the other hand, for u = 1 - 1, ch b(u) = a + a2 ak + ··· + + ···. 2 k!

Hence this element can serve as a generator in the ring of formal series H (BS1 ; Q). Consider the ideal of zero­dimensional virtual representations in R(G). The corresponding topology generates the completion R(G) of the ring R(G). It is easy to prove that homomorphism b : R(S1 ) Q-K (BS1 ) Q is an isomorphism.

The group G = Z.
The classifying space B Z can be taken to be homeomorphic to the circle S1 and K (S1 ) = Z. Hence linear representations of the group Z are not distinguished by the homomorphism b.

The group G = Z â Z.
In this case the classifying space can be taken to be homeomorphic to T 2 = S1 â S1 . The group K (T 2 ) can be calculated using the exact sequence of the


156

Chapter 3

pair (T 2 , S1 S1 ). Finally, K 0 (T 2 , s0 ) = Z, K 1 (T 2 , s0 ) = Z Z. Each irreducible representation of the commutative group Z â Z is one­ dimensional and homotopic to the trivial representation. Thus b() is trivial.

The group G = Zn and the case of a family of representations.
The case of G = Zn is similar to the case of G = Z2 . But one can obtain a nontrivial result by considering a continuous family of representations. Namely, let y , y Y be a continuous family of representations. Let G be the universal principal G­bundle over X = B G with the transition functions . We can consider the same transition functions over the base X â Y by taking them independent of the argument y Y . The compositions = y depend on both arguments x X, y Y . The latter defines a vector bundle G over the base X â Y . Then in spite of the fact that the restriction of vector bundle Gy0 over each X â y0 may be trivial the whole vector bundle G over X â Y may be nontrivial. For example, consider as the space Y , the character group of the group G, Y = G , and consider as representation y the corresponding character of the group G, y : G-S1 = U(1). When G = Zn = Gj , Gj = Z, G = T Then
n

and B G = T n .

X â Y = B G â G = T n â T n .

Theorem 30 Let be the family of al l characters of the group G = Zn . Then
c1 (G ) = x1 y1 + . . . + xn yn

where xk H 1 (B G; Z), yk H 1 (G ; Z) are corresponding generators.


Geometric constructions of bund les

157

Pro of. The two dimensional cohomology of the space X â Y is generated by monomials of degree 2 in the variables xk , yk . Hence
c1 (G ) =

ij xi yj +

µij xi xj +

ij yi yj

for some integers ij , µij , ij . It is clear that the second and the third sums are trivial by considering the characteristic classes of restrictions of the bundles over the factors. The coefficients ij can be interpreted as characteristic classes of restrictions to the subspaces B Gi â G Therefore, it is clear that ij = 0 j when i = j . Hence the proof reduces to the case where G = Z. In this case X = S1 and Y = S1 . There is an atlas consisting of two charts: the upper and lower semicircles, U1 and U2 . The intersection U12 = U1 U2 consists of two points: {1, -1}. The principal bundle G has one transition function 12 where 12 (1) = 0 Z, 12 (-1) = 1 Z. Let y Y = S1 be a character of the group G = Z. Then the corresponding transition function 12 = y 12 is defined by the formula 12 (1, y ) = 1, 12 (-1, y ) = y .
It is easy to see that the bundle G is trivial over the wedge of a parallel and a 1 1 meridian S S and on the quotient space T 2 /(S1 S1 ) = S2 it is isomorphic to the Hopf bundle. Hence c1 (G ) = xy H 2 (T 2 ; Z).

3.5

EQUIVARIANT 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 locally trivial bundle and also equivariant then f is called an equivariant locally 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


158

Chapter 3

sum and tensor product. In certain simple cases the description of equivariant vector bundles is reduced to the description of the usual vector bundles. Consider first the case where the action of the group G is free, then the natural pro jection onto the orbit space Y = X/G gives a locally trivial bundle. Theorem 31 The family of G-equivariant vector bund les over a space X with a free action of a group G is in one­to­one correspondence with the family of vector bund les over the orbit space Y = X/G. This correspondence is given by the projection : - with the natural action of the group G on the total space of vector bund le . Pro of. In fact, in the total space of the bundle ( ), there is natural action of the group G. By definition the charts of the bundle ( ) are inverse images of the charts U on Y . Then -1 (U ) U â G with a of the Then, action left action along the second summand. The corresponding trivialization vector bundle ( ) is homeomorphic to U â G â F where F is the fiber. by definition, the action of the group G on the total space is the left along the second summand.

Conversely, let -X be arbitrary G-equivariant vector bundle with a free G action. Consider a small chart U Y and put V = -1 (U ) = U â G. For U sufficiently small that the restriction of to U â g0 is trivial
U âg0

U â g0 â F.
U âG

Then the restriction of the bundle

also is trivial: -
U âG

U â g0 â G â F (u, g0 , g , f )-g (u, g0 , f ).

,

It is clear that the transition functions do not depend on the argument g G.

The second case occurs when the group G acts trivially on X .


Geometric constructions of bund les

159

Theorem 32 Suppose the group G acts trivial ly on the base X . Then for each G­equivariant vector bund le x there exist irreducible representations of the group G in vector spaces Vi and vector bund les i such that =
i

(i Vi )

with the natural action of the group G on the second factors. Pro of. The action of the group G preserves each fiber and acts there as a linear representation. Hence this representation can be split into a direct sum of irreducible representations. This means that the fiber x gives a direct sum x = i (
x,i

Vi ) .

(3.22)

Assume that locally the presentation (3.22) does not depend on the point x, that is, in some small neighborhood U we have
|U

= x0 â U = [i (x

0

,i

Vi )] â U .

(3.23)

Then using Schur's lemma the equivariant transition function decomposes into a direct sum = i ( ,i Vi ). The functions
,i

define transition functions for some vector bundles i . Thus = i (i Vi ).

Our problem is to prove the decomposition (3.23). We give here an elegant proof due to M.F. Atiyah. Consider a G­equivariant vector bundle E -X with the trivial action on the base X . Denote by E G the set of fixed points. Then E G -X also is a locally trivial vector bundle. To prove this it is sufficient to construct a fiberwise pro jection P onto the set of fixed points. Put P (v ) =
G

g (v )dµ(g )

where dµ(g ) is the Haar measure on the compact group G. It is clear that P is a pro jection since P (P (v )) =
G

h(P (v ))dµ(h) =
Gâ G

hg (v )dµ(h)dµ(g ) =


160

Chapter 3

=
GâG

g (v )dµ(g )dµ(h ) =
G

g (v )dµ(g ) ·
G

dµ(h ) =

=
G

g (v )dµ(g ) = P (v ).

On the other hand, g (P (v )) = P (v ), and if g (v ) = v then P (v ) = v . Hence the image of P coincides with E G . Using Theorem 7 it can be shown that E G is a locally trivial bundle. Consider the bundle HOM (1 , 2 ) (for the definition see section 1.3) . If both bundles 1 and 2 are G­equivariant then the bundle HOM (1 , 2 ) also is G­equivariant with the action defined as follows: -g g
-1

.

Then the fixed point set HOM G (1 , 2 ) gives a locally trivial bundle. Now we can construct the bundles i mentioned above. Let Vi be an irreducible G­ vector space and Vi be the corresponding trivial G­equivariant vector bundle. Put i = HOM G (Vi , ). Then there is an isomorphism : i Vi HOM G (Vi , ) - defined on each summand by formula (x, ) = (x), x Vi . (3.24)

The mapping (3.24) is equivariant. To prove that (3.24) is an isomorphism it is sufficient to check it on each fiber. Here it follows from Schur's lemma.

3.6

RELATIONS BETWEEN COMPLEX, SYMPLECTIC AND REAL BUNDLES

The category of G­ equivariant vector bundles is good place to give consistent descriptions of three different structures on vector bundles -- complex, real and symplectic.


Geometric constructions of bund les

161

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 reals and anticomplex over complex numbers, that is, if G = Z2 is the generator then ¯ (x) = (x), C, x E . (3.25)

A vector bundle with the action of the group G satisfying the condition (3.25) is called a K R­bund le . The operator is called the anticomplex 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. Prop osition 10 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 . Pro of. The complex bundle over X consists of a union of two bundles: 1 -- restriction of on Y â {1} and -- restriction of on Y â { }. Since interchanges the two summands -- Y â {1} and Y â { }, the involution gives an isomorphism : 1 - which is anticomplex. Hence = (1 ). Hence given a complex vector bundle 1 over Y â {1}, then we can take the same vector bundle over Y â { } with the conjugate complex structure. Then the identity involution is anticomplex and induces a K R­ bundle over X . Prop osition 11 Suppose the involution on X is trivial. Then K R(X ) KO (X ). (3.26)


162

Chapter 3

The isomorphism (3.26) associates to any K R­bund le the fixed points of the involution . Prop osition 12 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 ). Similar to the realification of vector bundles, we can construct, for any base X with involution , a homomorphism KU (X )-K R(X ) which coincides with realification r : KU (X )-KO (X ), when the involution on the base X is trivial. The homomorphism (3.27) asso¯ ciates to any complex vector bundle over X the bundle = ( ) with an involution which transposes the summands. As for complex vector bundles the operation of tensor product can be defined for K R­bundles making K R(X ) into a ring. The extension of the groups K R(X ) to a cohomology theory is more interesting. An equivariant analogue of suspension for Z2 ­spaces must be defined. Definition 8 Denote by Dp,q the disc Dp+q with involution which changes signs in the first p coordinates and denote by S p,q the boundary of Dp,q . Then the suspension of pair (X, Y ) with involution is given by the pair
p,q

(3.27)

(X, Y ) = (X â D

p,q

, (X â S

p,q

) (Y â D

p,q

))

Hence in a natural way we can define a bigraded K R­theory KR
-p,-q

(X, Y ) = K R (p,q (X, Y )) ,

(3.28)

where K R(X, Y ) is defined as the kernel of the mapping K R(X/ Y )-K R(pt).


Geometric constructions of bund les

163

Clearly the bigraded functor (3.28) is invariant with respect to homotopy and the following axiom holds: for any triple (X, Y , Z ) we have the exact sequence . . . -K R . . . -K R . . . -K R
p,q

(X, Y )-K Rp,q (X, Z )-K Rp,q (Y , Z )- . . . p,q +1 (X, Y )-K Rp,q+1 (X, Z )-K Rp,q+1 (Y , Z )- = ts p,0 (X, Y )-K Rp,0 (X, Z )-K Rp,0 (Y , Z )

for any p 0. As in classical K ­theory, the operation of tensor product extends to a graded multiplication KR
p,q

(X, Y ) K R

p ,q

(X, Y )-K R

p+p ,q +q

(X, Y Y ).

Moreover, the proof of Bott periodicity can be extended word by word to K R­ theory: Theorem 33 There is an element K R(D1,1 , S that the homomorphism given by multiplication by : KR is an isomorphism. The proof is completely similar to that of Theorem 27 with the observation that all constructions are equivariant. Further, the following facts are essential to our considerations. Lemma 9 Let X be space with involution. Then there exist equivariant homeomorphisms 1)X â S 2)X â S 3)X â S
1, 0 2, 0 4, 0 p,q 1, 1

) = KR

-1,-1

(pt) such (3.29)

(X, Y )-K R

p-1,q -1

(X.Y )

âD âD âD
p,0

0, 1 0, 2 0, 4


0,p

X âS X âS X âS

1,0 2,0 4,0

âD âD âD

1,0 2,0 4,0

, , .

In al l three cases, the homeomorphism µ:X âS is defined by the formula µ(x, s, u) = (x, s, su), x X, s S
p,0

âD

-X â S

p,0

âD

p,0

,u D

0,p

,

where we identify S p,0 , Dp,0 and D0,p , respectively, with the unit sphere or unit disk in the field of real, complex or quaternionic numbers, respectively.


164

Chapter 3

Thus we have the following isomorphisms: KR KR KR
p,q p,q p,q

(X â S (X â S (X â S

1, 0 2, 0 4, 0

) ) )

KR KR KR

p-1,q +1 p-2,q +2 p-4,q +4

(X â S (X â S (X â S

1, 0 2, 0 4, 0

), ), ).

(3.30) (3.31) (3.32)

In the case of (3.30) using the Bott periodicity (3.29) KR
p,q

(X â S

1, 0

) KR

p,q +2

(X â S

1, 0

),

that is, the classical Bott periodicity for complex vector bundles. The next step consists of the following theorem. Theorem 34 There is an element K R0,-8 (pt) = K R(D that the homomorphism given by multiplication by : KR is an isomorphism. In the case of the trivial action of an involution on the base we get the classical 8­periodicity in the real K ­theory. It turns out that this scheme can be modified so that it includes another type of K ­theory ­ that of quaternionic vector bundles. Let K be the(noncommutative) field of quaternions. As for real or complex vector bundles we can consider locally 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 anticomplex 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 anticomplex linear operator which acts on a complex vector bundles and satisfies J 4 = 1, I J + J I = 0. (3.33)
p,q 0,8

,S

0, 8

) such

(X, Y )-K R

p,q -8

(X, Y )


Geometric constructions of bund les

165

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

(3.34)

Hence the vector bundle 1 is the complexification of a real vector bundle and 2 is quaternionic vector bundle. Consider a similar situation over a base X with an involution such that the operator (3.33) commutes with . Such vector bundle will be called a K RS ­ bund le . Lemma 10 A K RS ­bund le is split into an equivariant direct sum = 1 such that J 2 = 1 on 1 and J 2 = -1 on 2 .

2

Lemma 10 shows that the Grothendieck group K RS (X ) generated by K RS ­ bundles has a Z2 ­grading, that is, K RS (X ) = K RS0 (X ) K RS1 (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 RS (X ) = KO (X ) KQ (X ) where KQ (X ) is the group generated by quaternionic bundles. The Z2 ­graded group K RS (X ) has a multiplicative structure induced by the tensor product of complex vector bundles. It is clear that the tensor product preserves the action of involution on the base and action of the operator J on the total space. Moreover, this action is compatible with the grading, that is, if 1 K RSi (X ), 2 K RSj (X ) then 1 2 K RSi+j (X ). As in K R­theory, p,q p,q the groups K RS p,q (X, Y ) = K RS0 (X, Y ) K RS1 (X, Y ) may be defined: K RS where K RS (X, Y ) = Ker (K RS (X/ Y )-K RS (pt)) . Bott periodicity generalizes: if K R D Bott element then
1, 1 -p,-q

(X, Y ) = K RS (D

p,q

â X , (S

p,q

â X ) (D

p,q

â Y )) ,

,S

1, 1

= K RS

-1,-1 0

(pt) is the


166

Chapter 3

Theorem 35 The homomorphism : K RS
p,q

(X, Y )-K RS

p-1,q -1

(X , Y )

given by multiplication by the element is an isomorphism. As in Theorem 34 we have the following Theorem 36 There is an element K RS
0,-4 1

(pt) = K RS1 (D

0, 4

,S

0,4

)

such that the homomorphism of given by multiplication by : K RS or : K RS or : K RS
p,q

(X, Y )-K RS (X, Y )-K RS (X, Y )-K RS

p,q -4

(X, Y ) (X, Y ) (X, Y )

p,q 0 p,q 1

p,q -4 1 p,q -4 0

are isomorphisms. Moreover, 2 = K RS
0,-8

(pt) = K R

0,-8

(pt) = K

-8 O

(pt).

