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 major 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 subject. 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.

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 objects 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.

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 Möbius 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 R^{N} and T M the space embedded in R^{N} × R^{N}, 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 T_{x}M, where each T_{x}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 T_{x}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.

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 locally trivial bundle 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 × F. Moreover, it is required that the homeomorphism

φ : U × F ⟶p^{−1}(U)

preserves fibers, it is a ‘fiberwise’ map, that is, the following equality holds:

p(φ(x, f)) = x, x ∈ U, f ∈ F.

The space E is called total space of the bundle or the fiberspace , the space B is called the base of the bundle , the space F is called the fiber of the bundle and the mapping p is called the projection . The requirement that the homeomorphism φ be fiberwise means in algebraic terms that the diagram

U × F ⟶^{φ} p^{−1}(U)
↓π  ↓p
U  =  U

where

π : U × F ⟶U, π(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 ψ : E⟶E′ such that the diagram

\begin{matrix}E & ⟶^{ψ} & E′ \\ ↓p & & ↓p′ \\ B & = & B \end{matrix}

Introduction to bundle theory 3

is commutative. It is clear that the homeomorphism ψ 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_{α}} such that each inverse image p^{−1}(U_{α}) is fiberwise homeomorphic to U_{α} × F. This gives a system of homeomorphisms

φ_{α} : U_{α} × F ⟶p^{−1}(U_{α}).

Since the homeomorphisms φ_{α} preserve fibers it is clear that for any open subset V ⊂ U_{α} the restriction of φ_{α} to V × F establishes the fiberwise homeomorphism of V × F onto p^{−1}(V). Hence on U_{α} × U_{β} there are two fiberwise homeomorphisms

φ_{α} : (U_{α} ∩ U_{β}) × F ⟶p^{−1}(U_{α} ∩ U_{β}),
φ_{β} : (U_{α} ∩ U_{β}) × F ⟶p^{−1}(U_{α} ∩ U_{β}).

Let φ_{αβ} denote the homeomorphism φ_{β}^{−1}φ_{α} which maps (U_{α} ∩ U_{β}) × F onto itself. The locally trivial bundle is uniquely determined by the following collection: the base B, the fiber F, the covering U_{α} and the homeomorphisms

φ_{αβ} : (U_{α} ∩ U_{β}) × F ⟶(U_{α} ∩ U_{β}) × F.

The total space E should be thought of as a union of the Cartesian products U_{α} × F with some identifications induced by the homeomorphisms φ_{αβ}. By analogy with the terminology for smooth manifolds, the open sets U_{α} are called charts , the family {U_{α}} is called the atlas of charts , the homeomorphisms φ_{α} are called the coordinate homeomorphisms and the φ_{αβ} are called the transition functions or the sewing functions . Sometimes the collection {U_{α}, φ_{α}} 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 φ_{α} forms an atlas. For the classification of locally trivial bundles, families of homeomorphisms φ_{αβ} that actually determine bundles should be selected and then separated into classes which determine isomorphic bundles. For the homeomorphisms φ_{αβ} to be transition functions for some locally trivial bundle:

φ_{βα} = φ_{β}^{−1}φ_{α}.  (1.1)

Then for any three indices α, β, γ on the intersection (U_{α} ∩ U_{β} ∩ U_{γ}) × F the following relation holds:

φ_{αγ}φ_{γβ}φ_{βα} = Id,

where Id is the identity homeomorphism and for each α,

φ_{αα} = Id.  (1.2)

In particular

φ_{αβ}φ_{βα} = Id,  (1.3)

thus

φ_{αβ} = φ_{βα}^{−1}.

Hence for an atlas the φ_{αβ} should satisfy

φ_{αα} = Id, φ_{αγ}φ_{γβ}φ_{βα} = Id.  (1.4)

These conditions are sufficient for a locally trivial bundle to be reconstructed from the base B, fiber F, atlas {U_{α}} and homeomorphisms φ_{βα}. To see this, let

E′ = ∪_{α}(U_{α} × F)

be the disjoint union of the spaces U_{α} × F. Introduce the following equivalence relation: the point (x, f) ∈ U_{α} × F is related to the point (y, g) ∈ U_{β} × F iff

x = y ∈ U_{α} ∩ U_{β}

and

(y, g) = φ_{βα}(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 projection

π : E′⟶E

which associates to a (x, f) its the equivalence class. In other words, the subset G ⊂ E is called open iff π^{−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

φ_{α} = π|_{U_{α}×F} : U_{α} × F ⟶E.

Each class z ∈ p^{−1}(U_{α}) has a unique representative (x, f) ∈ U_{α} × F. Hence φ_{α} is a one to one mapping onto p^{−1}(U_{α}). By virtue of the quotient topology on E the mapping φ_{α} is a homeomorphism. It is easy to check that (compare with (1.1))

φ_{β}^{−1}φ_{α} = φ_{βα}.

So we have shown that locally trivial bundles may be defined by atlas of charts {U_{α}} and a family of homeomorphisms {φ_{βα}} 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 {φ_{βα}} then these two bundles are isomorphic. Indeed, let

φ_{α} : U_{α}⟶p^{−1}(U_{α}).
ψ_{α} : U_{α}⟶p^{′−1}(U_{α}).

be the corresponding coordinate homeomorphisms and assume that

φ_{βα} = φ_{β}^{−1}φ_{α} = ψ_{β}^{−1}ψ_{α} = ψ_{βα}.

Then

φ_{α}ψ_{α}^{−1} = φ_{β}ψ_{β}^{−1}.

We construct a homeomorphism

ψ : E′⟶E.

Let x ∈ E′. The atlas {U_{α}} covers the base B and hence there is an index α such that x ∈ p^{′−1}(U_{α}). Set

ψ(x) = φ_{α}ψ_{α}^{−1}(x).

It is necessary to establish that the value of ψ(x) is independent of the choice of index α. If x ∈ p^{′−1}(U_{β}) also then

φ_{β}ψ_{β}^{−1}(x) = φ_{α}φ_{α}^{−1}φ_{β}ψ_{β}^{−1}ψ_{α}ψ_{α}^{−1}(x) =
= φ_{α}φ_{αβ}φ_{βα}ψ_{α}^{−1}(x) = φ_{α}ψ_{α}^{−1}(x).

Hence the definition of ψ(x) is independent of the choice of chart. Continuity and other necessary properties are evident. Further, given an atlas {U_{α}} and coordinate homomorphisms {φ_{α}}, if {V_{β}} is a finer atlas (that is, V_{β} ⊂ U_{α} for some α = α(β)) then for the atlas {V_{β}}, the coordinate homomorphisms are defined in a natural way

φ′_{β} = φ_{α(β)}|_{(V_{β}×F)} : V_{β} × F ⟶p^{−1}(V_{β}).

The transition functions φ′_{β_{1},β_{2}} for the new atlas {V_{β}} are defined using restrictions

φ′_{β_{1},β_{2}} = φ_{α(β_{1}),α(β_{2})}|_{(V_{β_{1}} ∩V_{β_{2}} )×F} : (V_{β_{1}} ∩ V_{β_{2}}) × F ⟶(V_{β_{1}} ∩ V_{β_{2}}) × F.

Thus if there are two atlases and transition functions for two bundles, with a common refinement, that is, a finer atlas with transition functions given by restrictions, it can be assumed that the two bundles have the same atlas. If φ_{βα}, φ′_{βα} are two systems of the transition functions (for the same atlas), giving isomorphic bundles then the transition functions φ_{βα}, φ′_{βα} must be related.

Theorem 1 Two systems of the transition functions φ_{βα}, and φ′_{βα} define isomorphic locally trivial bundles iff there exist fiber preserving homeomorphisms

h_{α} : U_{α} × F ⟶U_{α} × F

such that

φ_{βα} = h_{β}^{−1}φ′_{βα}h_{α}.  (1.5)

Proof.

Suppose that two bundles p : E⟶B andp′ : E′⟶B with the coordinate homeomorphisms φ_{α} and φ′_{α} are isomorphic. Then there is a homeomorphism ψ : E′⟶E. Let

h_{α} = φ_{α}^{′−1}ψ^{−1}φ_{α}.

Then

h_{β}^{−1}φ′_{βα}h_{α} = φ_{β}^{−1}ψφ′_{β}φ′_{βα}φ_{α}^{′−1}ψ^{−1}φ_{α} =
= φ_{β}^{−1}ψφ′_{β}φ_{β}^{′−1}φ′_{α}φ_{α}^{′−1}ψ^{−1}φ_{α} = φ_{βα}.

Conversely, if the relation (1.5) holds, put

ψ = φ_{α}h_{α}^{−1}φ_{α}^{′−1}.  (1.6)

The definition (1.6) is valid for the subspaces p^{′−1}(U_{α}) covering E′. To prove that the right hand sides of (1.6) coincide on the intersection p^{′−1}(U_{α} ∩U_{β}) the relations (1.5) are used:

φ_{β}h_{β}^{−1}φ_{β}^{′−1} = φ_{α}φ_{α}^{−1}φ_{β}h_{β}^{−1}φ_{β}^{′−1}φ′_{α}φ_{α}^{′−1} =
= φ_{α}φ_{αβ}^{−1}h_{β}^{−1}φ_{βα}^{′−1}φ_{α}^{′−1} = φ_{α}h_{α}^{−1}φ_{αβ}^{′−1}h_{β}h_{β}^{−1}φ_{βα}^{′−1}φ_{α}^{′−1} =
= φ_{α}h_{α}^{−1}φ_{α}^{′−1}.

Examples

1. Let E = B × F and p : E⟶B be projections onto the first factors. Then the atlas consists of one chart U_{α} = B and only one the transition function φ_{αα} = Id and the bundle is said to be trivial .

2. Let E be the Möbius 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 projection p maps the space E onto the segment I_{x} = {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_{α} = \{0 < x < 1\}, U_{β} = \{0 ≤ x < \frac{1}{2}\} ∪ \{\frac{1}{2} < x ≤ 1\}.

The coordinate homeomorphisms may be defined as following:

φ_{α} : U_{α} × I_{y}⟶E, φ_{α}(x, y) = (x, y),
φ_{β} : U_{β} × I_{y}⟶E
φ_{β}(x, y) = (x, y) for 0 ≤ x < \frac{1}{2},
φ_{β}(x, y) = (x, 1 − y) for \frac{1}{2} < x ≤ 1.

The intersection of two charts U_{α} ∩ U_{β} consists of union of two intervals U_{α} ∩ U_{β} = (0, \frac{1}{2}) ∪ (\frac{1}{2}, 1). The transition function φ_{βα} have the following form

φ_{βα} = (x, y) for 0 < x < \frac{1}{2},
φ_{βα} = (x, 1 − y) for \frac{1}{2} < x < 1.

The Möbius 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_{α} : U_{α} × I_{y}⟶U_{α} × I_{y},
h_{β} : U_{β} × I_{y}⟶U_{β} × I_{y},

such that

φ_{βα} = h_{β}^{−1}h_{α}

in its domain of definition (U_{α} ∩ U_{β}) × I_{y}. Then h_{α}, h_{β} are fiberwise homeomorphisms for fixed value of the first argument x giving homeomorphisms of interval I_{y} to itself. Each homeomorphism of the interval to itself maps end points to end points. So the functions

h_{α}(x, 0), h_{α}(x, 1), h_{β}(x, 0), h_{β}(x, 1)

are constant functions, with values equal to zero or one. The same is true for the functions h_{β}^{−1}h_{α}(x, 0). On the other hand the function φ_{βα}(x, 0) is not constant because it equals zero for each 0 < x < \frac{1}{2} and equals one for each \frac{1}{2} < x < 1.This contradiction shows that the Möbius band is not isomorphic to a trivial bundle.

3. Let E be the space of tangent vectors to two dimensional sphere S^{2} embedded in three dimensional Euclidean space R^{3}. Let

p : E⟶S^{2}

be the map associating each vector to its initial point. Let us show that p is a locally trivial bundle with fiber R^{2}. Fix a point s_{0} ∈ S^{2}. Choose a Cartesian system of coordinates in R^{3} such that the point s_{0} is the North Pole on the sphere (that is, the coordinates of s_{0} equal (0, 0, 1)). Let U be the open subset of the sphere S^{2} defined by inequality z > 0. If s ∈ U, s = (x, y, z), then

x^{2} + y^{2} + z^{2} = 1, z > 0.

Let e⃗ = (ξ, η, ζ) be a tangent vector to the sphere at the point s. Then

xξ + yη + zζ = 0,

that is,

ζ = −(xξ + yη)/z.

Define the map

φ : U × R^{2}⟶p^{−1}(U)

by the formula

φ(x, y, z, ξ, η) = (x, y, z, ξ, η, −(xξ + yη)/z)

giving the coordinate homomorphism for the chart U containing the point s_{0} ∈ S^{2}. Thus the map p gives a locally trivial bundle. This bundle is called the tangent bundle of the sphere S^{2}.

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

φ : U × F ⟶U × 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 φ defines a map

φ̄ : 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 φ̄ becomes continuous. Sometimes the opposite is true: the map (1.8) generates the fiberwise homeomorphism (1.7)with respect to the formula

φ(x, f) = (x, φ̄(x)f).

So instead of φ_{αβ} a family of functions

φ̄_{αβ} : U_{α} ∩ U_{β}⟶Homeo (F),

can be defined on the intersection U_{α} ∩ U_{β} and having values in the group Homeo (F). In homological algebra the family of functions φ̄_{αβ} 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

φ̄_{αα}(x) = Id,
φ̄_{αγ}(x)φ̄_{γβ}(x)φ̄_{βα}(x) = Id.
x ∈ U_{α} ∩ U_{β} ∩ U_{γ}.

and we say that the cochain {φ̄_{αβ}} is a cocycle. The condition (1.5) means that there is a zero dimensional cochain h_{α} : U_{α}⟶Homeo (F) such that

φ̄_{βα}(x) = h_{β}^{−1}(x)φ̄′_{βα}(x)h_{α}(x), x ∈ U_{α} ∩ U_{β}.

Using the language of homological algebra the condition (1.5) means that cocycles {φ̄_{βα}} and {φ̄′_{βα}} 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_{α}}. 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_{α}} and the functions {φ_{αβ}} 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 {φ_{αβ}} 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 φ̄_{αβ} and the functions h_{α} 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 locally trivial bundle with fiber F and the structure group G if there is an atlas {U_{α}} and the coordinate homeomorphisms

φ_{α} : U_{α} × F ⟶p^{−1}(U_{α})

such that the transition functions

φ_{βα} = φ_{β}^{−1}φ_{α} : (U_{α} ∩ U_{β}) × F ⟶(U_{α} ∩ U_{β}) × F

have the form

φ_{βα}(x, f) = (x, φ̄_{βα}(x)f),

where φ̄_{βα} : (U_{α} ∩ U_{β})⟶G are continuous functions satisfying the conditions

φ̄_{αα}(x) ≡ 1, x ∈ U_{α},
φ̄_{αβ}(x)φ̄_{βγ}(x)φ̄_{γα}(x) ≡ 1, x ∈ U_{α} ∩ U_{β} ∩ U_{γ}.  (1.9)

The functions φ̄_{αβ} are also called the transition functions

Let

ψ : E′⟶E

be an isomorphism of locally trivial bundles with the structure group G. Let φ_{α} and φα^{′} be the coordinate homeomorphisms of the bundles p : E⟶B and p′ : E′⟶B, respectively. One says that the isomorphism ψ is compatible with the structure group G if the homomorphisms

φ_{α}^{−1}ψφ′_{α} : U_{α} × F ⟶U_{α} × F

are determined by continuous functions

h_{α} : U_{α}⟶G,

defined by relation

φ_{α}^{−1}ψφ′_{α}(x, f) = (x, h_{α}(x)f).  (1.10)

Thus two bundles with the structure group G and transition functions φ̄_{βα} and φ′_{βα} are isomorphic, the isomorphism being compatible with the structure group G, if

φ̄_{βα}(x) = h_{β}(x)φ̄′_{βα}(x)h_{α}(x)  (1.11)

for some continuous functions h_{α} : U_{α}⟶G. So two bundles whose the transition functions satisfy the condition (1.11) are called equivalent bundles . 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

ρ : 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

φ_{αβ} : U_{α} ∩ U_{β}⟶G

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

φ′_{αβ}(x) = ρ(φ_{αβ}(x)).

This operation is called a change of the structure group (with respect to the homomorphism ρ).

Remark

Note that the fiberwise homeomorphism

φ : U × F ⟶U × F

in general is not induced by continuous map

φ̄ : 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) = gf, f ∈ F = G.

Definition 3 A locally trivial bundle with the structure group G is called principal G–bundle if F = G and action of the group G on F is defined by the left translations.

An important property of principal G–bundles 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.

Theorem 2 Let

p : E⟶B

be a principal G–bundle,

φ_{α} : U_{α} × G⟶p^{−1}(U_{α})

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 φ_{α}^{−1} transforms the right action of the group G on the total space into right translations on the second factor, that is,

φ_{α}(x, f)g = f_{α}(x, fg), x ∈ U_{α}, f, g ∈ G.  (1.13)

Proof.

According to the definitions 2 and 3, the transition functions φ_{βα} = φ_{β}^{−1}φ_{α} have the following form

φ_{βα}(x, f) = (x, φ̄_{βα}(x)f),

where

φ̄_{βα} : U_{α} ∩ U_{β}⟶G

are continuous functions satisfying the conditions (1.9). Since an arbitrary point z ∈ E can be represented in the form

z = φ_{α}(x, f)

for some index α, the formula (1.13) determines the continuous right action of the group G provided that this definition is independent of the choice of index α. So suppose that

z = φ_{α}(x, f) = φ_{β}(x, f′).

We need to show that the element zg does not depend on the choice of index, that is,

φ_{α}(x, fg) = φ_{β}(x, f′g).

or

(x, f′g) = φ_{α^{−1}}φ_{β}(x, fg) = φ_{βα}(x, fg)

or

f′g = φ̄_{βα}(x)fg.  (1.14)

However,

(x, f′) = φ_{β}^{−1}φ_{α}(x, f) = φ_{βα}(x, f) = (x, φ̄_{βα}f),

Hence

f′ = φ̄_{βα}(x)f.  (1.15)

Thus multiplying (1.15) by g on the right gives (1.14).

Theorem 2 allows us to consider principal G–bundles as having a right action on the total space.

Theorem 3 Let

ψ : E′⟶E  (1.16)

be a fiberwise map of principal G–bundles. The map (1.16) is the isomorphism of locally trivial bundles 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).

Proof.

Let

p : E⟶B,
p′ : E′⟶B

be locally trivial principal bundles both with the structure group G and let φ_{α}, φ′_{α} be coordinate homeomorphisms. Then by the definition (1.10), the map ψ is an isomorphism of locally trivial bundles with structure group G when

φ_{α}^{−1}ψφ′_{α}(x, g) = (x, h_{α}(x)g).  (1.17)

for some continuous functions

h_{α} : U_{α}⟶G.

It is clear that the maps defined by (1.17) are equivariant since

[φ_{α}^{−1}ψφ′_{α}(x, g)]g_{1} = (x, h_{α}(x)g)g_{1} =
= (x, h_{α}(x)gg_{1}) = φ_{α}^{−1}ψφ′_{α}(x, gg_{1}).

Hence the map ψ is equivariant with respect to the right actions of the group G on the total spaces E and E′. Conversely, let the map ψ be equivariant with respect to the right actions of the group G on the total spaces E and E′. By Theorem 2, the map φ_{α}^{−1}ψφ′_{α} is equivariant with respect to right translations of the second coordinate of the space U_{α} ×G. Since the map φ_{α}^{−1}ψφ′_{α} is fiberwise, it has the following form

φ_{α}^{−1}ψφ′_{α}(x, g) = (x, A_{α}(x, g)).  (1.18)

The equivariance of the map (1.18) implies that

A_{α}(x, gg_{1}) = A_{α}(x, g)g_{1}

for any x ∈ U_{α}, g, g_{1} ∈ G. In particular, putting g = e that

A_{α}(x, g_{1}) = A_{α}(x, e)g_{1}

So putting

h_{α}(x) = A_{α}(x, e),

it follows that

h_{α}(x)g = A_{α}(x, g).

and

φ_{α}^{−1}ψφ′_{α}(x, g) = (x, h_{α}(x)g).

The last identity means that ψ 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 × G, then to construct an equivariant map ψ : E′⟶E it is sufficient to define a continuous map ψ on the subspace {(x, e) : x ∈ B, } ⊂ E′ = B × G into E. Then using equivariance, the map ψ is extended by formula

ψ(x, g) = ψ(x, e)g.

The map{(x, e) : x ∈ B, }⟶^{ψ}E′ can be considered as a map

s : B⟶E  (1.19)

satisfying the property

ps(x) = x, x ∈ B.  (1.20)

The map (1.19) with the property (1.20) is called a cross–section of the bundle. So each trivial principal bundle has cross–sections. For instance, the map B⟶B × G defined by x⟶(x, e) is a cross–section. Conversely, if a principal bundle has a cross–section s then this bundle is isomorphic to the trivial principal bundle. The corresponding isomorphism ψ : B × G⟶E is defined by formula

ψ(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 G–bundles with arbitrary bases. Each fiber of a principal G–bundle is an orbit of the right action of the group G on the total space and hence for each equivariant mapping

ψ : E′⟶E

of total spaces, each fiber of the bundle

p′ : E′⟶B′

maps to a fiber of the bundle

p : E⟶B.  (1.21)

In other words, the mapping ψ induces a mapping of bases

χ : B′⟶B  (1.22)

and the following diagram is commutative

\begin{matrix}E′ & ⟶^{ψ} & E \\ ↓p′ & & ↓p \\ B′ & ⟶^{χ} & B \end{matrix}.  (1.23)

Let U_{α} ⊂ B be a chart in the base B and let U′_{β} be a chart such that

χ(U′_{β}) ⊂ U_{α}.

The mapping φ_{α}^{−1}ψφ′_{β} makes the following diagram commutative

\begin{matrix}U′_{β} × G & ⟶^{φ_{α}^{−1}ψφ′_{β}} & U_{α} × G \\ ↓p′φ′_{β} & & ↓pφ_{α} \\ U′_{β} & ⟶^{χ} & U_{α} \end{matrix}.  (1.24)

In diagram (1.24), the mappings p′φ′_{β} and pφ_{α} are projections onto the first factors. So one has

φ_{α}^{−1}ψφ′_{β}(x′, g) = (χ(x′), h_{β}(x′)g).

Hence the mapping (1.22) is continuous. Compare the transition functions of these two bundles. First we have

(x′, φ̄′_{β_{1}β_{2}}(x′)g) = φ′_{β_{1}β_{2}}(x′, g) = φ_{β_{1}}^{′−1}φ′_{β_{2}}(x′, g).