- Theorem 36 says that KQ 4 (pt). In particular, we have the following isomorphisms: q KO (X ) q KQ (X ) q KQ-4 (X ) q KO-4 (X ).


Geometric constructions of bund les

167

q K K

-8 Z

-7 0 0

-6 0 Z
2

-5 0 Z
2

-4 Z u Z

-3 0 0

-2 Z h
2

-1 Z2 h 0
2

0 Z u =4 Z u

O

= Z u
2

2

2

Q

h

2

h

0

Figure 3.1

A list of the groups K

O

and K

Q

are given in the figure 3.1.


168

Chapter 3


4
CALCULATION METHODS IN K ­THEORY

In this chapter we shall give some methods for describing the K ­groups of various concrete spaces by reducing them to a description in terms of the usual cohomology groups of spaces. The methods we discuss are spectral sequences, cohomology operations and direct images. These methods cannot be used universally but they allow us to use certain geometric properties of our concrete topological spaces and manifolds.

4.1

SPECTRAL SEQUENCES

Similar to classical cohomology theory, the spectral sequence for K ­theory is constructed using a filtration of a space X . The construction of the spectral sequence described below can be applied not only to K ­theory but to any generalized cohomology theory. Thus let X0 X1 . . . XN = X be a increasing filtration of the space X . Put D E =
p,q

(4.1)

D E
p,q

p,q

=
p,q

K K
p,q

p+q

(Xp ) (Xp , X ): (Xp , X
p-1 p-1

=

p,q

=

p+q

).

(4.2)

Consider the exact sequence of a pair (Xp , X ... -K -K
p-1+q p+q

p-1

(Xp

(Xp )-K 169

-1 i

)-K
p+q



p+q

)- (4.3)

j



(Xp-1 )- . . .


170

Chapter 4

After summation by all p and q , the sequences . . . - D
p-1,q

-

E

p,q j

-



D

p,q i

-



D

p-1,q +1

- . . . ,

can be written briefly as . . . -D-E -D-D-E - . . . , where the bigradings of the homomorphisms , j , i are as follows: deg i
j


i





(4.4)

deg j deg

= = =

(-1, 1) (0, 0) (1, 0).

The sequence (4.4) can be written as an exact triangle

D d s d d j

i

E

D

d

d

d

E

©



(4.5) Put d = j : E -E .

Then the bigrading of d is deg d = (1, 0) and it is clear that d2 = 0. Put D1 E1 i1 j1 1 d1 Now put D
2

= D, = E, = i , = j, = , = d.

= i1 (D1 ) D1 , = H (E1 , d1 ).

E2


The calculation methods in K ­theory

171

The grading of D2 is inherited from the grading as an image, that is, D D
2

= =

D i1 (D

p,q 2

, ).

p,q 2

p+1,q -1 1

The grading of E2 is inherited from E1 . Then we put i2 2 j2 = i1 |D2 : D2 -D2 , = 1 i-1 , 1 = j1 .

It is clear that i2 , j2 and 2 are well defined. In fact, if x D2 then i1 (x) D2 . If x D1 then x = i1 (y ) and 2 (x) = [1 (y )] H (E1 , d1 ). The latter inclusion follows from the identity d1 1 (y ) = 1 j1 1 (y ) = 0. If i1 (y ) = 0, the exactness of the triangle (4.5) gives y = j1 (x) and then 2 (x) = [1 j1 (z )] = [d1 (z )] = 0. Hence 2 is well defined. Finally, if x E1 , d1 (x) = 0 then 1 j1 (x) = 0. Hence j1 (x) = i1 (z ) D2 . If x = d1 (y ), x = 1 j1 (z ) then j1 (x) = j1 1 j1 (z ) = 0 Hence j2 is well defined. Thus there is a new exact triangle

D

2

i 2 d s d d j2

d

d

d E2

©



ED 2

2

This triangle is said to be derived from the triangle (4.5).


172

Chapter 4

Repeating the process one can construct a series of exact triangles

Dn d s d d jn

i n

d

d

d En

©



ED n

n

The bigradings of homomorphisms in , jn , n and dn = n jn are as follows: deg in deg jn deg n deg dn The sequence (En , dn ) (4.6) is called the spectral sequence in K ­theory associated with a filtration (4.1). Theorem 37 The spectral sequence (4.6) converges to the groups associated to the group K (X ) by the filtration (4.1). Pro of.
p,q According to the definition (4.2), E1 = 0 for p > N . Hence for n > N ,

= = = =

(-1, 1), (0, 0), (n, -n + 1), (n, -n + 1).

dn = 0, that is, E By definition, we have
p,q Dn = Im K p+q p,q n p,q p,q = En+1 = . . . = E .

(Xp

+n

)-K

p+q

(Xp ) .


The calculation methods in K ­theory

173

Hence for n > N ,
p,q Dn = Im K p+q

(X )-K

p+q

(Xp ) .

Hence the homomorphism in is an epimorphism. Thus we have the exact sequence p,q jn p,q in p 0-En -Dn -Dn-1,q+1 -0. Hence
p,q En = Ker in = = Ker K p+q (X )-K p+q p+q p+q

(Xp

-1

) /Ker K

(X )-K

(Xp )

Let p : Y -X be a locally trivial bundle with fibre F . Then the fundamental group 1 (X, x0 ) acts on the fiber F in the sense that there is natural homomorphism 1 (X, x0 )-[F, F ], (4.7) where [F, F ] is the family of homotopy equivalences of F . Then the homomorphism (4.7) induces an action of the group 1 (X, x0 ) on the groups K (F ). Theorem 38 Let p : trivial action of 1 (X, generated by filtration of X , converges to the fol lowing form: Y -X be a local ly trivial bund le with fiber F and the x0 ) on K ­groups of the fiber. Then the spectral sequence Yk = p-1 ([X ]k ), where [X ]k is k ­dimensional skeleton groups associated to K (Y ) and the second term has the
p,q E2 = H p (X, K q (F )) .

Corollary 6 If the filtration of X is formed by the k ­dimensional skeletons then the corresponding spectral sequence converges to the groups associated to K (X ) and H p (X ; Z ), for even q p,q E2 = (4.8) 0 for odd q .

Pro of. The first term of the spectral sequence E1 is defined to be
p,q E1 = K p+q

(Yp , Y

p-1

).


174

Chapter 4

Since the locally trivial bundle is trivial over each cell, the pair (Yp , Y the same K ­groups as the union
p p (j â F, j â F ), j

p-1

) has

that is,
p,q E1 j

K

p,q

p p (j â F, j â F ) = j

K q (F ).

Hence, we can identify the term E1 with the cochain group
p,q E1 = C p (X ; K q (F ))

(4.9)

with coefficients in the group Kq (F ). What we need to establish is that the differential d1 coincides with the coboundary homomorphism in the chain groups of the space X . This coincidence follows from the exact sequence (4.3). Notice that the coincidence (4.9) only holds if the fundamental group of the base X acts trivially in the K ­groups of the fiber F . In general, the term E1 is isomorphic to the chain group with a local system of coefficients defined by the action of the fundamental group 1 (X, x0 ) in the group K q (F ). We say that the spectral sequence is multiplicative if all groups Es = are bigraded rings, the differentials ds are derivations, that is, ds (xy ) = (ds x)y + (-1)
p+q p x(ds y ), x Es +q p,q p,q Es

,

and the homology of ds is isomorphic to E

s+1

as a ring.

Theorem 39 The spectral sequence from Theorem 38 is multiplicative. The isomorphism (4.8) is an isomorphism of rings. The ring structure of E is isomorphic to the ring structure of groups associated with the filtration generated by the skeletons of the base X . We leave the proof of this theorem to the reader.

Examples
1. Let X = Sn . Then the E2 term of the spectral sequence is
p,q E2 = H p (Sn ; K q ()) =

Z 0

for p = 0, n and for even q , (see fig.4.1.) in other cases.


The calculation methods in K ­theory

175

q6

Z

Z

@ @ @ @ @ @ @ @ @ @ @ @
Figure 4.1

Z

Z n

p

For a one point space all differentials are trivial. Hence ds |
E
o,q s

= 0.

Thus all differentials for a sphere are trivial and E2 = E . Finally, we have for even n, K 0 (Sn ) = Z Z, K 1 (Sn ) = 0, and for odd n, K 0 (Sn ) = Z, K 1 (Sn ) = Z.

2. Let X = CPn . Then
E2 = H (CPn ; K (pt)) = Z[t]/{t n+1

= 0}.

Nontrivial terms E trivial and we have

p,q 2

exist only when p + q is even. Hence all differentials are
E = Z[t]/{tn +1

= 0 }.
n

(4.10)

Hence the additive structure of the group K (CP ) is the same as in the term E . To show that the multiplicative structure of the group K (CPn ) coincides with (4.10), consider an element u K (CPn ) which corresponds to the element 2, t E0 . The element u should be trivial on the one dimensional skeleton of n CP . Further, the element uk , k n is trivial on the (2k - 1)­dimensional 2k skeleton of CPn and corresponds to the element tk E ,0 . Consider the n+1 n element u K (CP ). This element is trivial on the (2n + 1)­dimensional skeleton, that is, un+1 = 0. Hence K (CPn) = Z[u]/{un
+1

= 0}.


176

Chapter 4

It is easy to show that the element u can be chosen to be equal to [ ] - 1 where is the Hopf bundle. 3. Let us show that all differentials ds have finite order. To prove this consider the Chern character ch : K (X )-H (X ; Q). (4.11) The homomorphism (4.11) defines a natural transformation of cohomology theories and hence induces a homomorphism of spectral sequences generated by K ­theory and Z2 ­graded cohomology with rational coefficients induced by the filtration with respect to skeletons of X :
p,q p,q ch : (Es , ds )-( Es , ds ).

Since all differentials ds are trivial and ch QQ is an isomorphism, we have ds Q = 0.
p,q Thus the differential ds has finite order on each group Es .

For example, the differential
p,q d2 : E2 -E p+2,q -1 s

can be written as d2 : H p (X ; K q ()) -H
n+2

X; K

k-1

() .

p,q p,q Hence d2 = 0 and E2 = E3 . Thus the differential d3 has the form of a stable cohomology operation:

d3 : H p (X ; Z)-H

p+3

( X ; Z) .

Therefore, d3 may take one of only two possible values: d3 = 0 or d3 = Sq3 . To check which possibility is realized it suffices to consider a special example for the space X where the differential d3 can be calculated. 4. As in cohomology theory we can obtain a formula for the K ­groups of a Cartesian product X â Y . Assume firstly that the group K (Y ) has no torsion. Let X1 X be a subspace of X which differs from X in only one cell. Consider the exact sequences for pairs (X, X1 ) and (X â Y , X1 â Y ): . . . E (X, X ) K (Y ) E (X ) K (Y ) EK (X ) K (Y ) E . . . K K
1 1

f c . . . E (X â Y , X â Y ) K 1

g c E K (X â Y )

h c E K (X â Y ) 1

E ... (4.12)


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The upper row is exact since K (Y ) has no torsion. The vertical homomorphisms are defined by tensor multiplication. By Bott periodicity, the vertical homomorphism f is an isomorphism. Hence using induction and the Five Lemma for isomorphisms we get that g is an isomorphism. Now let Y1 be a space such that K (Y1 ) has no torsion and suppose there is a map f : Y -Y1 such that the homomorphism f : K (Y1 )-K (Y ) is an epimorphism. Then the exact sequence for the pair (Y1 , Y ) has the form of a free resolution 0-K (Y1 , Y )-K (Y1 )-K (Y )-0. Consider the diagram K . . . E (Y , Y ) K (X ) E (Y ) K (X ) K 1 1 g f c c . . . E (Y â X, Y â X ) E K (Y â X ) K 1 1 E K (Y ) K (X ) h c E K (Y â X ) E0

E ...

We will show that the homomorphism f is a monomorphism. If f (x) = 0 then there exists y such that (y ) = x. Then (g (y )) = 0. So there exists z such that g (y ) = (z ). Since h is an isomorphism, we have z = h(w) or g (y ) = g (w). Since g is also an isomorphism, y = (w). Hence x = (w) = 0. Now let us show that h(Ker ) = Ker . It is clear that h(Ker ) Ker . If (x) = 0, x = h(y ) then g (y ) = h(y ) = 0, that is, (y ) = 0 or y Ker . Hence there is an exact sequence 0-K (Y ) K (X )-K (Y â X )-Ker -0. The group Ker is denoted by tor (K (Y ), K (X )). So there is a Kunneth ¨ formula 0-K (Y ) K (X )-K (Y â X )-Tor (K (Y ), K (X )) -0.


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To complete the proof it remains for us to construct the space Y1 . The space Y1 one should regard as a Cartesian product of complex Grassmannian manifolds which classify finite families of generators of the group K (Y ). The cohomology of the Grassmannian manifolds has no torsion. Thus the proof is complete. 5. The splitting principle. If a functorial relation holds for vector bund les which split into a sum of one dimensional bund les then the same relation holds for al l vector bund les. This principle is based on the following theorem Theorem 40 For any vector bund le over the base X there is a continuous mapping f : Y -X such that f ( ) splits into a sum of one dimensional bund les and the homomorphism f : K (X )-K (Y ) is a monomorphism. Pro of.

Let p : E -X give the vector bundle . Denote by P (E ) the space of all one dimensional subspaces of fibers of the . Let : P (E )-X be the natural mapping. Then the inverse image ( ) split into sum ( ) = where is the Hopf bundle over P (E ). Hence dim = dim - 1. On the other hand, the homomorphism : K (X )-K (P (E )) is a monomorphism. In fact, consider the continuous mapping f : X -BU(n) and commutative diagram P (E ) E P E U(n)

c X

E

c BU(n)

(4.13)


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The diagram (4.13) induces a commutative diagram of spectral sequences
1 p,q Es

'

2

p,q Es

T

T

3

p,q Es

'

4

p,q Es

where
1 2 3 p,q Es p,q Es

= = = =

H H

p p p

X, K q (CPn-1 ) , BU(n), K q (CPn
q -1

),

p,q Es 4 p,q Es

H (X, K (pt)) , H p (BU(n), K q (pt)) .