Then

(χ(x′), h_{β_{1}}(x′)φ̄′_{β_{1}β_{2}}(x′)g) =
φ_{α_{1}}^{−1}ψφ′_{β_{1}}φ_{β_{1}}^{′−1}φ′_{β_{2}}(x′, g) = φ_{α_{1}}^{−1}ψφ′_{β_{2}}(x′, g) =
= φ_{α_{1}}^{−1}φ_{α_{1}}φ_{α_{2}}^{−1}ψφ′_{β_{2}}(x′, g) = (χ(x′), φ̄_{α_{1}α_{2}}(χ(x′))h_{β_{2}}(x′)g),

that is,

h_{β_{1}}(x′)φ̄_{β_{1}β_{2}}(x′) = φ̄_{α_{1}α_{2}}(χ(x′))h_{β_{2}}(x′),

or

h_{β_{1}}(x′)φ̄′_{β_{1}β_{2}}(x′)h_{β_{2}}^{−1}(x′) = φ̄_{α_{1}α_{2}}(χ(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

χ : 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

χ : B′⟶B

is a continuous mapping and

p : E⟶B

is a principal G–bundle then one can put

U′_{α} = χ^{−1}(U_{α}), φ̄′_{αβ}(x′) = φ̄_{αβ}(χ(x)).  (1.27)

Then the transition functions (1.27) define a principal G–bundle

p′ : ⟶B,

for which there exists an equivariant mapping

ψ : E′⟶E

with commutative diagram (1.23). The bundle defined by the transition functions (1.27) is called the inverse image of the bundle p : E⟶B with respect to the mapping χ . In the special case when the mapping χ : B′⟶B is an inclusion then we say that the inverse image of the bundle with respect to the mapping χ is the restriction of the bundle to the subspace B′′ = χ(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′⟶^{ψ}E

is an equivariant mapping of total spaces of principal G–bundles then the bundle p′ : E′⟶B′ is an inverse image of the bundle p : ⟶B with respect to the mapping χ : 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 × I

be a principal G–bundle, 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 bundle p to the subspaces B×{0} and B×{1} are isomorphic.

Proof.

Without of the loss of generality we can assume that the atlas {U_{α}} consists of an atlas {V_{β}} on the space B and a finite system of intervals [a_{k}, a_{k+1}] which cover the segment I, that is,

U_{α} = V_{β} × [a_{k}, a_{k+1}].

Further, it suffices to assume that there is only one interval [a_{k}, a_{k+1}] which equals I, so

U_{α} = V_{α} × I.

Then the transition functions φ_{αβ} depend on the two arguments x ∈ V_{α}∩V_{β} and t ∈ I. Further, we can assume that the transition functions φ_{αβ} are defined and continuous on the closures V̄_{α} ∩ V̄_{β} and thus are uniformly continuous. Hence, we can assume that there exists an open neighborhood O of the neutral element of the group G homeomorphic to a disc and such that

φ_{αβ}(x, t_{1})φ_{αβ}^{−1}(x, t_{2}) ∈ O, x ∈ V_{α} ∩ V_{β}, t_{1}, t_{2} ∈ I,

and the power O^{N} lies in a disk, where N is the number of charts. We construct the functions

h_{α} : V_{α}⟶G, 1 ≤ α ≤ N,

by induction. Let

h_{1}(x) ≡ 1.

Then the function h_{2}(x) is defined on the set V̄_{1} ∩ V̄_{2} by the formula

h_{2}(x) = φ_{12}^{−1}(x, 1)φ_{12}(x, 0) ∈ O.  (1.28)

Then the function (1.28) can be extended on the whole chart V_{2}, h_{2} : V_{2}⟶O. Further, notice that when α < β the function h_{β}(x) should satisfy the following condition

h_{β}(x) = φ_{αβ}^{−1}(x, 1)h_{α}(x)φ_{αβ}(x, 0).  (1.29)

on the set V̄_{α} ∩ V̄_{β}. Assume that the functions h_{α}(x) ∈ O^{α} are defined for all α < β and satisfy the condition (1.29). Then the function h_{β}(x) is well defined on the set

⋃_{α<β}(V̄_{α} ∩ V̄_{β})

and takes values in the set O^{β}. Hence, the function h_{β} can be extended on the chart V̄_{β} taking values in O^{β}.

Corollary 1 If the transition functions φ_{αβ}(x) and ψ_{αβ}(x) are homotopic within the class of the transition functions then corresponding bundles are isomorphic.

Examples

1. In section 1 we considered the Möbius band. The transition functions φ_{αβ} 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 Möbius band we can consider corresponding principal bundle with the structure group G = Z_{2}. 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 projection p : S^{1}⟶S^{1} is a two-sheeted covering. This bundle is nontrivial since the total space of a trivial bundle would have two connected components.

2. In example 3 of section 1 the tangent bundle of two dimensional sphere was considered. The coordinate homeomorphisms

φ : U × R^{2}⟶R^{3} × R^{3}

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 = R^{2}, 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 S^{2}. To define a coordinate homeomorphism means to define a basis of tangent vectors e_{1}(x), e_{2}(x) at each point x ∈ U_{α} such that functions e_{1}(x) and e_{2}(x) are continuous.

Let us choose two charts U_{α} = {(x, y, z) : z ≠ 1}, U_{β} = {(x, y, z) : z ≠ −1}. The south pole P_{0} = (0, 0, −1) belongs to the chart U_{α}. The north pole P_{1} = (0, 0, +1) belongs to the chart U_{β}. Consider the meridians. Choose an orthonormal basis for the tangent space of the point P_{0} and continue it along the meridians by parallel transfer with respect to the Riemannian metric of the sphere S^{2} to all of the chart U_{α}. Thus we obtain a continuous family of orthonormal bases e_{1}(x), e_{2}(x) defined at each point of U_{α}. In a similar way we construct a continuous family of orthonormal bases e′_{1}(x), e′_{2}(x) defined over U_{β}. Then the coordinate homeomorphisms are defined by the following formulas

φ_{α}(x, ξ, η) = ξe_{1}(x) + ηe_{2}(x)
φ_{β}(x, ξ′, η′) = ξ′e′_{1}(x) + η′e′_{2}(x).

The transition function f_{βα} = φ_{β}^{−1}φ_{α} expresses the coordinates of a tangent vector at a point x ∈ U_{α} ∩ U_{β} in terms of the basis e′_{1}(x), e′_{2}(x) by the coordinates of the same vector with respect to the basis e_{1}(x), e_{2}(x). As both bases are orthonormal, the change of coordinates (ξ′, η′) into coordinates (ξ, η) 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.

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 bundles . So, for example, the tangent bundle of two-dimensional sphere S^{2} 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 R^{n}, the vector bundle ξ is said to be finite dimensional and the dimension of the vector bundle is equal to n (dim ξ = 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

φ_{αβ} : (U_{α} ∩ U_{β}) × R^{n}⟶(U_{α} ∩ U_{β}) × R^{n}

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 Γ(ξ) the set of all sections of the vector bundle ξ. Then the set Γ(ξ) becomes an (infinite dimensional) vector space. To define the structure of vector space on the Γ(ξ) consider two sections s_{1}, s_{2}:

s_{1}, s_{2} : B⟶E.

Put

(s_{1} + s_{2})(x) = s_{1}(x) + s_{2}(x), x ∈ B,  (1.30)
(λs_{1})(x) = λ(s_{1}(x)), λ ∈ R, x ∈ B.  (1.31)

The formulas (1.30) and (1.31) define on the set Γ(ξ) the structure of vector space. Notice that an arbitrary section s : B⟶E can be described in local terms. Let {U_{α}} be an atlas, φ_{α} : U_{α} × R^{n}⟶p^{−1}(U_{α}) be coordinate homeomorphisms, φ_{αβ} = φ_{β}^{−1}φ_{α}. Then the compositions

φ_{α}^{−1}s : U_{α}⟶U_{α} × R^{n}

are sections of trivial bundles over U_{α} and determine vector valued functions s_{α} : U_{α}⟶R^{n} by the formula

(φ_{α}^{−1})(x) = (x, s_{α}), x ∈ U_{α}.

On the intersection of two charts U_{α} ∩ U_{β} the functions s_{α}(x) satisfy the following compatibility condition

s_{β}(x) = φ_{βα}(x)(s_{α}(x)).  (1.32)

Conversely, if one has a family of vector valued functions s_{α} : U_{α}⟶R^{n} which satisfy the compatibility condition (1.32) then the formula

s(x) = φ_{α}(x, s_{α}(x))

determines the mapping s : B⟶E uniquely (that is, independent of the choice of chart U_{α}).

The map s is a section of the bundle ξ.

1.3.1 Operations of direct sum and tensor product

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 ξ_{1} and ξ_{2} be two vector bundles with fibers V_{1} and V_{2}, respectively. Denote the transition functions of these bundles in a common atlas of charts by φ_{αβ}^{1}(x) and φ_{αβ}^{2}(x). Notice that values of the transition function φ_{αβ}^{1}(x) lie in the group GL (V_{1}) whereas the values of the transition function φ_{αβ}^{2}(x) lie in the group GL (V_{2}). Hence the transition functions φ_{αβ}^{1}(x) and φ_{αβ}^{2}(x) can be considered as matrix–values functions of orders n_{1} = dim V_{1} and n_{2} = dim V_{2}, respectively. Both of them should satisfy the conditions (1.9) from the section 2.

We form a new space V = V_{1} ⊕ V_{2}. The linear transformation group GL (V) is the group of matrices of order n = n_{1} + n_{2} which can be decomposed into blocks with respect to decomposition of the space V into the direct sum V_{1}⊕V_{2}. Then the group GL (V) has the subgroup GL (V_{1}) ⊕ GL (V_{2}) of matrices which have the following form:

A = ∥ \begin{matrix}A_{1} & 0 \\ 0 & A_{2} \end{matrix} ∥ = A_{1} ⊕ A_{2}, A_{1} = GL (V_{1}), A_{2} = GL (V_{2}).

Then we can construct new the transition functions

φ_{αβ}(x) = φ_{αβ}^{1}(x) ⊕ φ_{αβ}^{2}(x) = ∥ \begin{matrix}φ_{αβ}^{1}(x) & 0 \\ 0 & φ_{αβ}^{2}(x) \end{matrix} ∥ .  (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 = V_{1} ⊕ V_{2}. The bundle constructed above is called the direct sum of vector bundles ξ_{1} and ξ_{2} and is denoted by ξ = ξ_{1} ⊕ ξ_{2}. The direct sum operation can be constructed in a geometric way. Namely, let p_{1} : E_{1}⟶B be a vector bundle ξ_{1} and let p_{2} : E_{2}⟶B be a vector bundle ξ_{2}. Consider the Cartesian product of total spaces E_{1} × E_{2} and the projection

p_{3} = p_{1} × p_{2} : E_{1} × E_{2}⟶B × B.

It is clear that p is vector bundle with the fiber V = V_{1} ⊕ V_{2}.

Consider the diagonal Δ ⊂ B × B, that is, the subset Δ = {(x, x) : x ∈ B}. The diagonal Δ is canonically homeomorphic to the space B. The restriction of the bundle p_{3} to Δ ≈ B is a vector bundle over B. The total space E of this bundle is the subspace E ⊂ E_{1} × E_{2} that consists of the vectors (y_{1}, y_{2}) such that

p_{1}(y_{1}) = p_{2}(y_{2}).

It is easy to check that {U_{α_{1}} × U_{α_{2}}} gives an atlas of charts for the bundle p_{3}.

The transition functions φ_{(β_{1}β_{2})(α_{1}α_{2})}(x, y) on the intersection of two charts (U_{α_{1}} × U_{α_{2}}) ∩ (U_{β_{1}} × U_{β_{2}}) have the following form:

φ_{(β_{1}β_{2})(α_{1}α_{2})}(x, y) = ∥ \begin{matrix}φ_{β_{1}α_{1}}^{1}(x) & 0 \\ 0 & φ_{β_{2}α_{2}}^{2}(y) \end{matrix} ∥ .

Hence on the diagonal Δ ≈ B the atlas consists of sets U_{α} ≈ Δ ∩ (U_{α} × U_{α}).

Then the transition functions for the restriction of the bundle p_{3} on the diagonal have the following form:

φ_{(ββ)(αα)}(x, x) = ∥ \begin{matrix}φ_{βα}^{1}(x) & 0 \\ 0 & φ_{βα}^{2}(x) \end{matrix} ∥ .  (1.34)

So the transition functions (1.34) coincide with the transition functions defined for the direct sum of the bundles ξ_{1} and ξ_{2}. Now let us proceed to the definition of tensor product of vector bundles. As before, let ξ_{1} and ξ_{2} be two vector bundles with fibers V_{1} and V_{2} and let φ_{αβ}^{1}(x) and φ_{αβ}^{2}(x) be the transition functions of the vector bundles ξ_{1} and ξ_{2},

φ_{αβ}^{1}(x) ∈ GL (V_{1}), φ_{αβ}^{2}(x) ∈ GL (V_{2}), x ∈ V_{α} ∩ V_{β}.

Let V = V_{1} ⊗ V_{2}. Then form the tensor product A_{1} ⊗ A_{2} ∈ GL (V_{1} ⊗ V_{2}) of the two matrices A_{1} ∈ GL (V_{1}), A_{2} ∈ GL (V_{2}). Put

φ_{αβ}(x) = φ_{αβ}^{1}(x) ⊗ φ_{αβ}^{2}(x).

Now we have obtained a family of the matrix value functions φ_{αβ}(x) which satisfy the conditions (1.9) from the section 2. The corresponding vector bundle ξ with fiber V = V_{1} ⊗ V_{2} and transition functions φ_{αβ}(x) will be called the tensor product of bundles ξ_{1} and ξ_{2} and denoted by

ξ = ξ_{1} ⊗ ξ_{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 ξ_{1} and ξ_{2}:

1. Pass to the principal GL (V_{1})– and GL (V_{2})– bundles;

2. Construct the principal ( GL (V_{1}) × GL (V_{2}))– bundle over the Cartesian square B × B;

3. Restrict to the diagonal Δ, homeomorphic to the space B.

4. Finally, form a new principal bundle by means of the relevant representations of the structure group GL (V_{1}) × GL (V_{2}) in the groups GL (V_{1} ⊕ V_{2}) and GL (V_{1} ⊗ V_{2}), respectively.

The difference between the operations of direct sum and tensor product lies in choice of the representation of the group GL (V_{1}) × GL (V_{2}).

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

(ξ_{1} ⊕ ξ_{2}) ⊕ ξ_{3} = ξ_{1} ⊕ (ξ_{2} ⊕ ξ_{3}).

Introduction to bundle theory 25

This relation is a consequence of the commutative diagram

GL (V_{1} ⊕ V_{2}) × GL (V_{3})

↗ ρ_{1} ↘ ρ_{2}
GL (V_{1}) × GL (V_{2}) × GL (V_{3})  GL (V_{1} ⊕ V_{2} ⊕ V_{3})
↘ ρ_{3} ↗ ρ_{4}
GL (V_{1}) × GL (V_{2} ⊕ V_{3})

where

ρ_{1}(A_{1}, A_{2}, A_{3}) = (A_{1} ⊕ A_{2}, A_{3}),
ρ_{2}(B, A_{3}) = B ⊕ A_{3},
ρ_{3}(A_{1}, A_{2}, A_{3}) = (A_{1}, A_{2} ⊕ A_{3}),
ρ_{4}(A_{1}, C) = A_{1} ⊕ C.

Then

ρ_{2}ρ_{1}(A_{1}, A_{2}, A_{3}) = (A_{1} ⊕ A_{2}) ⊕ A_{3},
ρ_{4}ρ_{3}(A_{1}, A_{2}, A_{3}) = A_{1} ⊕ (A_{2} ⊕ A_{3}).

It is clear that

ρ_{2}ρ_{1} = ρ_{4}ρ_{3}

since the relation

(A_{1} ⊕ A_{2}) ⊕ A_{3} = A_{1} ⊕ (A_{2} ⊕ A_{3})

is true for matrices.

2. Associativity for tensor products

(ξ_{1} ⊗ ξ_{2}) ⊗ ξ_{3} = ξ_{1} ⊗ (ξ_{2} ⊗ ξ_{3}).

This relation is a consequence of the commutative diagram

GL (V_{1} ⊗ V_{2}) × GL (V_{3})

↗ ρ_{1} ↘ ρ_{2}
GL (V_{1}) × GL (V_{2}) × GL (V_{3})  GL (V_{1} ⊗ V_{2} ⊗ V_{3})
↘ ρ_{3} ↗ ρ_{4}

GL (V_{1}) × GL (V_{2} ⊗ V_{3})

The commutativity is implied from the relation

(A_{1} ⊗ A_{2}) ⊗ A_{3} = A_{1} ⊗ (A_{2} ⊗ A_{3})

for matrices.

3. Distributivity:

(ξ_{1} ⊕ ξ_{2}) ⊗ ξ_{3} = (ξ_{1} ⊗ ξ_{3}) ⊕ (ξ_{2} ⊗ ξ_{3}).

This property is implied by the corresponding relation

(A_{1} ⊕ A_{2}) ⊗ A_{3} = (A_{1} ⊗ A_{3}) ⊕ (A_{2} ⊗ A_{3}).

for matrices.

4. Denote the trivial vector bundle with the fiber R^{n} by n̄. The total space of trivial bundle is homeomorphic to the Cartesian product B × R_{n} and it follows that

n̄ = ¯1 ⊕ ¯1 ⊕ … ⊕ ¯1(n times).

and

ξ ⊗ ¯1 = ξ,
ξ ⊗ n̄ = ξ ⊕ ξ ⊕ … ⊕ ξ(n times).

1.3.2 Other operations with vector bundles

Let V = Hom (V_{1}, V_{2}) be the vector space of all linear mappings from the space V_{1} to the space V_{2}. 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 (V_{1})× GL (V_{2}) into the group GL (V) which to any pair A_{1} ∈ GL (V_{1}), A_{2} ∈ GL (V_{2}) associates the mapping

ρ(A_{1}, A_{2}) : Hom (V_{1}, V_{2})⟶Hom (V_{1}, V_{2})

by the formula

ρ(A_{1}, A_{2})(f) = A_{2} ∘ 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 ξ_{1} and ξ_{2} with fibers V_{1} and V_{2} and transition functions φ_{αβ}^{1}(x) and φ_{αβ}^{2}(x) a new vector bundle with fiber V and transition functions

φ_{αβ}(x) = ρ(φ_{αβ}^{1}(x), φ_{αβ}^{2}(x))  (1.36)

This bundle is denoted by HOM (ξ_{1}, ξ_{2}).

When V_{2} = R^{1}, the space Hom (V_{1}, R^{1}) is denoted by V_{1}^{∗}. Correspondingly, when ξ_{2} = ¯1 the bundle HOM (ξ, ¯1) will be denoted by ξ^{∗} and called the dual bundle . It is easy to check that the bundle ξ^{∗} can be constructed from ξ by means of the representation of the group GL (V) to itself by the formula

A⟶(A^{t})^{−1}, A ∈ GL (V).

There is a bilinear mapping

V × V^{∗}⟶^{β}R^{1},

which to each pair (x, h) associates the value h(x).

Consider the representation of the group GL (V) on the space V × V^{∗} defined by matrix

A⟶ ∥ \begin{matrix}A & 0 \\ 0 & ρ(A, 1) \end{matrix} ∥

(see (1.35)). Then the structure group GL (V × V^{∗}) of the bundle ξ ⊕ ξ^{∗} is reduced to the subgroup GL (V). The action of the group GL (V) on the V × V^{∗} has the property that the mapping β is equivariant with respect to trivial action of the group GL (V) on R^{1}. 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

β̄ : ξ ⊕ β^{∗}⟶¯1,

which coincides with β on each fiber.

Let Λ_{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

Λ_{k}(A) : Λ_{k}(V)⟶Λ_{k}(V),

that is, the representation

Λ_{k} : GL (V)⟶ GL (Λ_{k}(V)).

The corresponding operation for vector bundles will be called the operation of the k-th exterior power and the result denoted by Λ_{k}(ξ). Similar to vector spaces, for vector bundles one has

Λ_{1}(ξ) = ξ,
Λ_{k}(ξ) = 0 for k > dim ξ,
Λ_{k}(ξ_{1} ⊕ ξ_{2}) = ⊕_{α=0}^{k}Λ_{α}(ξ_{1}) ⊗ Λ_{k−α}(ξ_{2}),  (1.37)

where by definition

Λ_{0}(ξ) = ¯1.

It is convenient to write the relation (1.37) using the partition function. Let us introduce the polynomial

Λ_{t}(ξ) = Λ_{0}(ξ) + Λ_{1}(ξ)t + Λ_{2}(ξ)t^{2} + … + Λ_{n}(ξ)t^{n}.

Then

Λ_{t}(ξ_{1} ⊕ ξ_{2}) = Λ_{t}(ξ_{1}) ⊗ Λ_{t}(ξ_{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

Consider two vector bundles ξ_{1} and ξ_{2} where

ξ_{i} = {p_{i} : E_{i}⟶B, V_{i} is fiber}.

Consider a fiberwise continuous mapping

f : E_{1}⟶E_{2}.

The map f will be called a linear map of vector bundles or homomorphism of bundles if f is linear on each fiber. The family of all such linear mappings will be denoted by Hom (ξ_{1}, ξ_{2}). Then the following relation holds:

Hom (ξ_{1}, ξ_{2}) = Γ(HOM (ξ_{1}, ξ_{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 V_{1} to the fiber V_{2}, parametrized by points of the base B.

To prove the relation (1.39), let us express elements from both the left-hand and right-hand sides of (1.39) in terms of local coordinates. Consider an atlas {U_{α}} and coordinate homeomorphisms φ_{α}^{1}, φ_{α}^{2} for bundles ξ_{1}, ξ_{2}. By means of the mapping f : E_{1}⟶E_{2} we construct a family of mappings:

(φ_{α}^{2})^{−1}fφ_{α}^{1} : U_{α} × V_{1}⟶U_{α} × V_{2},

defined by the formula:

[(φ_{α}^{2})^{−1}fφ_{α}^{1}](x, h) = (x, f_{α}(x)h),

for the continuous family of linear mappings

f_{α}(x) : V_{1}⟶V_{2}.

On the intersection of two charts U_{α} ∩U_{β} two functions f_{α}(x) and f_{β}(x) satisfy the following condition

φ_{βα}^{2}(x)f_{α}(x) = f_{β}(x)φ_{βα}^{1}(x),

or

f_{β}(x) = φ_{βα}^{2}(x)f_{α}(x)φ_{αβ}^{1}(x).

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

f_{β}(x) = φ_{βα}(x)(f_{α}(x)).  (1.40)

In other words, the family of functions

f_{α}(x) ∈ V = Hom (V_{1}, V_{2}), x ∈ U_{α}

satisfies the condition (1.40), that is, determines a section of the bundle HOM (ξ_{1}, ξ_{2}). Conversely, given a section of the bundle HOM (ξ_{1}, ξ_{2}), that is, a family of functions f_{α}(x) satisfying condition (1.40) defines a linear mapping from the bundle ξ_{1} to the bundle ξ_{2}. In particular, if

ξ_{1} = ¯1, V_{1} = R^{1}

then

Hom (V_{1}, V_{2}) = V_{2}.

Hence

HOM (¯1, ξ_{2}) = ξ_{2}.