In the spectral sequences 2 E and 4 E , all differentials are trivial. Hence the differentials in 1 E are induced by the differentials in 3 E . Thus
1

E

, s

= 3E
3

, s

K (CPn
,

-1

).

Hence the homomorphism

, E -1 E

is a monomorphism. Therefore, is also a monomorphism. Now one can complete the proof by induction with respect to the dimension of vector bundle .

4.2

OPERATIONS IN K ­THEORY
: K (X )-K (X )

By definition, a cohomology operation is a transformation


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Chapter 4

which is functorial, that is, for any continuous mapping f : X -Y the following diagram K (X ) c K (X ) 'f


'f



K (Y ) c K (Y )

is commutative. In particular, if f : X -BU(N ), x = f (N ) then (x) = f ((N )). Here the element (N ) K (BU(N )) may be an arbitrary element and it completely defines the cohomology operation . This definition is not quite correct because the classifying space BU(N ) is not a finite cellular complex. Strictly speaking, we should consider the inverse sequence of the Grassmann manifolds G(N + n, N ), n- with natural inclusions jn : G(N + n, N )-G(N + n + 1, N ). Then the cohomology operation is defined as an inverse sequence of elements n K (G(N + n, N )) such that jn (n+1 ) = n . This means that the sequence {n } defines an element in the inverse limit lim K (G(N + n, N )) = K (BU(N )). -
n

(4.14)

The expression (4.14) can be considered as a definition of the K ­groups of an infinite cellular spaces. Let us continue the analysis of cohomology operations. Each cohomology operation should be independent of trivial direct summands. This means that a cohomology operation is a sequence of elements N K (BU(N )) such that N Then the natural mapping kN : BU(N )-BU(N + 1) sends the element N element
+1 +1

( 1) = N ( ).

to N . This means that the sequence {N } defines an lim K (BU(N )) . -
N


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181

Let us put BU = lim BU(N ). Then we can show that -
N

K (BU) = lim K (BU(N )). -
N

Definition 9 The Adams operation k : K (X )-K (X ) is the operation characterized by the fol lowing property
k k (1 . . . n ) = 1 . . . k n

for any one dimensional vector bund les j . We show now that such an operation k exists. Consider the symmetric polynomial sk (t1 , . . . , tn ) = tk + . . . + tk . Then the polynomial sk can be expressed n 1 as a polynomial n sk (t1 , . . . , tn ) = Pk (1 , . . . , n ).
n The polynomial Pk has integer coefficients and the k are the elementary symmetric polynomials: n Pk (1 , . . . , n ) =



k l1 ,l2 ,...,l

n

ll ln 11 22 · · · n .

Then

k ( ) =



k l l1 ,l2 ,...,ln

1

(2 ) 2 . . . (n )

l

ln

It is easy to check that k (1 2 ) = k (1 ) + k (2 ) . Theorem 41 The Adams operations have the fol lowing properties: 1. k (xy ) = k (x)k (y ); 2. k l = kl ; 3. p (x) xp ( mod p) for prime p. Pro of.


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As was shown earlier, it is sufficient to check the identities on vector bundles which can be split into one­dimensional summands. Let = 1 . . . n , = 1 . . . m and then
i,j

(i j ) .

Then we have k ( ) =
i,j

(i j ) =
k i

k

=
i,j

k j =


i k i k


k j = j

=
k



= ( ) ( ). Further,
l l k l (1 . . . n ) = k (1 . . . n ) = k k = 1 l . . . nl = kl (1 . . . n ).

Finally, if p is a prime integer then,
p p p (1 . . . n ) = 1 . . . n p (1 . . . n ) ( mod p) .

Examples
1. Consider the sphere S2n and the generator n K n is the image of the element ... K
0 0

S2

n

. The element

S2 â . . . â S

2

by the homomorphism induced by the continuous mapping f : S2 â . . . â S2 -S2
n

which contracts the (2n - 2)­dimensional skeleton to a point.


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183

Then

f k (n ) k ( . . . ) k . . . k .

Now = - 1 where is the Hopf bundle. Hence k = k - 1 = (1 + )k - 1 = k . Thus that is, f k n = k n ( . . . ) k n f (n ), k n = k n n .

2.

Division algebras.

Let V be a finite dimensional division algebra, that is, any x V , x = 0, has a multiplicative inverse x-1 V . Denote by P (V 3 ) the set of all subspaces of V 3 = V V V which are one dimensional V ­modules. The space P (V 3 ) is called the (2­dimensional) pro jective space over the algebra V . We know of three cases of division algebras: the real and complex fields, and the skew field of quaternions. We can calculate the cohomology of the space P (V 3 ). Let n = 2m = dim V > 1 be even. There is a locally trivial bundle S3 with the fiber S show that
n-1 n-1

-P (V 3 )

. As for CPn (see 18), we can use the spectral sequence to


H

P (V 3 ) = Z[v ]/{v 3 = 0}, dim v = n.

Let us use the K ­theory spectral sequence for the space P (V 3 ). We have K


P (V 3 ) = Z[u]/{u3 = 0}, u K

0

P (V 3 ) .

Moreover, the space P (V 3 ) is the union of the sphere Sn = P (V 2 ) and the disk D2n glued on by a continuous mapping : D2n = S2n-1 -Sn . In other words, we have the pair Sn P (V 3 )-P (V 3 )/Sn = S2n . The exact sequence for (4.15) has the following form 0-K
0

(4.15)

S2

n

-K

j



0

P (V 3 ) -K 0 (Sn ) -0.

i




184

Chapter 4

Let be generators and let

a K 0 (Sn ) ; b K

0

S2

n

i (x) = a; j (b) = y .

Let y = u + u2 . Then 0 = y 2 = (u + u2 )2 2 u2 . Hence = 0. Let x = u + u2 . The elements (x, y ) forms the basis in the group K 0 P (V 3 ) and so = ±1; = ±1. By choosing a and b appropriately we can assume that x2 = y . Now we can calculate the action of the Adams operations. Firstly, j k (x) = k (a) = k m a. Hence and Moreover, k (x) = k m x + y k (y ) = k j (b) = j k (b) = k k (x) xk ( mod k )
2m

y.

when k is prime. For example, when k = 2, the number must be odd; when k = 3, the number is divisible by 3. Consider the operation 6 = 2 3 = 3 2 . Then 2 3 (x) = (x) = or
3 2

2 (3m x + µy ) 6m x + 3m + 2 (2 x + y ) 6 x + 3
3 m m 2m

2m m

µ y,

+ 2 µ y,

2m (2m - 1) µ = 3m (3m - 1) ,

where is odd. This means that the number (3m - 1) is divisible by 2m . Let m = 2k l where l is odd. Then 3m - 1 = 3
2
k

l

- 1 32 - 1

k

3

2

k

l-1

+3

2

k

l -2

+ ... + 1

The second factor has an odd number of summands and, therefore, it is odd. k k Hence the number 32 - 1 is divisible by 2m and hence is divisible by 22 . On the other hand, 3
2
k

- 1 = 32

k-1

+1

3

2

k -2

+ 1 · . . . · (3 + 1) (3 - 1).

(4.16)


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185

It is easy to see that the number 32 + 1 is divisible by 2 but not divisible by k 4. Hence by the decomposition (4.16), the number 32 - 1 is divisible by 2k+2 but not divisible by 2k+3 . Hence 2k l k + 2. (4.17)

k

For l = 1, the inequality (4.17) holds for k = 0, 1, 2. For l 3, (4.17) has no solution. Hence division algebras exist only in dimensions n = 2, 4, 8. In fact, in addition to the real, complex and quaternionic fields, there is another 8­dimensional (nonassociative ) algebra, the Cayley numbers. Even though the algebra of the Cayley numbers is not associative, the pro jective space P (V 3 ) can be defined. It remains to consider an odd dimensional division algebra. But in this case, we must have dim V = 1 since on an even dimensional sphere there are no nonvanishing vector fields.

4.3

THE THOM ISOMORPHISM AND DIRECT IMAGE
p : E -X,

Consider a complex n­dimensional vector bundle

denoted by . The total space E is not compact. Denote by E the one­point compactification of E and call it the Thom space of the bund le . This space can be represented a quotient space D( )/S ( ) where D( ) is the subbundle of unit balls and S( ) is subbundle of unit spheres. Then we say that the group K 0 (D( )/S( )) = Kc (E ) is K ­group with compact supports of this noncompact space. This notation can be extended to suspensions. Hence there is a bigraded group 0 1 Kc (E ) = Kc (E ) Kc (E ). Consider the element ( ) K 0 (D( ), S( )) (4.18)

defined in the chapter 3 (3.11). Then the homomorphism given by multiplication (see (3.12 )) : K (X )-Kc (E ) is called the Thom homomorphism.


186

Chapter 4

Theorem 42 The Thom homomorphism is an isomorphism. Pro of. Actually Theorem 42 is a consequence of the Bott periodicity Theorem 27. The Bott periodicity theorem is a particular case of Theorem 42 when the bundle is trivial. To reduce Theorem 42 to the Bott periodicity theorem, it is sufficient to consider a pair of base spaces (Y X ) such that X \Y has only one cell . Then the vector bundle is trivial over and also over . Therefore, we can apply Bott periodicity to the pair (X, Y ) to obtain:
: K (X, Y )-Kc (E , E1 ),

where E1 = p-1 (Y ). Consider the commutative diagram . . . E K (X, Y ) E K (X ) E K (Y ) E ... c . . . E K (E , E ) 1 c c E K (E ) c c E K (E ) 1 c

E ...

By induction, two of the vertical homomorphisms are isomorphisms. Therefore, applying the Five Lemma for isomorphisms, we have the proof of the theorem 42. The Thom isomorphism can be extended to noncompact bases as a isomorphism of K ­groups with compact supports:
: Kc (X )-Kc (E ).

The homomorphism can be associated with the inclusion i : X -E of the base X into the total space E as the zero section. Let us denote the homomorphism by i . Consider another vector bundle over the base X . Then there is a commutative diagram of vector bundles: E
1

'

E2

c X

T p-1 ( ) j c ' EE i


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187

where i and j are inclusions as zero sections. Theorem 43 The inclusions i and j satisfy the fol lowing conditions: 1. (j i) = j i ,
2. i (xi (y )) = i (x)y for any y Kc (E ) and x Kc (X ).

Pro of. Since y = i (z ) = z , we have i (xi ( z )) = i ( )xz , i (x) z = 2 xz . Hence it is suffices to prove that 2 = i ( ) (4.19)

and it is suffices to prove this last equality for the special case when the vector bundle is the one dimensional Hopf bundle over CPn . Then the Thom n space CP is homeomorphic to CPn+1 . In this case, the element is equal to - 1. Let t H 2 (CPn ; Q) and H 2 CPn+1 ; Q be generators. Passing to cohomology via the Chern character in the formula (4.19), we get (ch ) = (ch ) (ch i ( )) . Therefore, we need to prove + 2 n+1 2 n+1 + ... + + + ... + 2! (n + 1)! 2! (n + 1)! 2 n+1 t2 tn = + + ... + t + + ... + 2! (n + 1)! 2! n! = . (4.20)
2

The equality (4.20) follows because i () t
kl

=t =
k+l

.


188

Chapter 4

The Thom isomorphism makes the K ­functor into a covariant functor (but not with respect to all continuous mappings), but rather the K ­functor can be defined as a covariant functor on the category of smooth manifolds and smooth mappings. Theorem 44 For smooth mappings f : X -Y of smooth manifolds, there is a homomorphism f : Kc (T X )-Kc (T Y ) such that 1. (f g ) = f g ,
2. f (xf (y )) = f (x)y for any y Kc (T Y ) and x Kc (T X ).

Pro of. Let : X -V be an inclusion into a Euclidean space of large dimension. Then the mapping f splits into a composition X -Y â V -Y ,
i p

(4.21)

where i(x) = (f (x), (x)) and p is the pro jection onto the first summand. Let U Y â V be a tubular neighborhood of submanifold i(X ). The total space of the tangent bundle T X is a noncompact manifold whose tangent bundle has natural complex structure. Therefore, the inclusion T i : T X -T U is the inclusion as the zero section of the complexification of the normal bundle of the inclusion i : X -U . Let j : T U -T Y â T V be the identity inclusion of an open domain. Let k : T Y -T Y â T V be the zero section inclusion. Definition 10 Define
- f (x)k 1 j i (x), x Kc (T X ), where j : Kc (T U )-Kc (T (Y â V )) is a natural extension of the difference construction from T U to T (Y â V ).


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189

It is clear that if dim V is sufficiently large then the definition 10 is independent of the choice of the inclusion . The property 1 of Theorem 44 follows from the following commutative diagram of inclusions and pro jections: Z âV âW B ¨ d rr ¨¨ (g f , , f¨¨ ) d rrr ¨ rr d ¨ j d E E E E Y âV Z âW X Y Z (g , ) (f , )
Let us prove the property 2 of Theorem 44. If x Kc (T X ), y Kc (T Y ), then - f (xf (y )) = k 1 j i (x (i p y )) - - = k 1 j (i (x)p (y )) k 1 ((j i (x)) p (y )) .

Since k (a) = p (a)k (1), it follows that Hence k j i (x) = p k
-1 -1 j i

(x) k (1). (x) y = f (x)y .

(j i (x)p (y )) k

-1 j i

Remark
Theorem 44 can be generalized to the case of almost complex smooth manifolds, that is, manifolds where the tangent bundle admits a complex structure. In particular, Theorem 44 holds for complex analytic manifolds.

4.4

THE RIEMANN­ROCH THEOREM

The Thom isomorphism considered in the section 4.3 is similar to the Thom isomorphism in cohomology. The latter is the operation of multiplication by the


190

Chapter 4

2 element (1) Hc n (E ; Z) where n = dim is the dimension of the complex vector bundle with total space E .

It is natural to compare these two Thom isomorphisms. Consider the diagram K (X ) ch c H (X ; Q) E
Kc (E )

E

ch c Hc (E ; Q)

(4.22)

The diagram (4.22) is not, in general, commutative. In fact, ch (x) = ch ( x) = ch ch x, ch x = (1)ch x.