Hence

Γ(ξ_{2}) = Hom (¯1, ξ_{2}),

that is, the space of all sections of vector bundle ξ_{2} is identified with the space of all linear mappings from the one dimensional trivial bundle ¯1 to the bundle ξ_{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 × V ⟶R^{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 : ξ ⊕ ξ⟶¯1  (1.41)

which is bilinear in each fiber and is called a bilinear form on the bundle ξ. Just as on a linear space, a bilinear form on a vector bundle (1.41) induces a linear mapping from the vector bundle ξ to its dual bundle ξ^{∗}

f̄: ξ⟶ξ^{∗},

such that f decomposes into the composition

ξ ⊕ ξ ⟶^{f̄⊕Id}ξ^{∗} ⊕ ξ⟶^{β}¯1,

where

Id : ξ⟶ξ

is the identity mapping and

ξ ⊕ ξ ⟶^{f̄⊕Id}ξ^{∗} ⊕ ξ

is the direct sum of mappings f̄ 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 bundle ξ.

Theorem 5 Let ξ be a finite dimensional vector bundle over a compact base space B. Then there exists a scalar product on the bundle ξ, that is, a nondegenerate, positive, symmetric bilinear form on the ξ.

Proof.

We must construct a fiberwise mapping (1.41) which is bilinear, symmetric, positive, nondegenerate form in each fiber. This means that if x ∈ B, v_{1}, v_{2} ∈ p^{−1}(x) then the value f(v_{1}, v_{2}) can be identified with a real number such that

f(v_{1}, v_{2}) = f(v_{2}, v_{1})

and

f(v, v) > 0 for any v ∈ p^{−1}(x), v ≠ 0.

Consider the weaker condition

f(v, v) ≥ 0.

Then we obtain a nonnegative bilinear form on the bundle ξ. If f_{1}, f_{2} are two nonnegative bilinear forms on the bundle ξ then the sum f_{1} + f_{2} and a linear combination φ_{1}f_{1} + φ_{2}f_{2} for any two nonnegative continuous functions φ_{1} and φ_{2} on the base B gives a nonnegative bilinear form as well.

Let {U_{α}} be an atlas for the bundle ξ. The restriction ξ|_{U_{α}} is a trivial bundle and is therefore isomorphic to a Cartesian product U_{α} × V where V is fiber of ξ. Therefore the bundle ξ|_{U_{α}} has a nondegenerate positive definite bilinear form

f_{α} : ξ|_{U_{α}} ⊕ ξ|_{U_{α}}⟶¯1.

In particular, if v ∈ p^{−1}(x), x ∈ U_{α} and v ≠ 0 then

f_{α}(v, v) > 0.

Consider a partition of unity {g_{α}} subordinate to the atlas {U_{α}}. Then

0 ≤ g_{α}(x) ≤ 1,

∑_{α} g_{α}(x) ≡ 1,

Supp g_{α} ⊂ U_{α}.

We extend the form f_{α} by formula

f̄_{α}(v_{1}, v_{2}) = \{ \begin{matrix}g_{α}(x)f_{α(v_{1},v_{2})} & v_{1}, v_{2} ∈ p^{−1}(x) & x ∈ U_{α}, \\ 0 & v_{1}, v_{2} ∈ p^{−1}(x) & x ∉ U_{α}. \end{matrix}  (1.42)

It is clear that the form (1.42) defines a continuous nonnegative form on the bundle ξ. Put

f(v_{1}, v_{2}) = ∑_{α} f_{α}(v_{1}, v_{2}).  (1.43)

The form (1.43) is then positive definite. Actually, let 0 ≠ v ∈ p^{−1}(x). Then there is an index α such that

g_{α}(x) > 0.

This means that

x ∈ U_{α} and f_{α}(v, v) > 0.

Hence

f̄_{α}(v, v) > 0

and

f(v, v) > 0.

Theorem 6 For any vector bundle ξ over a compact base space B with dim ξ = n, the structure group GL (n, R) reduces to subgroup O(n).

Proof.

Let us give another geometric interpretation of the property that the bundle ξ is locally trivial. Let U_{α} be a chart and let

φ_{α} : U_{α} × V ⟶p^{−1}(U_{α})

be a trivializing coordinate homeomorphism. Then any vector v ∈ V defines a section of the bundle ξ over the chart U_{α}

σ : U_{α}⟶p^{−1}(U_{α}),
σ(x) = φ_{α}(x, v) ∈ p^{−1}(U_{α}).

If v_{1}, …, v_{n} is a basis for the space V then corresponding sections

σ_{k}^{α}(x) = φ_{α}(x, v_{k})

form a system of sections such that for each point x ∈ U_{α} the family of vectors σ_{1}^{α}(x), …, σ_{n}^{α}(x) ∈ p^{−1}(x) is a basis in the fiber p^{−1}(x).

Conversely, if the system of sections

σ_{1}^{α}, …, σ_{n}^{α} : U_{α}⟶p^{−1}(U_{α})

forms basis in each fiber then we can recover a trivializing coordinate homeomorphism

φ_{α}(x, ∑_{i} λ_{i}v_{i}) = ∑_{i} λ_{i}σ_{i}^{α}(x) ∈ p^{−1}(U_{α}).

From this point of view, the transition function φ^{βα} = φ_{β}^{−1}φ_{α} has an interpretation as a change of basis matrix from the basis {σ_{1}^{α}(x), …, σ_{n}^{α}(x)} to {σ_{1}^{β}(x), …, σ_{n}^{β}(x)} in the fiber p^{−1}(x), x ∈ U_{α} ∩ U_{β}. Thus Theorem 6 will be proved if we construct in each chart U_{α} a system of sections {σ_{1}^{α}, …, σ_{n}^{α}} which form an orthonormal basis in each fiber with respect to a inner product in the bundle ξ. Then the transition matrices from one basis {σ_{1}^{α}(x), …, σ_{n}^{α}(x)} to another basis {σ_{1}^{β}(x), …, σ_{n}^{β}(x)} will be orthonormal, that is, φ_{βα}(x) ∈ O(n). The proof of Theorem 6 will be completed by the following lemma.

Lemma 1 Let ξ be a vector bundle, f a scalar product in the bundle ξ and {U_{α}} an atlas for the bundle ξ. Then for any chart U_{α} there is a system of sections {σ_{1}^{α}, …, σ_{n}^{α}} orthonormal in each fiber p^{−1}(x), x ∈ U_{α}.

Proof.

The proof of the lemma simply repeats the Gramm-Schmidt method of construction of orthonormal basis. Let

τ_{1}, …, τ_{n} : U_{α}⟶p^{−1}(U_{α})

be an arbitrary system of sections forming a basis in each fiber p^{−1}(x), x ∈ U_{α}. Since for any x ∈ U_{α},

τ_{1}(x) ≠ 0

one has

f(τ_{1}(x), τ_{1}(x)) > 0.

Put

τ′_{1}(x) = \frac{τ_{1}(x)}{\sqrt{f(τ_{1}(x), τ_{1}(x))}}.

The new system of sections τ′_{1}, τ_{2}, …, τ_{n} forms a basis in each fiber. Put

τ′′_{2}(x) = τ_{2}(x) − f(τ_{2}(x), τ′_{1}(x))τ′_{1}(x).

The new system of sections τ′_{1}, τ′′_{2}, τ_{3}(x), …, τ_{n} forms a basis in each fiber. The vectors τ′_{1}(x) have unit length and are orthogonal to the vectors τ′′_{2}(x) at each point x ∈ U_{α}. Put

τ′_{2}(x) = \frac{τ′′_{2}(x)}{\sqrt{f(τ′′_{2}(x), τ′′_{2}(x))}}.

Again, the system of sections τ′_{1}, τ′_{2}, τ_{3}(x), …, τ_{n} forms a basis in each fiber and, moreover, the vectors τ′_{1}, τ′_{2} are orthonormal.

Then we rebuild the system of sections by induction. Let the sections τ′_{1}, …, τ′_{k}, τ_{k+1}(x), …, τ_{n} form a basis in each fiber and suppose that the sections τ′_{1}, …, τ′_{k} be are orthonormal in each fiber. Put

τ′′_{k+1}(x) = τ_{k+1}(x) − ∑_{i=1}^{k} f(τ_{k+1}(x), τ′_{i}(x))τ′_{i}(x),

τ′_{k+1}(x) = \frac{τ′′_{k+1}(x)}{\sqrt{f(τ′′_{k+1}(x), τ′′_{k+1}(x))}}.

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

Remarks

1. In Lemma 1 we proved a stronger statement: if {τ_{1}, …, τ_{n}} is a system of sections of the bundle ξ in the chart U_{α} which is a basis in each fiber p^{−1}(x) and if in addition vectors {τ_{1}, …, τ_{k}} are orthonormal then there are sections {τ′_{k+1}, …, τ′_{n}} such that the system

{τ_{1}, …, τ_{k}, τ′_{k+1}, …, τ′_{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 ξ with the base B equipped with a scalar product. The value of the scalar product on a pair of vectors v_{1}, v_{2} ∈ p^{−1}(x) will be denoted by ⟨v_{1}, v_{2}⟩.

Consider a homomorphism f : ξ⟶ξ of the vector bundle ξ to itself. The space of all such homomorphisms Hom (ξ, ξ) has the natural operations of summation and multiplication by continuous functions. Thus the space Hom (ξ, ξ) is a module over the algebra C(B) of continuous functions on the base B. Further, the operation of composition equips the space Hom (ξ, ξ) with the structure of algebra. Using a scalar product in the bundle ξ one can introduce a norm in the algebra Hom (ξ, ξ) and hence equip it with a structure of a Banach algebra: for each homomorphism f : ξ⟶ξ put

∥f∥ = sup_{v≠0} \frac{∥f(v)∥}{∥v∥},  (1.44)

where

∥v∥ = \sqrt{⟨v, v⟩}.

The space Hom (ξ, ξ) is complete with respect to the norm (1.44). Indeed, if a sequence of homomorphisms f_{n} : ξ⟶ξ is a Cauchy sequence with respect to this norm, that is,

lim_{n,m⟶∞} ∥f_{n} − f_{m}∥ = 0

then for any fixed vector v ∈ p^{−1}(x) the sequence f_{n}(v) ∈ p^{−1}(x) is a Cauchy sequence as well since

∥f_{n}(v) − f_{m}(v)∥leq∥f_{n} − f_{m}∥ · ∥v∥.

Hence there exists a limit

f(v) = lim_{n⟶∞} f_{n}(v).

The mapping f : ξ⟶ξ is evidently linear. To show its continuity one should consider the mappings

h_{n,α} = φ_{α}^{−1}f_{n}φ_{α} : U_{α} × V ⟶U_{α} × V,

which are defined by the matrix valued functions on the U_{α}. 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 ξ defines an adjoint linear mapping f^{∗} by the formula

⟨f^{∗}(v_{1}), v_{2}⟩ = ⟨v_{1}, f(v_{2})⟩, v_{1}, v_{2} ∈ 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 ξ to itself will be denoted by 1. Consider a homomorphism

I : ξ⟶ξ

which satisfies the condition

I^{2} = −1.

The restriction

I_{x} = I_{|p^{−1}(x)}

is an automorphism of the fiber p^{−1}(x) with the property

I_{x}^{2} = −1.

Hence the transformation I_{x} 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 {v_{1}, …, v_{n}} has the property that the system {v_{1}, …, v_{n}, Iv_{1}, …, Iv_{n}} is a real basis in the space V, dim V = 2n, then the system {v_{1}, …, v_{n}} is a complex basis of V. Fix a point x_{0}. The space p^{−1}(x) is a complex vector space with respect to the operator I_{x_{0}}, and hence there is a complex basis {v_{1}, …, v_{n}} ⊂ p^{−1}(x).