(4.23)

Therefore, the left hand sides of (4.22) differ from each of other by the factor T ( ) = -1 (ch ): ch (x) = T ( ) (ch (x)) The class T ( ) is a characteristic class and, therefore, can be expressed in terms of symmetric functions. Let = 1 . . . n be a splitting into one­dimensional summands. Then it is clear that = 1 . . . n , where the k the Bott elements of the one dimensional vector bundles. Similarly, (1) = 1 (1) · . . . · n (1),
where each k : H (X ; Z)-Hc (Ek ; Z) is the Thom isomorphism for the bundle k . Then

T ( ) = -1 (ch ) =

-1 1

(ch 1 ) · . . . · -1 (ch n ) = T (1 ) · . . . · T (n ). n

Hence it is sufficient to calculate T ( ) for a one dimensional vector bundle, particularly for the Hopf bundle over CPn . Let t H 2 (CPn ; Z) be the generator. Then et - 1 . T ( ) = -1 (ch ) = t


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191

Thus for an n­dimensional vector bundle we have
n

T ( ) =
k=1

e

t

k

-1

tk

.

The class T ( ) is a nonhomogeneous cohomology class which begins with 1. The inverse class T -1 ( ) is called the Todd class of the vector bundle . Theorem 45 Let f : X -Y be a smooth mapping of smooth manifolds. Then ch f (x)T Pro of. Let T X -U -T Y â R2n -T Y
i j p -1

(T Y ) = f



ch xT

-1

(T X ) , x Kc (T X ).

be the decomposition induced by (4.21). Then
- ch f (x) = ch k 1 j i (x) = - - = k 1 ch j i (x) = k 1 j ch i (x) = -1 = k j (i ch (x)T ( )) = f (ch (x)T ( )) ,

where is the normal bundle to the inclusion i. Then up to a trivial summand we have = -[T X ] + f [T Y ]. Therefore, T ( ) = T and thus, ch f (x) = f
-1

(T X )f T (T Y ) f T (T Y ) T (T X ) . (4.24)

Examples
1. Let X be a compact almost complex 2n­dimensional manifold, Y a one point space, and f : X -Y be the natural mapping. Then the direct image in cohomology f : H (X ; Z)-H (pt) = Z


192

Chapter 4

maps the group H 2n (X ; Z) = Z isomorphically and the other cohomology groups map to zero. In more detail, we have f (x) = x, [X ] Z. Applying the formula (4.24) we get F (1) K (pt) = Z. Hence ch f (1) is an integer. Therefore, f T -1 (T X ) is also an integer. In other words, T -1 (T X ), [T X ] Z. (4.25) This number is called the Todd genus . The formula (4.25) gives an integer theorem for almost complex manifolds. In general, the Todd genus is rational number, but in the case of an almost complex manifolds the Todd genus is an integer. 2. Assume that the n­dimensional almost complex manifold F has Todd genus T -1 (F ), [F ] = 1. Consider a locally trivial bundle f : E -X with fiber F and structure group of the diffeomorphisms which preserve the almost complex structure. Then ch f (x) = f


ch xT

-1

(T F ) , x Kc (E ),

where T F means the complex bundle over E of tangent vectors to the fibers. In the case n = 4 we have ch f (1) = f T
-1

(T F ) .

The direct image f in cohomology decreases the dimension exactly by 2n. Hence the zero­dimensional component of f T -1 (T F ) equals 1. Therefore, f (1) starts with 1 and hence is invertible. Corollary 7 The homomorphism f : K (X )-K (E ) is a monomorphism onto a direct summand. Now we can continue our study of the homomorphism (3.20) b : R(G)-K (B G).


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193

from the ring of virtual representations of the compact group G. Let T G be a maximal torus. Then there is the commutative diagram R(G) i1 c R(T ) bE K (B G) i2 c K (B T )

b E

The lower horizontal homomorphism b becomes an isomorphism after completion of the rings. The homomorphism i1 becomes a monomorphism onto the subring of virtual representations of the torus T which are invariant with respect to action of the Weyl group of inner automorphisms. The homomorphism i2 also maps the group K (B G) into subring of K (B T ) of elements which are invariant with respect to the action of the Weyl group. It is known that the fiber of i2 , G/T , admits an almost complex structure with unit Todd genus (A.Borel, F.Hirzebruch [12]). Then Corollary 7 says that i2 is a monomorphism. Hence the upper horizontal homomorphism b should be an isomorphism after completion of the rings with respect to the topology induced by inclusion (see M.F.Atiyah, F.Hirzebruch [11]).


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Chapter 4


5
ELLIPTIC OPERATORS ON SMOOTH MANIFOLDS AND K ­THEORY

In this chapter we describe some of the most fruitful applications of vector bundles, namely, in elliptic operator theory. We study some of the geometrical constructions which appear naturally in the analysis of differential and pseudodifferential operators on smooth manifolds.

5.1

SYMBOLS OF PSEUDODIFFERENTIAL OPERATORS

Consider a linear differential operator A which acts on the space of smooth functions of n real variables: A : C (Rn )-C (Rn ). The operator A is a finite linear combination of partial derivatives A=


a (x)

| | , x

(5.1)

where the = (1 , . . . , n ) is multi-index, a (x) are smooth functions and || | | = , 1 )1 ( x2 )2 . . . ( xn )n x ( x || = 1 + . . . + n 195


196

Chapter 5

is the operator given by the partial derivatives. The maximal value of || is called the order of the differential operator differential operator , so the formula (5.1) can be written A=
||m

a (x)

| | , x

Let us introduce a new set of variables = (1 , 2 , . . . , n ). Put a(x, ) =
||m where = 1 1 2 2 · · · n n . The function a(x, ) is called the symbol of a differential operator A. The operator A can be reconstructed from its symbol by substitution of the operators 1 k for the variables k , that is, ix

a (x) i|| ,

A = a x,

1 i x

.

Since the symbol is polynomial with respect to variables , it can be split into homogeneous summands a(x, ) = am (x, ) + a
m-1

(x, ) + . . . + a0 (x. ).

The highest term am (x, x) is called the principal symbol of the operator A while whole symbol sometimes is called the ful l symbol . The reason for singling out the principal symbol is as follows: Prop osition 13 Let yk = y
k

x1 , . . . , x

n

( or 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 . (5.2)


El liptic operators and K ­theory

197

The formula (5.2) shows that 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 smooth 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 does 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 if F and F
-x x-

(u)( ) =

1 (2 )

n/2 Rn

e

-i(x, )

u(x)dx

(v )(x) =

1 (2 )n/

2 Rn

e

i(x, )

v ( )d

are the direct and inverse Fourier transformations then (Au) (x) = F
-x

(a(x, ) (F

x-

(u)( )))

(5.3)

Hence we can enlarge the family of symbols to include some functions which are not polinomials. Namely, suppose a function a defined on the cotangent bundle T M satisfies the condition || | | a(x, ) C x
,

(1 + | |)

m-||

(5.4)

for some constants C, . Denote by S the Schwartz space of functions on Rn which satisfy the condition x


|| u(x) C x

,

for any multiindexes and . Then the operator A defined by formula (5.3) is called a pseudodifferential operator of order m (more exactly, not greater than m). The pseudodifferential operator A acts in the Schwartz space S. This definition of a pseudodifferential operator can be extended to the Schwartz space of functions on an arbitrary compact manifold M . Let {U } be an atlas


198

Chapter 5

of charts with a local coordinate system x = x1 , . . . , xn . Without loss of generality we can assume that the local coordinate system x maps the chart U onto the space Rn . Let = (1 , . . . , n ) be the corresponding components of a cotangent vector. Let { } be a partition of unity subordinate to the atlas of charts, that is, 0 (x) 1,


(x) 1, Supp U .

Finally, let (x) be functions such that Supp U , (x) (x) (x). Then we can define an operator A by the formula A(u)(x) =


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

(5.5)

where A is a pseudodifferential operator on the chart U (which is diffeomorphic to Rn ) with principal symbol a (x , ) = a(x, ). When the function a(x, ) is polynomial (of order m) the operator A defined by formula (5.5) is a differential operator not depending on the choice of functions . In general, the operator A depends on the choice of functions , and the local coordinate system x , uniquely up to the addition of a pseudodifferential operator of order strictly less than m. The next useful generalization consists of a change from functions on the manifold M to smooth sections of vector bundles. Let 1 and 2 be two vector bundles over the manifold M . Consider a linear mapping a : (1 )- 2 , (5.6)

where : T M -M is the natural pro jection. Then in any local coordinate system (x , ) the mapping (5.6) defines a matrix valued function which we require to satisfy the condition (5.4). Then the mapping (5.6) defines a pseudodifferential operator A = a(D) : (1 )- (2 ) by formulas similar to (5.5), again uniquely up to the addition of a pseudodifferential operator of the order less than m. The crucial property of the definition (5.5) is the following


El liptic operators and K ­theory

199

Prop osition 14 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 14 leads to a way of solving equations of the form Au = f (5.7)

for certain pseudodifferential operators A. To find a solution of (5.7), it sufficient to construct a left inverse operator B , that is, B A = 1. If B is the pseudodifferential operator B = b(D) then 1 = b(D)a(D) = c(D) + (b(D)a(A) - c(D)) . Then by 14, the operator c(D) differs from identity by an operator of order -1. Hence the symbol c has the form c(x, ) = 1 + symbol of order (-1). Hence for existence of the left inverse operator B , it is necessary that symbol b satisfies the condition a(x, )b(x, ) = 1 + symbol of order (-1). In particular, the condition (5.8) holds if Condition 1 a(x, ) is invertible for sufficiently large | | C . In fact, if the condition (5.8) holds then we could put b(x, ) = a where (x, ) is a function such that (x, ) (x, ) 1 for | | 2C, 0 for | | C.
-1

(5.8)

(x, )(x, ),


200

Chapter 5

Then the pseudodifferential operator a(D) is called an el liptic holds.

if Condition 1

The final generalization for elliptic operators is the substitution of a sequence of pseudodifferential 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 k -1

a

a

a

(5.9)

be a sequence of symbols of order (m1 , . . . , mk-1 ). Suppose the sequence (5.9) forms a complex, that is, as as-1 = 0. Then the sequence of operators 0- (1 ) - (2 )- . . . - (k )-0
a1 (D )

(5.10)

in general, does not form a complex because we can only know that the composition ak (D)ak-1 (D) is a pseudodifferential operator of the order less then ms + ms-1 . If the sequence of pseudodifferential operators forms a complex and the sequence of symbols (5.9) is exact away from a neighborhood of zero section in T M then the sequence (5.10) is called an el liptic complex of pseudodifferential operators.

5.2

FREDHOLM OPERATORS
F : H -H

A bounded linear operator on a Hilbert space H is called a Fredholm operator if dim Ker F < , dim Coker F < and the image, Im F , is closed. The number index F = dim Ker F - dim Coker F is called the index of the Fredholm operator F . The index can be obtained as index F = dim Ker F - dim Ker F , where F is the adjoint operator. The bounded operator K : H -H is said to be a compact if any bounded subset X H is mapped to a precompact set, that is, the set F (X ) is compact.


El liptic operators and K ­theory

201

Theorem 46 Let F be a Fredholm operator. Then 1. there exists > 0 such that if F - G < then G is a Fredholm operator and index F = index G, 2. if K is compact then F + K is also Fredholm and index (F + K ) = index F. (5.11)

The operator F is Fredholm if and only if there is an operator G such that both K = F G - 1 and K = GF - 1 are compact. If F and G are Fredholm operators then the composition F G is Fredholm and index (F G) = index F + index G. Pro of. Consider first an operator of the form F = 1 + K , where K is compact. Let us show that F is Fredholm. The kernel Ker F consists of vectors x such that x + K x = 0, that is, K x = -x. Then the unit ball B Ker F is bounded and hence compact, since K (B ) = B . Hence dim Ker F is finite. Let us show that Im F is a closed subset. Consider xn Im F , lim xn = x. Then xn = F (yn ) = yn + K yn . Without loss of generality we can assume that yn Ker F. If the set of numbers { y
n

} is unbounded then we can assume that y
n

-.
n n

Then
n-

lim

y y

n n

+K

y y

= lim

n-

x =0 yn

The sequence yn / yn is bounded and by passing to subsequence we can assume that there is a limit yn = z. lim K n- yn


202

Chapter 5

Hence
n-

lim

y y

n n

= -z ,

(5.12)

that is, F ( z ) = z + K z = 0. On the other hand, we have that z is orthogonal to Ker F , that is, z = 0. The latter contradicts (5.12). This means that sequence {yn } is bounded and hence there is a subsequence such that
n-

lim K yn = z .

Therefore,
n-

lim yn = lim (xn - K yn ) = x - z .
n-

Hence x = lim xn = lim F (yn ) = F (x - z ),
n- n-

that is, x Im F . Thus Im F is closed and hence Coker F = Ker F . The operator F = 1 + K has a finite dimensional kernel and, therefore, dim Coker F = dim Ker F < . This means that F is Fredholm. Assume that for compact operators K and K F G = 1 + K, GF = 1 + K . Then we have Ker F Ker (1 + K ), that is, dim Ker F < . Similarly, we have Im F Im (1 + K ). Hence the image Im F is closed and has finite codimension. This means that both F and G are Fredholm. Suppose again that F is a Fredholm operator. Consider two splittings of the space H into direct sums: H H = Ker F (Ker F ) = (Im F ) Im F.


(5.13)


El liptic operators and K ­theory

203

Then the operator F can be represented as a matrix with respect to the splittings (5.13): 00 , (5.14) F= 0 F1 where F1 is an isomorphism. We define a new operator G by the following matrix 0 0 , G= - 0 F1 1 Then the operators K = FG - 1 = 10 00 and K = GF - 1 = 10 00

are finite dimensional and hence compact. Let > 0 be such that G < 1 . Suppose that the operator is such that < and F1 = F + . Notice that both operators 1 + G and 1 + G are invertible. Then (1 + G)-1 GF F1 G(1 + G)-
1 1

= (1 + G)-1 (1 + D + K ) = 1 + (1 + G)-1 K, = (1 + G + K )(1 + G)-1 = 1 + K (1 + G)-1 .

Since operators (1 + G)-1 K and K (1 + G)-1 are compact, it follows that the operator F is Fredholm. Let us estimate the index of the Fredholm operator. If the operator F has matrix form F 0 F= 1 (5.15) 0 F2 then, evidently, index F = index F1 + index F2 . If both Hilbert spaces H1 and H2 are finite dimensional and F : H1 -H2 is bounded then F is Fredholm and index F = dim H1 + dim H2 . If F is invertible then index F = 0. If in the operator (5.15), F1 = 0 is finite dimensional and F2 is invertible then F is Fredholm and index F = 0.