Let U_{α} ∋ x_{0} be a chart for the bundle ξ. There are sections {τ_{1}, …, τ_{n}} in the chart U_{α} such that

τ_{k}(x_{0}) = v_{k}, 1 ≤ k ≤ n.

Then the system of sections {τ_{1}, …, τ_{n}, Iτ_{1}, …, Iτ_{n}} forms a basis in the fiber p^{−1}(x_{0}) and therefore forms basis in each fiber p^{−1}(x) in sufficiently small neighborhood U ∋ x_{0}. Hence the system {τ_{1}(x), …, τ_{n}(x)} forms a complex basis in each fiber p^{−1}(x) in the neighborhood U ∋ x_{0}. This means that there is a sufficient fine atlas {U_{α}} and a system of sections {τ_{1}^{α}(x), …, τ_{n}^{α}(x)} on each chart U_{α} giving a complex basis in each fiber p^{−1}(x) in the neighborhood U_{α} ∋ x_{0}. Fix a complex basis {e_{1}, …, e_{n}} in the complex vector space C^{n}. Put

φ_{α} : U_{α} × C^{n} ⟶ p^{−1}(U_{α}),
φ_{α}(x, ∑_{k=1}^{n} z_{k}e_{k}) = ∑_{k=1}^{n} u_{k}τ_{k}^{α}(x) + ∑_{k=1}^{n} v_{k}I_{x}(τ_{k}^{α}(x))

= ∑_{k=1}^{n} z_{k}τ_{k}^{α}(x),
z_{k} = u_{k} + iv_{k}, 1 ≤ k ≤ n.

Then the transition functions φ_{βα} = φ_{β}^{−1}φ_{α} are determined by the transition matrices from the complex basis {τ_{1}^{α}(x), …, τ_{n}^{α}(x)} to the complex basis {τ_{1}^{β}(x), …, τ_{n}^{β}(x)}. These matrices are complex, that is, belong to the group GL (n, C). A vector bundle with the structure group GL (n, C) is called a complex vector bundle .

Let ξ be a real vector bundle. In the vector bundle ξ⊕ξ, introduce the structure of a complex vector bundle by means the homomorphism

I : ξ ⊕ ξ⟶ξ ⊕ ξ

given by

I(v_{1}, v_{2}) = (−v_{2}, v_{1}), v_{1}, v_{2} ∈ p^{−1}(x).  (1.45)

The complex vector bundle defined by (1.45) is called the complexification of the bundle ξ and is denoted by cξ. Conversely, forgetting of the structure of complex bundle on the complex vector bundle ξ turns it into a real vector bundle. This operation is called the realification of a complex vector bundle ξ and is denoted by rξ. It is clear that

rcξ = ξ ⊕ ξ.

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 crξ. If ξ is a complex bundle, that is, ξ is a real vector bundle with a homomorphism I : ξ⟶ξ giving the structure of complex bundle on it. By definition the vector bundle crξ is a new real vector bundle η = ξ ⊕ ξ with the structure of complex vector bundle defined by a homomorphism

I_{1} : ξ ⊕ ξ ⟶ ξ ⊕ ξ,
I_{1}(v_{1}, v_{2}) = (−v_{2}, v_{1}).  (1.46)

The mapping (1.46) defines a new complex structure in the vector bundle η which is, in general, different from the complex structure defined by the mapping I.

Let us split the bundle η in another way:

f : ξ ⊕ ξ ⟶ ξ ⊕ ξ,
f(v_{1}, v_{2}) = (I(v_{1} + v_{2}), v_{1} − v_{2}).

and define in the inverse image a new homomorphism

I_{2}(v_{1}, v_{2}) = (Iv_{1}, −Iv_{2}).

Then

fI_{2} = I_{1}f  (1.47)

because

fI_{2}(v_{1}, v_{2}) = f(Iv_{1}, −Iv_{2}) = (v_{2} − v_{1}, I(v_{1} + v_{2})),  (1.48)
I_{1}f(v_{1}, v_{2}) = I_{1}(I(v_{1} + v_{2}), v_{1} − v_{2}) = (v_{2} − v_{1}, I(v_{1} + v_{2})).  (1.49)

Comparing (1.49) and (1.48) we obtain (1.47). Thus the mapping f gives an isomorphism of the bundle crξ (in the image) with the sum of two complex vector bundles: the first is ξ and the second summand is homeomorphic to ξ but with different complex structure defined by the mapping I′,

I′(v) = −I(v).

This new complex structure on the bundle ξ is denoted by ξ̄. The vector bundle ξ̄ is called the complex conjugate of the complex bundle ξ. Note that the vector bundles ξ and ξ̄ 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

crξ = ξ ⊕ ξ̄.

The next proposition gives a description of the complex conjugate vector bundle in term of the transition functions.

Proposition 1 Let the

φ_{βα} : U_{αβ}⟶ GL (n, C)

be transition functions of a complex vector bundle ξ. Then the complex conjugate vector bundle ξ̄ is defined by the complex conjugate matrices φ̄_{βα}.

Proof.

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 {e_{1}, …, e_{n}} of the space V is simultaneously the complex basis of V′. But if the vector v ∈ V has complex coordinates (z_{1}, …, z_{n}) then the same vector v in V′ has coordinates (z̄_{1}, …, z̄_{n}).

Hence if {f_{1}, …, f_{n}} is another complex basis then the complex matrix of change from the first coordinate system to another is defined by expanding vectors {f_{1}, …, f_{n}} in terms of the basis {e_{1}, …, e_{n}}.

For the space V these expansions are

f_{k} = ∑_{j} z_{jk}e_{j},  (1.50)

and in the space V′ these expansions are

f_{k} = ∑_{j} z̄_{jk}e_{j}.  (1.51)

Relations (1.50) and (1.51) prove the proposition (1).

Proposition 2 A complex vector bundle ξ has the form ξ = cη if and only if there is a real linear mapping

∗ : ξ⟶ξ  (1.52)

such that

∗^{2} = 1, ∗I = −I∗,  (1.53)

where I is the multiplication by the imaginary unit.

Proof.

First we prove the necessity. As a real vector bundle the bundle ξ has the form ξ = η ⊕ η. Then define a mapping ∗ : η ⊕ η⟶η ⊕ η by formula

∗(v_{1}, v_{2}) = (v_{1}, −v_{2}).

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 τ is called real if ∗(τ(x)) = τ(x) for any x ∈ B.

A homomorphism f : ξ⟶ξ 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.

1.4.2 Subbundles

Let f : ξ_{1}⟶ξ_{2} be a homomorphism of vector bundles with a common base B and assume that the fiberwise mappings f_{x} : (ξ_{1})_{x}⟶(ξ_{2})_{x} have constant rank. Let

p_{1} : E_{1}⟶B
p_{2} : E_{2}⟶B

be the projections of the vector bundles ξ_{1} and ξ_{2}. Put

E_{0} = {y ∈ E_{1} : f(y) = 0 ∈ p_{2}^{−1}(x), x = p_{1}(y)}
E = f(E_{1}).

Theorem 7 1. The mapping

p_{0} = p_{1}|E_{0} : E_{0}⟶B

is a locally trivial bundle which admits a unique vector bundle structure ξ_{0} such that natural inclusion E_{0} ⊂ E_{1} is a homomorphism of vector bundles.

2. The mapping

p = p_{2}|E : E⟶B  (1.54)

is a locally trivial bundle which admits a unique vector bundle structure ξ such that the inclusion E ⊂ E_{2} and the mapping f : E_{1}⟶E are homomorphisms of vector bundles.

3. There exist isomorphisms

φ : ξ_{1} ⟶ ξ_{0} ⊕ ξ,
ψ : ξ_{2} ⟶ ξ ⊕ η,

such that composition

ψ ∘ f ∘ φ^{−1} : ξ_{0} ⊕ ξ⟶ξ ⊕ η

has the matrix form

ψ ∘ f ∘ φ^{−1} = ( \begin{matrix}0 & Id \\ 0 & 0 \end{matrix} ) .

The bundle ξ_{0} is called the kernel of the mapping f and denoted Ker f, the bundle ξ is called the image of the mapping f and denoted Im f. So we have

dim ξ_{1} = dim Ker f + dim Im f.

Proof.

Assume that vector bundles ξ_{1} and ξ_{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 x_{0} ∈ B there is a chart U ∋ x_{0} and a system of continuous sections σ_{1}, …, σ_{k} : U⟶E which form a basis in each subspace f_{x}(p_{1}^{−1}(x)) ⊂ p_{2}^{−1}(x). For this fix a basis {e_{1}, …, e_{k}} in the subspace f_{x_{0}}(p_{1}^{−1}(x_{0})). The mapping

f_{x_{0}} : p_{1}^{−1}(x_{0})⟶f_{x_{0}}(p_{1}^{−1}(x_{0}))

is an epimorphism and hence we may choose vectors g_{1}, …, g_{k} ∈ p_{1}^{−1}(x_{0}) such that

f_{x_{0}}(g_{j}) = e_{j}, 1 ≤ j ≤ k.

Since ξ_{1} is a locally trivial bundle for each x_{0} there is a chart U ∋ x_{0} such that over U the bundle ξ_{1} is isomorphic to a Cartesian product U × p_{1}^{−1}(x_{0}). Therefore there are continuous sections

τ_{1}, …, τ_{k} : U⟶E_{1}

such that

τ_{j}(x_{0}) = g_{j}, 1 ≤ j ≤ k.

Consider the sections

σ_{j} = f(τ_{j}), 1 ≤ j ≤ k.

Then

σ_{j}(x_{0}) = f(τ_{j}(x_{0})) = f(g_{j}) = e_{j}, 1 ≤ j ≤ k.

The sections σ_{j} are continuous and hence there is a smaller neighborhood U′ ∋ x_{0} such that for any point x ∈ U′ the system {σ_{1}(x), …, σ_{k}(x)} forms a basis of the subspace f_{x}(p_{1}^{−1}(x)). Thus we have shown that E is the total space of a vector bundle and the inclusion E ⊂ E_{2} is a homomorphism. Uniqueness of the vector bundle structure on the E is clear. Now we pass to the statement 1. Denote by P : ξ_{1}⟶ξ_{1} the fiberwise mapping which in each fiber p_{1}^{−1}(x) is the orthonormal projection onto the kernel Ker f_{x} ⊂ p_{1}^{−1}(x). It sufficient to check that P is continuous over each separate chart U. Notice that the kernel Ker f_{x} is the orthogonal complement of the subspace Im f_{x}^{∗}. The rank of f_{x}^{∗} equals the rank of f_{x}. Hence we can apply part 2 of Theorem 7 to the mapping f^{∗}.

Thus the image Im f^{∗} is a locally trivial vector bundle. Now let {σ_{1}, …, σ_{k}} be a system of sections,

σ_{j} : U⟶Im f^{∗} ⊂ E_{1},

forming a basis of the subspace Im f_{x}^{∗} ⊂ p_{1}^{−1}(x) for any x ∈ U. Then applying the projection P to a vector y ∈ p_{1}^{−1}(x) gives a decomposition of the vector y into a linear combination

y = P (y) + ∑_{j=1}^{k} λ_{j}σ_{j}(x)

satisfying

⟨P (y), σ_{j}(x)⟩ = 0, 1 ≤ j ≤ k

or

⟨y − ∑_{j=1}^{k} λ_{j}σ_{j}(x), σ_{l}(x)⟩ = 0, 1 ≤ l ≤ k.