204

Chapter 5

Hence, by adding a compact summand we can change the Fredholm operator so that the index does not change and a given finite dimensional subspace is included in the kernel of the resulting operator. Suppose that F has the splitting (5.14) and let G be another operator such that F - G < . Let 12 G = 11 , (5.16) 21 F2 + 22 where 22 < . For sufficiently small the operator F2 + 22 is invertible. Then for X = -(F2 + 22 )-1 21 we have G= 1 X 0 0
-1

11 0 ,

12 . F2 + 22

Similarly, for Y = -12 (F2 + 22 ) G= 1 X 0 1

11 0

0 F2 + 22

1Y . 01

Thus index G = index F. Now let us prove (5.11). We can represent the operators F and K as a matrices F= F1 0 0 K ;K= F2 K
22 11 21

K12 K22

such that F1 is finite dimensional and K situation of the condition (5.16). Hence

is sufficiently small. Then we are in

index (F + K ) = index F.

The notion of a Fredholm operator has an interpretation in terms of the finite dimensional 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.

d

d

dn

-1

(5.17)
-1

We say that the sequence (5.17) is Fredholm complex if dk dk closed subspace and dim (Ker dk /Coker dk
-1

0, Im dk is a

) = dim H (Ck , dk ) < .


El liptic operators and K ­theory

205

Then the index of Fredholm complex (5.17) is defined by the following formula: index (C, d) =
k

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

Theorem 47 Let
0 1 0-C0 -C1 - . . . - Cn -0

d

d

d

n-1

(5.18) is compact. Then the

be a sequence satisfying the condition that each dk dk fol lowing conditions are equivalent: 1. There exist operators fk : Ck -C where each rk is compact.

-1

k -1

such that f

k+1 dk

+ dk

-1 fk

= 1+r

k

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 . We leave the proof to the reader. Theorem 47 allows us to generalize the notion of a Fredholm complex using one of the equivalent conditions from Theorem 47.

Exercise
Consider the Fredholm complex (5.18). Form the new short complex C C
ev

=
k

C2k , C
k 2k+1

odd

= d0 0 0

, 0 d 3 d4 . . . . . . . . . . . . 0 0 0

D

=

d 1 d2 0

(5.19)

...

Prove that the complex (5.19) is Fredholm with the same index.


206

Chapter 5

5.3

THE SOBOLEV NORMS

Consider the Schwartz space S. Define the Sobolev norm by the formula u where =
k=1 2 s

=
Rn

u(x)(1 + )s u(x)dx, ¯

n

i

x

2 k

is the Laplace operator. Using the Fourier transformation this becomes u
2 s

=
Rn

(1 + | |2 )s |u( )|2 d ^

(5.20)

where the index s need not be an integer. The Sobolev space Hs (Rn ) is the completion of S with respect to the Sobolev norm (5.20). Prop osition 15 The Sobolev norms (5.20) for different coordinate systems are al l equivalent on any compact K Rn , that is, there are two constants C1 > 0 and C2 > 0 such that for Supp u K u
1,s

C1 u

2,s

C2 u

1,s

.

Proposition 15 can easily be checked when s is an integer. For arbitrary s we leave it to the reader. This proposition allows us to generalize the Sobolev spaces to an arbitrary compact manifold M and vector bundle . Let {U } be an atlas of charts where the vector bundle is trivial over each U . Let { } be a partition of unity subordinate to the atlas. Let u (M , ) be a section. Put u 2 . (5.21) u 2= s s


Proposition 15 says that the definition (5.21) defines a Sobolev norm, well defined up to equivalent norms. Hence the completion of the space of sections (M , ) does not depend on the choice of partition of unity or the choice of local coordinate system in each chart U . We shall denote this completion by Hs (M , ).


El liptic operators and K ­theory

207

Theorem 48 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 48 is called the Sobolev inclusion theorem. Pro of. For the proof it is sufficient to work in a local chart since any section u (M , ) can be split into a sum u=


(5.22)

u,

where each summand u has support in the chart U . So let u be a function defined on Rn with support in the unit cube In . These functions can be considered as functions on the torus Tn . Then a function u can be expanded in a convergent Fourier series u(x) = al ei(l,x) .
l

Partial differentiation transforms to multiplication of each coefficient al by the number lk , where l is a multiindex l = (l1 , . . . , ln ). Therefore, u
2 s

=
l

|al |

2

1 + |l1 |2 + . . . + |ln |

2s

.

(5.23)

By formula (5.23), the space Hs (Tn ) is isomorphic to the Hilbert space l2 of the square summable sequences by the correspondence u(x) bl = a
2 l s/2

(1 + |l1 | + . . . + |ln |2 )

Then the inclusion (5.22) becomes an operator l2 -l2 defined by the correspondence bl {bl } (s -s )/2 (1 + |l1 |2 + . . . + |ln |2 ) 2 1 which is clearly compact.


208

Chapter 5

Theorem 49 Let

a(D) : (M , 1 )- (M , 2 )

(5.24)

be a pseudodifferential operator of order m. Then there is a constant C such that a(D)u s-m C u s , (5.25) that is, the operator a(D) can be extended to a bounded operator on Sobolev spaces: a(D) : Hs (M , 1 )-Hs-m (M , 2 ). (5.26)

Pro of. The theorem is clear for differential operators. Indeed, it is sufficient to obtain estimates in a chart, as the symbol has compact support. If a(D) = j then x u(x) xj =
Rn 2 s

=
R
n

1 + | |
2s

2s

u d = xj 1 + | |
Rn 2 s+1

2

1 + | |
2 s+1

|j u( )|2 d ^

|u( )|2 d = ^ (5.27)

= u(x)

.

Hence the inequality (5.25) follows from (5.27) by induction. For pseudodifferential operators, the required inequality can also be obtained locally using a more complicated technique. Using theorems 48 and 49 it can be shown that an elliptic operator is Fredholm for appropriate choices of Sobolev spaces. Theorem 50 Let a(D) be an el liptic pseudodifferential operator of order m as in (5.24). Then its extension (5.26) is Fredholm. The index of the operator (5.26) is independent of the choice of the number s. Pro of.


El liptic operators and K ­theory

209

As in section 5.1 we can construct a new symbol b of order -m such that both a(D)b(D) - 1 and b(D)a(D) - 1 are pseudodifferential operators of order -1. Hence by Theorem 46, a(D) gives a Fredholm operator (5.26). To prove that the index of a(D) does not depend of the number s, consider the k special operator (1 + )k with symbol 1 + | |2 . Since the norm (1 + )k u s is equivalent to the norm u s+2k , the operator (1 + )k : H
s+2k

( )-Hs ( )

is an isomorphism. Then the operator A = (1 + )-k (D)(1 + )k : Hs+2k (1 )-Hs (1 )- -Hs-m (2 )-Hs+2k-m (2 ) differs from (D) by a compact operator and therefore has the same index.

Corollary 8 The kernel of an el liptic operator (D) consists of infinitely smooth sections. Pro of. Indeed, by increasing the number s we have a commutative diagram (D) EH Hs (1 ) s-m (2 ) T H
s+1

T (D) EH
s+1-m

(1 )

(2 )

(5.28)

An increase in the number s can only decrease the dimension of the kernel, Ker (D). Similarly, the dimension of the cokernel may only decrease since the cokernel is isomorphic to the kernel of the adjoint operator (D) . Since the index does not change, the dimension of kernel cannot change. Hence the kernel does not change in the diagram (5.28). Thus the kernel belongs to the Hs (1 ) = (1 ).
s


210

Chapter 5

5.4

THE ATIYAH­SINGER FORMULA FOR THE INDEX OF AN ELLIPTIC OPERATOR

In the sense explained in previous sections, an elliptic operator (D) is defined by a symbol : (1 )- (2 ) (5.29) which is an isomorphism away from a neighborhood of the zero section of the cotangent bundle T M . Since M is a compact manifold, the symbol (5.29) defines a triple ( (1 ), , (2 )) which in turn defines an element [ ] Kc (T M ). Theorem 51 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. Pro of. A homotopy of an elliptic symbol gives a homotopy of Fredholm operators and, under homotopy, the index does not change. Assume that the symbol is an isomorphism not only away from zero section but everywhere. The operator (D) can be decomposed into a composition of an invertible operator (1 + )m/2 and a operator 1 (D) of the order 0. The symbol 1 is again invertible everywhere and is therefore homotopic to a symbol 2 which is independent of the cotangent vector and also invertible everywhere. Then the operator 2 (D) is multiplication by the invertible function 2 . Therefore, 2 (D) is invertible. Thus index (D) = 0. Finally, if = 1 2 then (D) = 1 (D) 2 (D) and hence index (D) = index 1 (D) + index 2 (D).


El liptic operators and K ­theory

211

The total space of the cotangent bundle T M has a natural almost complex structure. Therefore, the trivial mapping p : M -pt induces the direct image homomorphism p : Kc (T M )-Kc (pt) = Z, as described in Section 4.3 Theorem 52 Let (D) be an el liptic pseudodifferential operator. Then index (D) = p [ ]. (5.30)

Using the Riemann-Roch theorem 45, the formula (5.30) can be written as index (D) = ch [ ]T
-1

(T M ), [T M ]

where [T M ] is the fundamental (open) cycle of the manifold T M . For an oriented manifold M we have the Thom isomorphism
: H (M )-Hc (T M ).

(5.31)

Therefore, the formula (5.31) has the form index (D) =
-1

ch [ ]T

-1

(T M ), [M ]

(5.32)

The formula (5.32) was proved by M.F.Atiyah and I.M.Singer [9] and is known as the Atiyah­Singer formula. Pro of. The proof of the Atiyah­Singer formula is technically complicated. Known proofs are based on studying the algebraic and topological properties of both the left and right hand sides of (5.30). In particular, Theorem 51 shows that both the left and right hand sides of the (5.30) are homomorphisms on the group Kc (T M ). Let j : M1 -M2 be a smooth inclusion of compact manifolds and let 1 be an elliptic symbol on the manifold M1 . Assume that 2 is a symbol on the manifold M2 such that [2 ] = j [1 ]. (5.33)


212

Chapter 5

Theorem 53 If two symbols 1 and 2 satisfy the condition (5.33) then the el liptic operators 1 (D) and 2 (D) have the same index, index 1 (D) = index 2 (D). Theorem 52 follows from Theorem 53. In fact, Theorem 52 holds for M = pt. Then the inclusion j : pt-Sn . gives the direct image j : Kc (pt)-K
c T Sn , Ts0

which is an isomorphism.Therefore, for a symbol defining a class [ ] K we have [ ] = j [ ], [ ] = p [ ], index (D) = index (D) = [ ] = p ([ ]). Finally, let j : M -Sn an inclusion and q : Sn -pt be the natural pro jection. Then p = q j, j ([ ]) = [ ], and by Theorem 53 index (D) = index (D) = q [ ] = q j [ ]p [ ]. Thus the Atiyah­Singer formula follows from Theorem 53. Let us explain Theorem 53 for the example when the normal bundle of the inclusion M1 -M2 is trivial. The symbol [2 ] = j [1 ] is invertible outside of a neighborhood of submanifold M1 T M2 . Suppose that the symbols are of order 0. Then the symbol 2 can be chosen so that it is independent of the cotangent vector outside of a neighborhood U of submanifold M1 . Then we can chose the operator 2 (D) such that if Supp U = then 2 (D)(u) is multiplication by a function. Hence we can substitute for the manifold M2 , the Cartesian product M1 â T k , where T k is torus, equipped with an elliptic
c T Sn , Ts0


El liptic operators and K ­theory

213

operator of the same index. Now we can use induction with respect to the integer k and that means that it is sufficient consider the case when k = 1. The problem is now reduced to the existence of an elliptic operator (D) on the circle such that index (D) = 1 and [ ] K is a generator.
c

T S1 , T

s0


214

Chapter 5


6
SOME APPLICATIONS OF VECTOR BUNDLE THEORY

Here we describe some problems where vector bundles appear in a natural way. We do not pretend to give a complete list but rather they correspond to the author's interest. Nevertheless, we hope that these examples will demonstrate the usefulness of vector bundle theory as a geometric technique.

6.1

SIGNATURES OF MANIFOLDS

The signature of a compact oriented 4n-dimensional manifold X is the signature of a quadratic form on the space H 2n (X ; Q) of 2n­dimensional rational cohomology given by the formula (x, y ) = xy , [X ] , (6.1)

where x, y H 2n (X ; Q), [X ] is fundamental 4n­dimensional cycle and , is the natural value of a cocycle on a cycle. By Poincar´ duality we see that the quadratic form (6.1) is nondegenerate. This e form can be reduced to a form with a diagonal matrix with respect to a basis of the vector space H 2n (X ; Q). Notice that the signature of the quadratic form is the same as the signature of a similar quadratic form on the space H 2n (X ; R). This allows us to describe the cohomology groups using de Rham differential forms. The notion of signature can be generalized to a bilinear form on the whole cohomology group H (X ) =
4n k=0

H k (X ) using the same definition (6.1). This 215


216

Chapter 6

is also a nondegenerate form but it is not symmetric. Nevertheless, consider the splitting H (X ) = H ev (X ) H odd (X ). Then the bilinear form (6.1) splits into a direct sum of a symmetric form on H ev (X ) and a skewsymmetric form on H odd (X ). Let us agree that the signature of a skewsymmetric form equals zero. Then the signature of the manifold X is equal to the signature of the bilinear form (6.1) on the H (X ). In fact, the form (6.1) decomposes into a direct sum of the subspaces H (X ) = H
2 odd

(X ) H 0 (X ) H
4n-2

4n

(X )
2n-2

H (X ) H

(X ) . . . H

(X ) H

2n+2

(X ) H

2n

(X ).

The matrix of the bilinear form then has the form 0 At A 0 (6.2)

on each summand except the first and last. The signature of the matrix (6.2) is trivial. Hence the signature of the form on the H coincides with the signature on the H 2n (X ). Theorem 54 If a compact oriented 4n­dimensional manifold X is the boundary of an oriented compact 4n + 1­dimensional manifold Y then sign X = 0. Pro of. The homology group (with rational or real coefficients) can be thought of as Hk (X ) = hom H k (X ); R . Then the bilinear form (6.1) can be considered as an isomorphism D : H k (X )-H
4n-k

(X )

coinciding with ­product with the fundamental cycle [X ] H4n (X ) inducing Poincar´ duality. The manifold Y also has a Poincar´ duality induced by ­ e e product with the fundamental cycle [Y ] H4k+1 (Y , Y ): D = [Y ] : D = [Y ] : H k (Y )-H
k 4n+1-k

(Y , X ), (Y ).

H (Y , X )-H

4n+1-k


Applications of vector bund le theory

217

The exact homology sequence for the pair (Y , X ) fits into a commutative diagram j i E ... E Ek E E ... H k (Y , X ) H (Y ) H k (X ) H k+1 (Y , X ) D c j . . . EH EH 4n+1-k (Y ) D c E H n+1-k (Y , X ) D c n-k (X ) D c i EH 4n-k (Y )

4

4

E ... (6.3)

Let k = 2n. Then the dimension of the image Im i H 2n (X ) can be calculated as an alternating sum of dimensions of all spaces which lie on the left side of upper line of the diagram (6.3): dim Im i = dim H 2n (Y ) - dim H 2n (Y , X ) + dim H 2n-1 (X ) - - dim H 2n-1 (Y ) + dim H 2n-1 (Y , X ) - dim H 2n-2 (X ) + . . . Using the right lower part of (6.3) we have codim Im i = codim Im = codim Ker i = = dim H2n (Y ) - dim H2n (Y , X ) + dim H2n-1 (X ) - - dim H2n-1 (Y ) + dim H2n-1 (Y , X ) - dim H2n-2 (X ) + . . . Since dim H k (X ) = dim Hk (X ), dim Im i = codim Im i = Then the group H H
2n

1 dim H 2

2n

(X ).

(X ) decomposes into two summands: (X ) = A B , A = Im i , B = (Ker i )


2n

By commutativity of the diagram, D(A) B and hence D= 0 D3 D D
2 4 t , D3 = D2 .

Thus the signature of D is trivial. Theorem 54 shows that the signature of the manifold X depends only on the bordism class and hence can be expressed in terms of characteristic classes of


218

Chapter 6

the manifold X . This means that there should exist a polynomial L(X ) of the Pontryagin classes such that sign X = L(X ), [X ] . This class L(X ) is called the Hirzebruch­Pontryagin class.

6.1.1

Calculation of the Hirzebruch­Pontryagin class

The Hirzebruch­Pontryagin class and the signature can be given an interpretation as the index of an elliptic operator on the manifold X . Such operator exists and is called the Hirzebruch operator. To construct the Hirzebruch operator, consider the de Rham complex of differential forms on the manifold X : 0-0 -1 - . . . -4n -0
d d d

(6.4)

The space k is the space of sections of the vector bundle k over X , the exterior power of the cotangent bundle T X = 1 . The complex (6.4) is an elliptic differential complex for which the symbols are given by exterior multiplication by the cotangent vector T X . Notice that the signature of the manifold X is defined in terms of multiplication of differential forms (1 , 2 ) = 1 2 .
X

Fix a Riemannian metric g on the manifold X . The metric g induces a scalar product on the vector bundles k . Let e1 , . . . , e4k be an orthogonal basis in the fiber of T X . Then the corresponding orthogonal basis in the fiber of k consists of the vectors ei1 . . . eik . Consider the mapping : k - which to each basis vector ei1 . ej4n-k where the ejl complement basis and is the parity of the 2 = (-1)k on k . Thus we have
4n-k

,

. . eik associates the vector (-1) ej1 . . . the system of vectors ei1 , . . . , eik to form the permutation (i1 , . . . , ik , j1 , . . . , j4n-k ). Then a homomorphism
4n-k

: (k ) - (

).


Applications of vector bund le theory

219

Let µ be the differential form which defines the measure on the manifold X induced by the Riemannian metric. In terms of the local coordinate system, µ= det gij dx1 . . . dx4n .

It is clear that µ = (1) and for any function f we have (f ) = f ( ). Denote by 1 , 2 the function whose value is equal to the scalar product of the forms 1 and 2 . Then 1 , 2 µ = 1 2 = (-1)k 1 2 . (6.5)

In the space
k

k we have the positive scalar product 1 , = 1 , 2 µ. (6.6)

2

Then taking into account (6.5), (1 , 2 ) = 1 , 2 for 1 (k ), 2 4
n-k

(6.7)

).

Let d denote the adjoint operator with respect to the scalar product (6.6). This means that 1 , d2 d 1 , 2 . Using (6.7), d 1 ,
2

= 1 , d
k X

2

=
X

1 d2 = d(1 2 ) -
X X

(-1) -
X

1 2 =

d 1 2 =

d 1 2 = -
X

d 1 2 = - d 1 , 2 .

Hence

d = - d .

(6.8)


220

Chapter 6

Now we can identify the de Rham cohomology with the kernel of the Laplace operator = (d + d )2 = d d + dd . Note that the Laplace operator commutes with both d and d . Theorem 55 The operator D = d + d is an el liptic differential operator and in the space () Ker d = Ker D Im d. Pro of. The first statement of Theorem 55 follows directly from the previous chapter since the symbol of the operator D is an invertible matrix away from zero section of the cotangent bundle. Let


(k ). Then Hs ( ). Decompose the element into a sum = 1 + 2 ,

where On the other hand, (D (Hs Hence that is, 2 = D(), H Let s > s and
s+1 +1

1 Ker D Hs ( ) , 2 (Ker D) . ( ))) = Ker D
s+1



(Ker D) = D (H

( )) , ( ) .

= 1 + 2 , 1 Ker D Hs ( ) , 2 (Ker D) . Then 1 - 1 = 2 - 2 (Ker D) . Hence 1 = 1 , 2 = 2 and 2 ( ). If Ker d then = 1 + D2 , 1 Ker D, that is, 0 = d = d1 + dD2 .





Applications of vector bund le theory

221

Hence Thus

0 = dD2 = Dd 2 , dd 2 = 0, d 2 = 0. = 1 + d 2 .

Theorem 55 says that cohomology group H k (X ) is isomorphic to Ker D in the space (k ). Notice that by formula (6.8), the operator preserves the kernel Ker D. Consider a new operator = ik Then
(k-1)+2n

: k -

4n-k

.

2 = 1, d = -d, D = -D. = +
-

Thus there is a splitting

into the two eigen subspaces corresponding to the eigenvalues +1 and -1 of the operator . Then D D Hence where D
+



+ -







- +

, .





Ker D = Ker D+ Ker D- , and D
-

are restrictions of D to



(+ ) and



(- ). Thus

sign X = dim Ker D+ - dim Ker D- . Since the operator D is selfadjoint, the operator D D- . Hence sign X = index D+ .
+

is adjoint to the operator

The operator D+ is called the Hirzebruch operator. To calculate the index of the Hirzebruch operator, let be the symbol of the Hirzebruch operator. The symbol is a homomorphism :


c+ -



c- ,

(6.9)


222

Chapter 6

which is an isomorphism away from the zero section of the cotangent bundle. Hence index D+ = ch []T (X ), [T X ] , where T (X ) is the Todd class of the complexification of the tangent bundle. The construction of the element [] defined by the triple (6.9) can be extended to arbitrary orientable real bundles over the base X . Indeed, let p : E -X be the pro jection on the total space of the bundle . Let 0-p 0 (c )-p 1 (c )- . . . -p n (c )-0 be the complex of vector bundles where is the exterior multiplication by the vector E . Similar to the tangent bundle, the bundle ( ) = k ( ) has an operation : k ( )-n-k ( ) which induces a splitting (c ) = + ( ) - ( ) and a homomorphism = - mapping : + ( )-- ( ) (6.10)


isomorphically away from the zero section of the total space E . Hence the triple (6.10) defines an element [] Kc (E ). Notice that if X = X1 â X2 and = 1 2 , where k are vector bundles over Xk , then [( )] = [1 ] [2 ]. Then the problem can be reduced to the universal two dimensional vector bundle over BSO(2) = BU(1) = CP . In this case, the bundle is isomorphic to the complex Hopf bundle and c = . Hence + ( ) = 1 ( ) = 1.
-

Then the symbol : + ( )-- ( ) has the following form = z 0 0 , z


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223

where z is a vector in the fiber of the bundle . Hence [] = ([ ] - 1) (1 - [ ]) = [ ] - [ ], where is the Hopf bundle over the Thom complex ch [] = e - e
- CP


= CP . Hence

, H2 (CP ; Z) .

Thus if the bundle is decomposed into a direct sum of two dimensional bundles = 1 . . . n , then ch (( )) =
k=1 n n

e
X

k

- e-



k

=
k=1

ek - e- k



k

,

where : H (X )-H

c

is the Thom isomorphism. Then
n

-1 (ch ()T (X )) =
k=1

e

(-k ) e k k - 1 e-k - 1
n



k

-a k

-

k

=

=2

n k=1

k /2 . tanh(k /2)

Thus for a 4n­dimensional manifold X ,
n

L(X ) = 2

2n k=1

tk /2 , tanh(tk /2)

where the tk are formal generators for the tangent bundle of the manifold X .

6.2

C ­ALGEBRAS AND K ­THEORY

The space ( ) of sections of a vector bundle over a base B is a module over the ring C(B ) of continuous functions on the base B . The ring C(B ) also can be considered as a section space (1) of the one dimensional trivial vector bundle 1. Hence (1) is a one dimensional free module. Hence for an arbitrary vector bundle , the module ( ) is a finitely generated pro jective module. Theorem 56 The correspondence ( ) is an equivalence between the category of complex vector bund les over a compact base X and the category of finitely generated projective modules over the algebra C(B ).


224

Chapter 6

Pro of. If two vector bundles and are isomorphic then the modules ( ) and ( ) of sections are isomorphic. Therefore, it is sufficient to prove that any finitely generated pro jective module P over the C ­algebra C (B ) is isomorphic to a module ( ) for a vector bundle , and any homomorphism : ( )-( ) is induced by a homomorphism of bundles f : - . We start by proving the second statement. Let (6.11) be a homomorphism of modules. Assume first that both vector bundles and are trivial, that is, = B â Cn , = B â Cm . Let a1 , . . . , an Cn b1 , . . . , bm Cm be bases. The vectors ak , bk induces constant sections ak ( ), ¯k ( ). ¯ b Hence any sections ( ), ( ) decompose into linear combinations =
k

(6.11)

k (x)ak , ¯ =
l

k (x) C (B ), l (x) C (B ).

l¯l , b

Therefore, the homomorphism has the following form ( ) =
k

k (x)(ak ) = ¯
k

k (x)
l

l (x)¯l b k

=

=
k,l

k (x)l (x)¯l . b k

The matrix valued function l (x) defines a fiberwise homomorphism of bundles k f = l (x) : - . k (6.12)


Applications of vector bund le theory

225

Clearly, the homomorphism (6.12) induces the initial homomorphism (6.11). Now let and be arbitrary vector bundles. Complement them to form trivial bundles, that is, 1 = n, 1 = m. ¯ ¯ Let p : n-n, q : m-m ¯ ¯ ¯ ¯ be pro jections onto the first summands and , respectively. Define a new homomorphism : (n)-(m) ¯ ¯ by the matrix = It is clear that = , where : (n) and : (m) are homomorphisms induced be p and q . We see ¯ ¯ that is induced by a homomorphism of trivial vector bundles g : n-m. ¯ ¯ Hence the homomorphism g has the matrix form g= f 0 0 , 0 0 0 . 0

where f = q g p is the homomorphism from the bundle to the vector bundle which induces the homomorphism (6.11). Now we prove the first statement. Let P be a finitely generated pro jective module over the algebra C (B ). By definition the module P is the direct summand of the free module C (B )n = (n) ¯ P Q = (n). ¯ The module P is the image of a pro jection : (n)-(n), 2 = , P = Im . ¯ ¯ The pro jection , a homomorphism of modules, is induced by a homomorphism g of the trivial vector bundle n. Hence g 2 = g , that is, g is a fiberwise pro jection. ¯ Therefore, the image of g is a locally trivial vector bundle, n = , = Im g . ¯


226

Chapter 6

Thus P = ( ). Theorem 56 says that the group K (B ) is naturally isomorphic to the group K0 (C (B )) in the algebraic K ­theory for the ring C (B ) : K (B )-K0 (C (B )). (6.13)

We can generalize the situation considered in Theorem 56 to the case of a Cartesian product B = X â Y . Let be a vector bundle over a base B . Fix a point y0 Y . Then the section space ( , (X â y0 )) is a pro jective module over the ring C (X ). Thus we have a family of pro jective modules parametrized by points of Y . This leads us to the definition of a vector bundle over a base Y with fiber a pro jective module P = ( , (X â y0 )) and structure group the group of automorphisms of the module P . More generally, let A be an arbitrary C ­algebra and P a pro jective A­module. Then a locally trivial bundle over the base Y with the structure group G = AutA (P ) will be called a vector A­bund le. The corresponding Grothendieck group will be denoted by KA (X ). If X = pt is a one point space then the group KA (pt) coincides with K0 (A) in algebraic K ­theory. If B = X â Y then there is a natural isomorphism K (X â Y )-K
C (Y )

(X ).

(6.14)

If Y is a one point space then the isomorphism (6.14) coincides with the isomorphism (6.13) from Theorem 56. Theorem 56 is a vector module over in A. Hence can also be generalized to the case of vector A­bundles. Namely, if A­bundle over a base Y , then the section space ( ) is a pro jective the algebra C (Y , A) of continuous functions on the Y with values there is a natural isomorphism KA (Y )-K0 (C (Y , A)). When B = X â Y , C (X â Y ) = C (Y , C (X )) and there is a commutative diagram K (X Y ) â K
C (X )

-

K0 (C (X â Y ))

(Y )

- K0 (C (Y , C (X )))


Applications of vector bund le theory

227

The groups KA (X ) satisfy many properties similar to those of classical complex K ­theory. The tensor product induces a natural homomorphism KA (X ) K (Y )-KA (X â Y ). In particular, multiplication by the Bott element K 0 (S 2 , s0 ) defines an isomorphism KA (X, Y )-KA (X â D2 , (X â S 1 ) (Y â D2 )), and an isomorphism
n KA (X, Y )-K n-2 A

(X, Y ).

Consider the homomorphism : KA (pt) K (X )-KA (X ), induced by tensor multiplication. After tensoring with the rational number field Q, there is an isomorphism Define ch
A -1 : KA (X ) Q = K (X ) KA (pt) Q.