The coefficients λ_{j} satisfy the following system of linear equations

∑_{j=1}^{k} λ_{j}⟨σ_{j}(x), σ_{l}(x)⟩ = ⟨y, σ_{l}(x)⟩, 1 ≤ l ≤ k.

The matrix ∥⟨σ_{j}(x), σ_{l}(x)⟩∥ depends continuously on x. Hence the inverse matrix also depends continuously and, as a consequence, the numbers λ_{j} depend continuously on x. It follows that the projection 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

ξ_{1} = Ker f ⊕ Im f^{∗},
ξ_{2} = Ker f^{∗} ⊕ Im f.  (1.55)

Now we can complete the proof of Proposition 2. Consider a vector bundle ξ with mapping (1.52) satisfying the condition (1.53). Consider a mapping

P = \frac{1}{2}(1 + ∗) : ξ⟶ξ.  (1.56)

The mapping P is a projection in each fiber and has a constant rank. Applying Theorem 7 we get

ξ = Im P ⊕ Ker P = ξ_{0} ⊕ ξ_{1}

and

I(ξ_{0}) = ξ_{1}, I(ξ_{1}) = ξ_{0}.

Hence

ξ = cξ_{0}.

The proof of Proposition 2 is now completed.

Theorem 8 Let ξ be a vector bundle over a compact base B. Then there is a vector bundle η over B such that

ξ ⊕ η = N̄ = a trivial bundle.

Proof.

Let us use Theorem 7. It is sufficient to construct a homomorphism

f : ξ⟶N̄,

where the rank of f equals dim ξ in each fiber. Notice that if ξ is trivial then such an f exists. Hence for any chart U_{α} there is a homomorphism

f_{α} : ξ|U_{α}⟶N̄_{α}, rank f_{α} = dim ξ.

Let {φ_{α}} be a partition of unity subordinate to the atlas {U_{α}}. Then each mapping φ_{α}f_{α} can be extended by the zero trivial mapping to a mapping

g_{α} : ξ⟶N̄_{α}, g_{α}|U_{α} = φ_{α}f_{α}.

The mapping g_{α} has the following property: if φ_{α}(x) ≠ 0 then rank g_{α}|_{x} = dim ξ. Let

g : ξ ⟶ ⊕_{α}N̄_{α} = N̄,
g = ⊕_{α}g_{α} that is
g(y) = (g_{1}(y), …, g_{α}(y), …).

It is clear that the rank of g at each point satisfies the relation

rank g_{α} ≤ rank g ≤ dim ξ.

Further, for each point x ∈ B there is such an α that φ_{α}(x) ≠ 0. Hence,

rank g ≡ dim ξ.  (1.57)

Therefore we can apply Theorem 7. By (1.57) we get

Ker g = 0.

Hence the bundle ξ is isomorphic to Im g and N̄ = Im g ⊕ η.

Theorem 9 Let ξ_{1}, ξ_{2} be two vector bundles over a base B and let B_{0} ⊂ B a closed subspace. Let

f_{0} : ξ_{1}|B_{0}⟶ξ_{2}|B_{0}

be a homomorphism of the restrictions of the bundles to the subspace B_{0}. Then the mapping f_{0} can be extended to a homomorphism

f : ξ_{1}⟶ξ_{2}, f|B_{0} = f_{0}.

Proof.

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 ξ_{1} ⊕ η_{1} and ξ_{2} ⊕ η_{2} be trivial bundles, let P : ξ_{2} ⊕ η_{2}⟶ξ_{2} be a natural projection, and let Q : ξ_{1}⟶ξ_{1} ⊕ η_{1} be the natural inclusion. Finally, let

h_{0} = f_{0} ⊕ 0 : (ξ_{1} ⊕ η_{1})|B_{0}⟶(ξ_{2} ⊕ η_{2})|B_{0}

be the direct sum of f_{0} and the trivial mapping. Then there is an extension of h_{0} to a homomorphism

h : ξ_{1} ⊕ η_{1}⟶ξ_{2} ⊕ η_{2}.

Let

f = P hQ : ξ_{1}⟶ξ_{2}.  (1.58)

It is clear that f|B_{0} = f_{0}.

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 f_{0} : ξ_{1}|B_{0}⟶ξ_{2}|B_{0} is a fiberwise monomorphism, prove that f_{0} can be extended to a fiberwise monomorphism f : ξ_{1}|U⟶ξ_{2}|U in some neighborhood U ⊃ B_{0}.

2. Under the conditions of previous exercise prove there is a trivial bundle N̄ and a fiberwise monomorphism

g : ξ_{1}⟶ξ_{2} ⊕ N̄,

extending f_{0}, that is,

g|B_{0} = (f_{0}, 0).

1.5 VECTOR BUNDLES RELATED TO MANIFOLDS

The most natural vector bundles arise from the theory of smooth manifolds. Recall that by an n–dimensional 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 n–dimensional linear space R^{n}. A homeomorphism

φ : U⟶V ⊂ R^{n}

is called a coordinate homeomorphism . The coordinate functions on the linear space R^{n} pulled back to points of the neighborhood U, that is, the compositions

x^{j} = x^{j} ∘ φ : U⟶R^{1}

are called coordinate functions on the manifold X in the neighborhood U. This system of functions {x^{1}, …, x^{n}} 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 {x^{1}, …, x^{n}} is called a chart . The system of charts {U_{α}, {x_{α}^{1}, …, x_{α}^{n}}} is called an atlas if {U_{α}} covers the manifold X, that is,

X = ∪_{α}U_{α}.

So each n–dimensional manifold has an atlas. If a point x ∈ X belongs to two charts,

x ∈ U_{α} ∩ U_{β},

then in a neighborhood of x there are two local systems of coordinates. In this case, the local coordinates x_{α}^{j} can expressed as functions of values of the local coordinates {x_{β}^{1}, …, x_{β}^{n}}, that is, there are functions f_{αβ}^{k} such that

x_{α}^{k} = φ_{αβ}^{k}(x_{β}^{1}, …, x_{β}^{n}).  (1.59)

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

x_{α}^{k} = x_{α}^{k}(x_{β}^{1}, …, x_{β}^{n}).

If an atlas {U_{α}, {x_{α}^{1}, …, x_{α}^{n}}} is taken such that all the transition functions are differentiable functions of the class C^{k}, 1 ≤ k ≤ ∞ then one says X has the structure of differentiable manifold of the class C^{k}. If all the transition functions are analytic functions then one says that X has the structure of an analytic manifold . In the case

n = 2m,
u_{α}^{k} = x_{α}^{k}, 1 ≤ k ≤ m,
v_{α}^{k} = x_{α}^{m+k}, 1 ≤ k ≤ m,
z_{α}^{k} = u_{α}^{k} + iv_{α}^{k}, 1 ≤ k ≤ m,

and the functions

z_{α}^{k} = φ_{αβ}^{k}(z_{β}^{1}, …, z_{β}^{m}) + iφ_{αβ}^{m+k}(z_{β}^{1}, …, z_{β}^{m})

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 n–dimensional manifold, {U_{α}, {x_{α}^{1}, …, x_{α}^{n}}} be an atlas. Fix a point x_{0} ∈ X. A tangent vector ξ to the manifold X at the point x_{0} is a system of numbers (ξ_{α}^{1}, …, ξ_{α}^{n}) satisfying the relations:

ξ_{α}^{k} = ∑_{j=1}^{n} ξ_{β}^{j} \frac{∂x_{α}^{k}}{∂x_{β}^{j}} (x_{0}).  (1.60)

The numbers (ξ_{α}^{1}, …, ξ_{α}^{n}) are called the coordinates or components of the vector ξ with respect to the chart {U_{α}, {x_{α}^{1}, …, x_{α}^{n}}}. The formula (1.60) gives the transformation law of the components of the tangent vector ξ 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 valency (1, 0). Consider a smooth curve γ passing through a point x_{0}, that is, a smooth mapping of the interval

γ : (−1, 1)⟶X, γ(0) = x_{0}.

In terms of a local coordinate system {x_{α}^{1}, …, x_{α}^{n}} the curve γ is determined by a family of smooth functions

x_{α}^{k}(t) = x_{α}^{k}(γ(t)), t ∈ (−1, 1).

Let

ξ_{α}^{k} = \frac{∂}{∂t}(x_{α}^{k}(t))|_{t=0}.  (1.61)

Clearly the numbers (1.61) satisfy a tensor law (1.60), that is, they define a tangent vector ξ at the point x_{0} to manifold X. This vector is called the tangent vector to the curve γ and is denoted by \frac{dγ}{dt} (0), that is,

ξ = \frac{dγ}{dt} (0).

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 ξ_{0} at the point x_{0} contains all vectors η in points x such that x ∈ U_{α} for some chart U_{α} and for some ε,

ρ(x, x_{0}) < ε,
∑_{k=1}^{n}(ξ_{0α}^{k} − η_{α}^{k})^{2} < ε.

The verification that the system of the neighborhoods V forms a base of a topology is left to the reader.

Let

π : T X⟶X  (1.62)

be the mapping which to any vector ξ associates its point x of tangency. Clearly, the mapping π 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 R^{n}. If U_{α} is a chart on the manifold X, we define a homeomorphism

φ_{α} : U_{α} × R^{n}⟶π^{−1}(U_{α})

which to the system (x_{0}, ξ^{1}, …, ξ^{n}) associates the tangent vector ξ whose components are defined by the formula

ξ_{β}^{k} = ∑_{j=1}^{n} ξ^{j} \frac{∂x_{β}^{k}}{∂x_{α}^{j}} (x_{0}).  (1.63)

It is easy to check that the definition (1.63) gives the components of a vector ξ, 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:

φ_{α}^{−1}(ξ) = (π(ξ), ξ_{α}^{1}, …, ξ_{α}^{n}),

where ξ_{α}^{k} are components of the vector ξ. Therefore the transition functions φ_{βα} = φ_{β}^{−1}φ_{α} are determined by the formula

φ_{βα}(x_{0}, ξ^{1}, …, ξ^{n}) = (x_{0}, ∑ ξ^{j} \frac{∂x_{β}^{1}}{∂x_{α}^{j}} (x_{0}), …, ∑ ξ^{j} \frac{∂x_{β}^{n}}{∂x_{α}^{j}} (x_{0})) . (1.64)

Formula (1.64) shows that the transition functions are fiberwise linear mappings. Hence the mapping π defines a vector bundle. The vector bundle π : T X⟶X is called the tangent bundle of the manifold X. The fiber T_{x}X is called the tangent space at the point x to manifold X.

The terminology described above is justified by the following. Let f : X⟶R^{N} be an inclusion of the manifold X in the Euclidean space R^{N}. In a local coordinate system (x_{α}^{1}, …, x_{α}^{n}) in a neighborhood of the point x_{0∈X}, the inclusion f is determined as a vector valued function of the variables (x_{α}^{1}, …, x_{α}^{n}) :

f(x) = f(x_{α}^{1}, …, x_{α}^{n}).  (1.65)

If we expand the function (1.65) by a Taylor expansion at the point x_{0} = (x_{0α}^{1}, …, x_{0α}^{n}) :

f(x_{α}^{1}, …, x_{α}^{n}) = f(x_{0α}^{1}, …, x_{0α}^{n})+
+ ∑_{j=1}^{n} \frac{∂f}{∂x_{α}^{j}} (x_{0α}^{1}, …, x_{0α}^{n})Δx_{α}^{j} + o(Δx_{α}^{k}).

Ignoring the remainder term o(Δx_{α}^{k}) we obtain a function g which is close to f :

g(x_{α}^{1}, …, x_{α}^{n}) = f(x_{0α}^{1}, …, x_{0α}^{n})+
+ ∑_{j=1}^{n} \frac{∂f}{∂x_{α}^{j}} (x_{0α}^{1}, …, x_{0α}^{n})Δx_{α}^{j}.