= (Id ch )

-1

: KA (X )-H (X ; KA (pt) Q)

(6.15)

The homomorphism (6.15) is an analogue of the Chern character and fits into the following commutative diagram
KA (X ) K (X ) - KAX ) ( ch A ch ch A H (X ; KA (pt) Q) H (X ; Q) - H (X ; KA (pt) Q)

Example
The Calkin algebra. The Calkin algebra is a quotient algebra C (H) = B (H )/Comp(H ) where B (H ) is the algebra of bounded operators on the Hilbert space H and Comp(H ) is the ideal of compact operators. To study the group KC (H) (X ) it is sufficient to consider locally trivial bundles with free C (H)­module as fibers. In this case the


228

Chapter 6

structure group is GL(n, C (H)) -- the group of invertible matrices with entries n in C (H). The algebra End(C (H) ) of n­dimensional matrices is isomorphic to the algebra C (H) since the infinite dimensional Hilbert space H is isomorphic to the direct sum of n copies of H . Hence GL(n, C (H)) GL(1, C (H)) = GL(C (H)). Let : B (H )-C (H) be the natural pro jection. Then the inverse image with the set F of Fredholm operators. The pro jection : F -GL(C (H)) is a fibration with the fiber Comp(H ) which satisfies the covering homotopy axiom. The space F of Fredholm operators is the union of an infinite number of connected components Fk , F=
k -1

(GL(C (H)) coincides

where Fk is the space of Fredholm operators of index = k . Hence the group GL(C (H)) also splits into a union of connected components GL(C (H)) =
k

GL(C (H))k .

Consider the group GL(H ) of all invertible operators on the Hilbert space H and the map : GL(H )-GL(C (H))0 . This map is clearly an isomorphism. The kernel consists of invertible operators of the form 1 + K where K is compact and is denoted by 1 + Comp. The group 1 + Comp is the closure of the invertible finite dimensional operators. Each finite dimensional invertible matrix F can be extended to an operator ~ F= Hence there is an inclusion GL(n, C) 1 + Comp, F 0 0 1 1 + Comp.


Applications of vector bund le theory

229

and an inclusion of the direct limit GL(, C) 1 + Comp. (6.16)

The inclusion (6.16) is a weak homotopy equivalence. Hence classical K ­theory can be represented by locally trivial bundles with the structure group 1 + Comp. Theorem 57 There is a natural isomorphism f :K Pro of. Consider the suspension of X as a union of two cones over X . Then any vector bundle over the algebra C (H) can be described by two charts -- two copies of the cone. Hence the vector bundle is defined by a single transition function F : X -F0 . Let {U } be an atlas of sufficiently small charts on X . Then there are functions : U -Comp(H ) (6.17)
-1 C (H)

(X, x0 )-K 0 (X, x0 ).

such that for any and x U the operator F (x) + (x) is invertible. Put (x) = (F (x) + (x)) (F (x) + (x))
-1

1 + Comp.

(6.18)

The functions (6.18) define transition functions with values in the structure group 1 + Comp. Hence the transition functions (6.18) define an element of K 0 (X ). This element is independent of the choice of functions (6.17). Therefore, we have a well defined homomorphism f :K
-1 C (H)

(X, x0 )-K 0 (X, x0 ).

(6.19)

The homomorphism (6.19) is a monomorphism. Indeed, if the transition functions (6.18) define a trivial bundle then there are functions u : U -1 + Comp such that Using (6.18), u-1 (x) (F (x) + (x)) = u-1 (x) (F (x) + (x)) = (x). (6.20)


(x) = u

-1

(x)u (x).


230

Chapter 6

Hence the function defined by (6.20) is invertible and differs from F (x) by a compact summand. Since the group GL(H ) is contractible, there is a homotopy (x, t), x X, t [0, 1] such that (x, 1) = (x), (x, 0) 1. (6.21)

Thus the function (6.21) is defined on the cone on X and can serve on the boundary as the transition function on the suspension S X . Now let us prove that (6.19) is surjective. Consider a system of the transition functions : U -1 + Comp. Using the contractibility of the group GL(H ), we can find a system of functions h : U -1 + Comp such that This means that h (x) = h (x) + Comp. Hence there is a Fredholm operator valued function F (x) such that F (x) - h (x) Comp.


(x) = h (x)h

-1

(x).

There is another interpretation of the homomorphism (6.19). Theorem 58 For any continuous map F : X -F there is a homotopic map F : X -F such that dim Ker F (x) Const, dim Coker F (x) Const. The unions Ker F (x) and Coker F (x) form local ly trivial vector bund les [Ker F ] and [Coker F ]. The element index F = [Ker F ] - [Coker F ] K (X ) is independent of the choice of F . If index F = 0 then the map index : [X, F ]-K (X ) defined by (6.22) coincides with (6.19). (6.22)


Applications of vector bund le theory

231

6.3

FAMILIES OF ELLIPTIC OPERATORS
Fx : H -H, x X,

Consider a continuous family of Fredholm operators

parametrized by a compact space X . Then index Fx is a locally constant function and hence gives no new invariant. But if dim ker Fx const, the kernels ker Fx form a vector bundle Ker F . Similarly, Coker F is a vector bundle. In this case, the index of the family of Fredholm operators may be interpreted as an element def index F = [Ker F ] - [Coker F ] K (X ). Clearly, the classical index gives dim index F = dim[Ker F ] - dim[Coker F ]. The generalized index of a family of elliptic operators x (D), index (D) K (X ) can be expressed in terms of the symbol of the operator (D). Let p : M -X
be a locally trivial bundle for which fiber is a manifold Y . Let TY M be the vector bundle annihilating the tangent vectors to the fibers and : TY M -M

be the correspondent pro jection. Let 1 , 2 be two vector bundles over M . Consider the infinite dimensional vector bundles p! (1 ) and p! (2 ) over X with fibers 1 , p-1 (x) and 2 , p-1 (x) , respectively, over the point x X . Notice that p-1 (x) is diffeomorphic to manifold Y . Let p! (i )s denote the bundle associated to p! (i ) by substituting for the fiber i , p-1 (x) its completion by s-Sobolev norm. Then a homomorphism of bundles (D) : p! (1 )-p! (2 ) (6.23) is said to be a family of pseudodifferential operators if at each point x X the homomorphism of fibers is a pseudodifferential operator of order m on the manifold p-1 (x) Y . Assume that the symbols of these operators together = give a continuous homomorphism : (1 )- (2 ). (6.24)


232

Chapter 6

and that the homomorphism (6.24) is an isomorphism away from the zero section of the bundle TY (M ). In this case we say that the family (6.23) is an elliptic family of pseudodifferential operators. Then, if (6.23) is an elliptic family, it defines both a continuous family of Fredholm operators (D) : p! (1 )s -p! (2 ) and, using (6.24), an element
[ ] Kc (TY (M )) . s-m

Theorem 59 For an el liptic family (6.23), index (D) = p! [ ] K (X ) Q. Pro of. Let us prove this theorem for the case when M = X â Y . Then there is a second pro jection q : M -Y . The crucial observation is that ( , M ) = (q! ( ), Y ) = (p! ( ), X ) . The family of pseudodifferential operators (D) : p! (1 ) = (1 , Y ) â X -(2 , Y ) â X generates the homomorphism ( (D)) : ((1 , Y ), X ) = (1 , X â Y )-(2 , X â Y ) = ((1 , Y ), X ) or q! ( (D)) : ((1 , X ), Y ) = (q! (1 ), Y ) -(q! (2 ), Y ) = ((1 , X ), Y ) . (6.26) (6.25)

The last representation (6.26) of the family (6.25) is a pseudodifferential operator over the C ­algebra C (X ). In the case where the family of elliptic operators has both constant dimensional kernel and cokernel, both the kernel and cokernel of the operator (6.26) over


Applications of vector bund le theory

233

algebra C (X ) are pro jective modules since they can be identified with spaces of sections of finite dimensional vector bundles: Ker q! ( (D)) = (Ker (D)) Coker q! ( (D)) = (Coker (D)) Hence the index of the family of elliptic operators (D) as an element of the group K (X ) is identical to the index of the elliptic operator q! ( (D)) as an element of the group KC (X ) (pt) = K (X ). On the other hand, the symbol of the family (6.25) generates a homomorphism of vector bundles q! ( ) : (q! (1 ))- (q! (2 )) which is the symbol of a pseudodifferential operator (6.26) over C ­algebra C (X ). Hence, in the elliptic case, the element [q! ( )] KC (X ),c (T Y ) is identified with [ ] by the isomorphism Kc (T Y â X ) K =
C (X ),c

(T Y ).

Thus to prove the theorem it is sufficient to generalize the Atiyah­Singer formula to the case of an elliptic operator over the C ­algebra C (X ). This pro ject can be realized (see for example [?]). Firstly, we need to investigate the category of vector bundles over C ­algebras and study the category of Hilbert modules and so called Fredholm operators over C ­algebras. We shall follow here ideas of Paschke [?] (see also Rieffel [?]). Let A be an arbitrary C ­algebra (with unit). Then a Hilbert A­module is a Banach space M which is also a (unital) A-module with a sequilinear form with values in A . Assume that 1. (x, y ) = (y , x) A, 2. (x, x) 0, x M; x, y M , x M. A; x, y M ;

3. (x, y ) = (x, y ), 4. x
2

= (x, x) ,

A model (infinite dimensional) Hilbert A­module l2 (A) can constructed as follows. By definition, a point x l2 (A) is a sequence x = (x1 , x2 , . . . , xn , . . .), xn A


234

Chapter 6

such that the series there is a limit

k=1

xk x converges in norm in the algebra A, that is, k
N 2

x

= lim

N -

xk x A. k
k=1

We define a `scalar' product by (x, y ) = The following inequality
xk yk k 2 def xk yk A. k=1

(6.27)


k

xk x · k
k

yk yk .

implies that the series on the right hand side of (6.27) converges. The space l2 (A) has a decreasing sequence of subspaces [l2 (A)]n = {x : x1 = x2 = . . . = xn = 0}. A bounded A­linear operator K : l2 (A)-l2 (A) is said to be A­compact if
n-

lim

K|

[l2 (A)]

n

= 0.

A bounded A­linear operator F : l2 (A)-l2 (A) is said to be A­Fredholm if there exists a bounded A­linear operator G : l2 (A)-l2 (A) such that both F G - 1 and GF - 1 are A­compact. This terminology is justified by the following properties: 1. Any finitely generated pro jective A­module admits the structure of a Hilbert module. 2. Let be a A­bundle over a compact manifold X with fiber a finitely generated pro jective A­module. Then the Sobolev completion H s ( ) of the section space is isomorphic to a direct summand of l2 (A).


Applications of vector bund le theory

235

3. A pseudodifferential A­operator (D) of order m that maps from the section space (1 ) to the section space (2 ) can be extended to a bounded operator of the Sobolev spaces (D) : Hs (1 )-H 4. The natural inclusion H is an A­compact operator. 5. If the operator (6.28) is elliptic then it is A­Fredholm. 6. If F is an A­Fredholm operator then it defines a homotopy invariant index F K0 (A). In fact, for such an operator F , F : H1 = l2 (A)-H2 = l2 (A), there exist decompositions H1 = V
10 s-m

(2 ).

(6.28)

s+1

( ) H s ( )

V11 , F0 0

H2 = V 0 F1
10

20

V21 ,

such that the corresponding matrix of the operator F has the form F=

where F1 is an isomorphism, the modules V pro jective modules and

and V

20

are finitely generated

index F = [V10 ] - [V20 ] K0 (A). 7. If (A) is an elliptic pseudodifferential A­operator on a compact manifold X then index (D) = p ([ ]) K0 Q, where p : X -pt and

p : KA,c (T X )-KA (pt) = K0 (A)

is the direct image in KA ­theory. The proofs of these properties are very similar to those for classical pseudodifferential operators.


236

Chapter 6

6.4

FREDHOLM REPRESENTATIONS AND ASYMPTOTIC REPRESENTATIONS OF DISCRETE GROUPS
: G-A

Let be a unitary representation of a group G in a C ­algebra A. Then by section 3.4 there is an associated vector bundle b() KA (B G) giving a homomorphism b : R(G; A)-KA (B G), (6.29)

where R(G; A) is the ring of virtual representations of the group G in the algebra A. When A is the algebra of all bounded operators on an infinite dimensional Hilbert space H we do not obtain an interesting homomorphism (6.29): the homomorphism is trivial since all infinite dimensional vector bundles are trivial. For finite dimensional representations it has been shown that, in some sense, the homomorphism (6.29) is an isomorphism when the group G is compact. But in the case of a discrete group such as the fundamental group of a manifold this homomorphism is usually trivial. Therefore, the problem arises to find new examples of geometric representations which give richer information about the group K (B G).

6.4.1

Fredholm representations

There is a so called relative version of representation theory associated with a homomorphism : A1 -A2 . We use the term relative representation for a triple = (1 , F, 2 ) where 1 , 2 : G-A
1

are two representations and F is an operator interlacing the representations 1 and 2 . In other words, F is an invertible element in A2 such that F
-1

1 (g )F = 2 (g ),

g G.

(6.30)


Applications of vector bund le theory

237

The representations 1 and 2 define vector A1 ­bundles over B G, 1 = b(1 ) and 2 = b(2 ). The associated A2 ­bundles 1 A2 and 2 A2 are isomorphic: b(F ) : 1 A1 -2 A2 . The triple (1 , b(F ), 2 ) defines an element of the relative K ­theory b() = (1 , b(F ), 2 ) K (B G). (6.31)

On the other hand, if 1 = 2 then the condition (6.30) means that the triple (1 , F, 2 ) defines a classical representation of the group G â Z into the algebra ¯ A2 : (g ) = (g ) A2 , g G; ¯ (a) = F A2 , where a Z is the generator. ¯ Hence the representation defines an element ¯ b() K ¯
A
2

(B (G â Z)) = K

A

2

BG â S

1

.

(6.32)

The two elements (6.31) and (6.32) are connected in the classical exact sequence for the homomorphism : A1 -A2 : K
A
2

B G â S1 b() ¯

- K

A

2

(S(B G)) =
2

b() - K (B G) - K
A

K

1 A

(B G)

1

(B G)

For an illustration let : B (H )-B (H )/Comp(H ) = C (H) (6.33)

be the natural pro jection to the Calkin algebra. As the bundles 1 and 2 are trivial the isomorphism b(F ) generates a continuous family of Fredholm operators, that is, an element of the group K (B G): K (B G) K (B G). Thus we have a homomorphism b : R(G; )-K (B G). (6.34)


238

Chapter 6

Relative representations for the pro jection (6.33) are called Fredholm representations of the group G. Since the group K (B G) is nontrivial, we might expect that the homomorphism (6.34) is nontrivial. The classical finite dimensional representations can be considered as a special cases of Fredholm representations. In fact, let us substitute for the condition (6.30) the stronger one : 1 F = F 2 ; g G.

Then the representations 1 and 2 induce two finite dimensional representations in Ker F and Coker F . It is clear that b(Ker F ) - b(Coker F ) = b(1 , F, 2 ) K (B G). For example, consider the discrete group G = Zn . We showed earlier that for any finite dimensional representation the associated vector bundle b() is trivial, that is, the homomorphism b : R(Zn )-K (B Zn ) is trivial. However, if we enlarge the group of virtual representations by adding Fredholm representations the homomorphism (6.35) becomes an epimorphism. Moreover, the following theorem is true. Theorem 60 Let G be a group such that B G can be represented by a smooth complete Riemannian manifold with nonpositive section curvature. Then the homomorphism b : R(G, )-K (B G) is an epimorphism. When G = Zn we have a special case of Theorem 60 since B Zn is the torus with trivial sectional curvature. (see for example Mishchenko [?], Solov'yov [?], Kasparov [?]) (6.35)


Applications of vector bund le theory

239

6.4.2

Asymptotic finite dimensional representations

Another example of a very geometric variation of representation theory that may lead to a nontrivial homomorphism (6.29) is given by the so called asymptotic representations . Here we follow the ideas of Connes, Gromov and Moscovici [?]. Assume that : G-U (N ) is a map that is not actually a representation but is an approximation defined by the conditions: 1. (g 2.
-1

) = ((g ))

-1

, g G,

(g1 g2 ) - (g1 )(g2 ) for g1 , g2 K G. finite subset including a set of generators of the group G and the is sufficiently small we can say that is an almost representation of G. Then in the following section 3.4 we construct candidates for the functions on B G,


If K is a number the group transition

=



,

(6.36)

where are the transition functions of the universal covering of B G. The problem here is that the functions (6.36) do not satisfy the cocycle condition (1.4). To rescue this situation we should make a small deformation of the functions (6.36) to make them into a cocycle. To avoid certain technical difficulties let us change the definition slightly. Let
K

=

sup
g1 ,g2 K

(g1 )(g2 ) - (g1 g2 ) .

Consider an increasing sequence of integers Nk : Nk < N and a sequence of maps = {k : G-U (Nk )}. (6.37)
k+1

, Nk -


240

Chapter 6

The norms of matrices for different Nk can be compared by using the natural inclusions U (Nk )-U (Nk
+1

) : U (Nk )

X X 1

N

k+1

-N

k

U (Nk

+1

).

Then the sequence (6.37) is called an asymptotic representation if the following conditions hold: 1. k (g 2. 3.
-1

) = (k (g ))
kK k+1

-1

, g G,

k- k-

lim

= 0,

lim

(g ) - k (g ) = 0, g K

It can be shown that an asymptotic representation is actually a representation in a special C ­algebra.

6.5

CONCLUSION

The applications of vector bundles are not exhausted by the examples considered above. We summarize further outstanding examples of their use.

6.5.1

Fib erwise homotopy equivalence of vector bundles

Let S( ) be the bundle formed by all unit vectors in the total space of a vector bundle over a base X . Two vector bundles and are said to be fiberwise homotopy equivalent if there are two fiberwise maps f : S( )-S( ); such that the two compositions g f : S( )-S( ); are homotopic to the identity maps. This definition is justified by the following theorem (Atiyah, [4]): f g : S( )-S( ) g : S( )-S( ),


Applications of vector bund le theory

241

Theorem 61 If f : X -Y is a homotopy equivalence of closed compact manifolds, (X ) and (Y ) are normal bund les ( of the same dimension) of X and Y , respectively, then the bund les (X ) and f ( (Y )) are fiberwise homotopy equivalent. In particular, Theorem 61 is important in the classification of smooth structures of a given homotopy type. The classes of fiberwise homotopy equivalent bundles form a group J (X ) which is a quotient of the group K (X ). The problem of describing the group J (X ) or the kernel of the pro jection K (X )-J (X ) can be solved by using the Adams cohomology operations k . This description is reduced to the so called the Adams conjecture. Namely, let f (k ) Z be an arbitrary positive function , where Wf (X ) K (X ) is the subgroup generated by elements of the form k If W (X ) =
f f (k)

k - 1 y , y K (X ). Wf (X ).

Theorem 62 (Quillen [?], Sullivan [?]) J (X ) = K (X )/W (X ). In particular, the group J (X ) is finite.

6.5.2
Let

Immersions of manifolds
f : X -Y

be an immersion of a compact manifold. Then, clearly, the normal bundle (X -Y ) has dimension dim (X -Y ) = dim Y - dim X. Hence, by the dimension restriction, some characteristic classes must be trivial. This observation leads to some criteria showing when one manifold cannot be immersed or embedded in another given manifold.


242

Chapter 6

For example, for n = 2r , the real pro jective space RPn cannot be immersed into RP2n-2 since then the normal bundle = (RPn -RP2n-2 ) would have a nontrivial Stieffel­Whitney class wn-1 ( ). Hence, the dimension cannot equal n - 2. More generally, let (X ) be a normal bundle for which the dimension is determined by the dimension of Euclidean space RN in which the manifold X is embedded. Put k = min dim 0 (X ) where the minimum is taken over all possible decompositions = 0 (X ) n, ¯ where n is a trivial bundle. The number k is called the geometric dimension of ¯ the normal bundle (X ). It is clear that the manifold X cannot be embedded in the Euclidean space Rdim X +k-1 .

6.5.3

Poincare duality in K ­theory

To define Poincare duality we need to understand which homology theory is dual to K ­theory. The trivial definition for the homology groups K (X ), K (X ) = Hom (K (X ), Z) , is incorrect since we do not have exactness of the homology pair sequence. We need a more delicate definition. Consider a natural transformation of functors : K (X â Y )-K (Y ) (6.38) which is a homomorphism of modules over the ring K (Y ). Clearly, all such transformations form a Z2 ­graded group, which we denote by K (X ) = K0 (X ) K1 (X ). (6.39)
def

It can be shown that the groups (6.39) form a homology theory. When Y = pt, the homomorphism (6.38) can be understood as the value of a homology element on the cohomology element: , y = (y ).


Applications of vector bund le theory

243

Theorem 63 Let X be an almost complex manifold. Then there is a homomorphism (Poincare duality)
D : Kc (X )-K (X )

satisfying the condition D(x), y = xy , D(1) Atiyah ([5]) noticed that each elliptic operator (D) defines an element of the homology K ­theory dual to the element [] Kc (T X ). The construction is as follows. A continuous family of elliptic operators (D) : (1 y )-(2 y ), yY

is constructed for each vector bundle over X â Y . Then the index of this family induces a homomorphism index : K (X â Y )-K (Y ) which is by definition an element of the homology group K (X ). The notion of an elliptic Fredholm operator F : H Hilbert space H in such a compact operators. Let (F, ) gives a well defined operator can be extended as follows. Consider a -H . Assume that the algebra C (X ) acts on the way such that F commutes with this action, modulo denote the action of the algebra. Clearly, the pair family of Fredholm operators (y )-H (y ), y Y ,

Fy = F : H

C (X )

C (X )

and hence defines a homomorphism index : K (X â Y )-K (Y ), that is, an element [F, ] K (X ). Moreover, if f : X1 -X2 is a continuous map then the pair (F, f ) defines an element of K (X2 ) for which f ([F, ]) = [F, f ]. When Y2 = pt, K (X2 ) = Z


244

Chapter 6

and

[F, f ] = index F.

This last formula in some sense explains why, in the Atiyah­Singer formula, the direct image appears.

6.5.4

Historical notes

The study of bundles was probably started by Poincar´ with the study of none trivial coverings. Fibrations appeared in connection with the study of smooth manifolds , then characteristic classes were defined and for a long time they were the main tool of investigation. The Stieffel­Whitney characteristic classes were introduced by Stieffel [?] and Whitney [?] in 1935 for tangent bundles of smooth manifolds. Whitney [?] also considered arbitrary sphere bundles. These were mod 2 characteristic classes. The integer characteristic classes were constructed by Pontryagin [?]. He also proved the classification theorem for general bundles with the structure group O(n) and SO(n) using a universal bundle over a universal base such as a Grassmannian. For complex bundles, characteristic classes were constructed by Chern [?]. This was the beginning of the general study of bundles and the period was well described in the book of Steenrod [?] A crucial point of time came with the which were applied to the calculation other was at the discovery by Bott groups of the unitary and orthogonal discovery by Leray of spectral sequences of the homology groups of bundles. An[13] of the periodicity of the homotopy groups.

From this point, vector bundles occupied an important place in the theory of bundles as it was through them that nontrivial cohomology theories were constructed. It was through Bott periodicity that the Grothendieck constructions became so significant in their influence on the development of algebraic K ­theory. This period was very rich: the problem of vector fields on the spheres was solved, J -functor was calculated, a homotopy description of the group representations was achieved. Among the mathematicians important for their contributions in


Applications of vector bund le theory

245

this period are: Adams, Atiyah, Bott, Hirzebruch, Borel, Godement, Milnor and Novikov. The most brilliant example of an application of K ­theory was the Atiyah­ Singer formula for the index of elliptic operator. This theory generated new wave of work and applications of K ­theory. The contemporary period may be characterized as a time in which K ­theory became a suitable language for theoretical physics and related mathematical branches - representation theory, operator and Banach algebras, quantizations, and so on. The name of this new circle of ideas is "non commutative geometry". But it is not possible here to give an exposition that would do it justice .



INDEX

Atiyah­Singer formula, 213 Atlas of charts, 3 Atlas, 46 Base of the bundle, 2 Bilinear form on the bundle, 30 Bott homomorphism, 148 Bundle, 2 K R­bundle, 163 K RS ­bundle, 167 atlas of charts, 3 base, 2 bundle complex, 142 charts, 3 coordinate homeomorphism, 3 cross­section, 15 equivalent bundles, 11 fiber, 2 fiberspace, 2 inverse image, 18 isomorphism compatible with the structure group, 11 locally trivial, 2 principal, 77 pro jection, 2 restriction to subspace, 18 sewing function, 3 structure group, 10 change of, 12 reduced to subgroup, 11 tangent bundle, 8 total space, 2 transition function, 3 trivial, 7 vector bundle, 21 Change of coordinates, 46

Characteristic class, 100 Chern character, 134 Chern class, 113 Euler class, 119 of manifold, 113 Pontryagin class, 119 stable, 101 Stieffel­Whitney class, 119 Todd class, 193 Todd genus, 194 Chart, 45 Charts, 3 Chern character, 134 Chern classes, 113 Classifying space, 77 Cocycle, 9 Compact operator, 202 Complex conjugate of the complex bundle, 38 Complex conjugation, 40 Complex vector bundle, 37 Complexification of the bundle, 37 Components of vector, 47 Coordinate functions, 45 Coordinate homeomorphism, 45 Coordinate homeomorphisms, 3 Coordinates of vector, 47 Critical point, 63 Cross­section, 15 Difference construction, 144 Differential of the mapping, 50 Differential operator, 198 full symbol, 198 order, 198 principal symbol, 198


248

Chapter 6

symbol, 198 Direct sum of vector bundles, 23 Division algebras, 185 Dual bundle, 26 Elliptic complex, 202 Elliptic pseudodifferential operator, 202 Euler class, 119 Exterior power, 27 Fiberspace, 2 Fiberwise homotopy equivalent, 243 First obstruction, 124 Fredholm complex, 206 index, 207 Fredholm operator, 202 index, 202 Full symbol, 198 Geometric dimension, 244 Grassmann manifolds, 72 Grothendieck group, 127 Hirzebruch operator, 220, 224 Hirzebruch­Pontryagin class, 220 Homomorphism of bundles, 28 Homotopy stability of orthogonal groups, 86 Hopf bundles, 68 Image, 41 Index of Fredholm complex, 207 Index of the Fredholm operator, 202 Inverse image of the bundle, 18 Join, 95 Kernel, 41 K ­group with compact supports, 187 Linear map of vector bundles, 28 Local system of coordinates, 45 Locally trivial bundle, 2 Manifold, 45 analytic structure, 46 atlas, 46

change of coordinate, 46 chart, 45 class of differentiability, 46 complex analytic structure, 46 coordinate function, 45 coordinate homeomorphism, 45 local system of coordinate, 45 normal bundle, 52 orientation, 64 smooth structure, 46 tangent bundle, 48 tangent space, 48­49 tangent vector, 47 the transition function, 46 Mapping transversal along submanifold, 63 Morse function, 64 Multiplicative spectral sequence, 176 Normal bundle, 52 Obstruction, 124 first, 124 One dimensional cochain, 9 Orientation, 64 Oriented bundle, 120 Principal G­bundle principal, 77 Principal symbol, 198 Pro jection, 2 Pro jective coordinates, 66 Pro jective space, 65 pro jective coordinates, 66 real, 65 Pseudodifferential operator, 199 elliptic complex, 202 elliptic, 202 Rational Pontryagin class, 119 Real homomorphism, 40 Real section, 40 Real vector, 40 Realification of a complex vector bundle, 37


249

Regular value of the mapping, 63 Restriction of the bundle to the subspace, 18 Scalar product on the bundle, 30 Sewing functions, 3 Signature, 217 Skeleton, 73 Sobolev inclusion theorem, 209 Sobolev norm, 208 Sobolev space, 208 Spectral sequence in K ­theory, 174 Splitting principle, 180 Stable isomorphism of vector bundles, 128 Stieffel manifolds, 72 Stieffel­Whitney characteristic class, 119 Structure group, 10 change of, 12 reduced to subgroup, 11 Structure of a complex analytic manifold, 46 Structure of an analytic manifold, 46 Structure of differentiable manifold, 46 Symbol of a differential operator, 198 Tangent bundle, 8, 48 Tangent space, 48­49 Tangent vector to the curve, 47 Tangent vector, 47 Tensor law of transformation of components, 47 Tensor product, 24 The fiber of the bundle, 2 The transition functions, 3 Thom space, 187 Todd class, 193 Todd genus, 194 Total space of the bundle, 2 Transition function, 46

Universal principal G­bundle, 77 Vector A­bundle, 228 Vector bundle, 21 bilinear form, 30 complex conjugate, 38 complex conjugation, 40 complex, 37 complexification, 37 direct sum, 23 dual, 27 exterior power, 27 finite dimensional, 21 homomorphism, 28 infinite dimensional, 21 linear map, 28 oriented, 120 realification, 37 scalar product, 30 stable isomorphism, 128 tensor product, 24 Thom space, 187 virtual, 131 Virtual vector bundle, 131


250


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