Then if the vectors

\{\frac{∂f}{∂x_{α}^{j}} (x_{0α}^{1}, …, x_{0α}^{n})Δx_{α}^{j}\} , 1 ≤ k ≤ n

are linearly independent the function g defines a linear n–dimensional subspace in R^{n}. It is natural to call this space the tangent space to the manifold X. Any vector ξ which lies in the tangent space to manifold X (having the initial point at x_{0}) can be uniquely decomposed into a linear combination of the basis vectors:

ξ = ∑_{j=1}^{n} \frac{∂f}{∂x_{α}^{j}} ξ_{α}^{j}.  (1.66)

The coordinate s {ξ_{α}^{k}} 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 definition of tangent vector as a system of components {ξ_{α}^{k}} determines by the formula (1.66) a tangent vector to the submanifold X in R^{N}. 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 ξ ∈ T X be a tangent vector at the point x_{0} and let γ be a smooth curve which goes through the point x_{0},

γ(0) = x_{0},

and which has tangent vector ξ, that is,

ξ = \frac{dγ}{dt} (0).

Then the curve f(γ(t)) in the manifold Y goes through the point y_{0} = f(x_{0}). Put

Df(ξ) = \frac{d(f(γ))}{dt}(0).  (1.67)

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 T_{x_{0}}X and T_{y_{0}}Y. Let {x_{α}^{1}, …, x_{α}^{n}} and {y_{β}^{1}, …, y_{β}^{m}} be local systems of coordinates in neighborhoods of points the x_{0} and y_{0}, respectively . Then the mapping f is defined as a family of functions

y_{β}^{j} = y_{β}^{j}(x_{α}^{1}, …, x_{α}^{n}).

If

x_{α}^{k} = x_{α}^{k}(t)

are the functions defining the curve γ(t) then

ξ_{α}^{k} = \frac{dx_{α}^{k}}{dt} (0).

Hence the curve f(γ(t)) is defined by the functions

y_{β}^{j} = y_{β}^{j}(x_{α}^{1}(t), …, x_{α}^{n}(t))

and the vector Df(ξ) is defined by the components

η^{j} = \frac{dy_{β}^{j}}{dt} (0) = ∑_{k=1}^{n} \frac{∂y_{β}^{j}}{∂x_{α}^{k}} (x_{0})\frac{dx_{α}^{k}}{dt} (0) =

∑_{k=1}^{n} \frac{∂y_{β}^{j}}{∂x_{α}^{k}} (x_{0})ξ^{k}.  (1.68)

Formula (1.68) shows firstly that the mapping Df is well- defined since the definition does not depend on the choice of curve γ but only on the tangent vector at the point x_{0}. Secondly, the mapping Df is fiberwise linear. The mapping Df is 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 R^{1} into itself:

f : R^{1}⟶R^{1}.

The tangent bundle of the manifold R^{1} is isomorphic to the Cartesian product R^{1} × R^{1} = R^{2}. Hence the differential

Df : R^{1} × R^{1}⟶R^{1} × R^{1}

in the coordinates (x, ξ) is defined by the formula

Df(x, ξ) = (x, f′(x)ξ).

The classical differential has the form

df = f′(x)dx.

So

Df(x, dx) = (x, df).

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 : X ⊂ Y

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 ⊕ η.

The complement η is called the normal bundle to the submanifold X of the manifold Y ^{1}. Each fiber of the bundle η over the point x_{0} consists of those tangent vectors to the manifold Y which are orthogonal to the tangent space T_{x_{0}}(X). The normal bundle will be denoted by ν(X ⊂ Y) or more briefly by ν(X).

It is clear that the notion of a normal bundle can be defined not only for submanifolds but for any immersion j : X⟶Y of the manifold X into the manifold Y. It is known that any compact manifold X has an inclusion in a Euclidean space R^{N} for some sufficiently large number N. Let j : X⟶R^{N} be such an inclusion. Then

j^{∗}(T R^{N}) = T X ⊕ ν(X ⊂ R^{N}).

1By construction the bundle η depends on the choice of metric on T Y. However, different metrics yield isomorphic complementary summands

The bundle T R^{N} is trivial and so

T X ⊕ ν(X) = N̄.  (1.69)

In this case the bundle ν(X) is called the normal bundle for manifold X (irrespective of the inclusion). Note, the normal bundle ν(X) of the manifold X is not uniquely defined. It depends on inclusion into the space R^{N} and on the dimension N. But the equation (1.69) shows that the bundle is not very far from being unique.

Let ν_{1}(X) be another such normal bundle, that is,

T X ⊕ ν_{1}(X) = N̄_{1}.

Then

ν(X) ⊕ T (X) ⊕ ν_{1}(X) = ν(X) ⊕ N̄_{1} = N̄ ⊕ ν_{1}(X).

The last relation means that bundles ν(X) and ν_{1}(X) became isomorphic after the addition of trivial summands.

2. Let us study the tangent bundle of the one dimensional manifold S^{1}, the circle. Define two charts on the S^{1}:

U_{1} = {−π < φ < π},
U_{2} = {0 < φ < 2π},

where φ is angular parameter in the polar system of coordinates on the plane. On the U_{1} take the function

x_{1} = φ, −π < x_{1} < π,

as coordinate, whereas on the U_{2} take the function

x_{2} = φ, 0 < x_{2} < 2π.

The intersection U_{1} ∩ U_{2} consists of the two connected components

V_{1} = {0 < φ < π},
V_{2} = {π < φ < 2π}.

Then the transition function has the form

x_{1} = x_{1}(x_{2}) = \{ \begin{matrix}x_{2}, & 0 < x_{2} < π, \\ x_{2} − 2π, & π < x_{2} < 2π. \end{matrix}

Then by (1.64), the transition function φ_{12} for the tangent bundle has the form

φ_{12}(x, ξ) = ξ \frac{∂x_{1}}{∂x_{2}} = ξ.

This means that the transition function is the identity. Hence the tangent bundle T S^{1} is isomorphic to Cartesian product

T S^{1} = S^{1} × R^{1},

in other words, it is the trivial one dimensional bundle.

3. Consider the two-dimensional sphere S^{2}. It is convenient to consider it as the extended complex plane

S^{2} = C^{1} ∪ {∞}.

We define two charts on the S^{2}

U_{1} = C^{1}
U_{2} = (C^{1}\{0}) ∪ {∞}.

Define the complex coordinate z_{1} = z on the chart U_{1} and z_{2} = \frac{1}{z} on the chart U_{2} extended by zero at the infinity ∞. Then the transition function on the intersection U_{1} ∩ U_{2} has the form

z_{1} ≡ \frac{1}{z_{2}}

and the tangent bundle has the corresponding transition function of the form

φ_{12}(z, ξ) = ξ \frac{∂z_{1}}{∂z_{2}} = −ξ \frac{1}{z_{2}^{2}} = −ξz^{2}.

The real form of the matrix φ_{12} is given by

φ_{12}(x, y) = ∥ \begin{matrix}−ℜz^{2} & −ℑz_{2} \\ ℑx_{2} & −ℜz_{2} \end{matrix} ∥ = ∥ \begin{matrix}y_{2} − x_{2} & −2xy \\ 2xy & y_{2} − x_{2} \end{matrix} ∥ ,

where z = x + iy. In polar coordinates z = ρe^{iα} this becomes

φ_{12}(ρ, α) = ρ^{2} ∥ \begin{matrix}cos 2α & − sin 2α \\ sin 2α & cos 2α \end{matrix} ∥ .

Let us show that the tangent bundle T S^{2} is not isomorphic to a trivial bundle. If the bundle T S^{2} were trivial then there would be matrix valued functions

h_{1} : U_{1} ⟶ GL (2, R)
h_{2} : U_{2} ⟶ GL (2, R),  (1.70)

such that

φ_{12}(ρ, α) = h_{1}(ρ, α)h_{2}^{−1}(ρ, α).

The charts U_{1}, U_{2} are contractible and so the functions h_{1}, h_{2} are homotopic to constant mappings.

Hence the transition function φ_{12}(ρ, α) must be homotopic to a constant function. On the other hand, for fixed ρ the function φ_{12} defines a mapping of the circle S^{1} with the parameter α into the group SO(2) = S^{1} 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

φ_{αβ} : U_{α} ∩ U_{β}⟶ 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_{α}} is an atlas of charts for the manifold X then an atlas of the manifold E can be defined by

V_{α} = p^{−1}(U_{α}) = U_{α} × R^{n}.

The local coordinates on V_{α} can be defined as the family of local coordinates on the chart U_{α} with Cartesian coordinates on the fiber. The smoothness of the functions φ_{αβ} 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 φ_{αβ}. The answer lies in the following theorem.

Theorem 10 Let p : E⟶X be an n–dimensional vector bundle and X a compact smooth manifold. Then there exists an atlas {U_{α}} on X and coordinate homeomorphisms

φ_{α} : U_{α} × R^{n}⟶p^{−1}(U_{α})

such that the transition functions

φ_{αβ} : U_{α} ∩ U_{β}⟶ GL (n, R)

are smooth.

Proof.

Consider a sufficiently fine atlas for the bundle p and coordinate homeomorphisms

ψ_{α} : U_{α} × R^{n}⟶p^{−1}(U_{α}).

The transition functions

ψ_{αβ} : U_{α} ∩ U_{β}⟶ GL (n, R)

are in general only required to be continuous. The problem is to change the homeomorphisms ψ_{α} to new coordinate homeomorphisms φ_{α} such that the new transition functions φ_{αβ} are smooth. In other words, one should find functions

h_{α} : U_{α}⟶ GL (n, R)

such that the compositions h_{β}^{−1}(x)ψ_{βα}(x)h_{α}(x) are smooth. Let {U′_{α}} be a new atlas such that

U′_{α} ⊂ Ū′_{α} ⊂ U_{α}.

We construct the functions h_{α}(x) by induction on the index α, 1 ≤ α ≤ N. Without loss of generality, we can assume that all the functions ψ_{αβ}(x) are uniformly bounded, that is,

∥ψ_{αβ}(x)∥ ≤ C.

Let

0 < ε < \frac{1}{C} , ε < 1.

Choose smooth functions φ_{αβ}(x), x ∈ U_{α} ∩ U_{β} such that

∥ψ_{αβ}(x) − φ_{αβ}(x)∥ < ε

and which form a cocycle, that is,

φ_{αβ}φ_{βγ}φ_{γα} ≡ 1, φ_{αβ}φ_{βα} ≡ 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 ε > 0 there is a smooth function g(x) defined on V such that

|g(x) − f(x)| < ε, x ∈ V.

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 φ_{αβ}. By the lemma there exists a smooth function φ_{12} defined in a neighborhood of the set Ū′_{1} ∩ Ū′_{2} such that

∥φ_{12}(x) − ψ_{12}(x)∥ < ε.

Assume that we have chosen smooth functions φ_{αβ}(x) defined in neighborhoods of the subsets Ū′_{α} ∩ Ū′_{β} for all indices α < β ≤ β_{0} such that

∥φ_{αβ}(x) − ψ_{αβ}(x)∥ < ε

and

φ_{αβ}(x)φ_{βγ}(x)φ_{γα}(x) ≡ 1, α, β, γ ≤ β_{0}.

The function φ_{1,β_{0}+1}(x) can be constructed in the same way as φ_{12} using the lemma (2).

Assume that functions φ_{α,β_{0}+1}(x) are constructed for all α ≤ α_{0} ≤ β_{0} and they satisfy the condition of cocycle:

φ_{α,β_{0}+1}(x) ≡ φ_{α,α′}(x)φ_{α′,β_{0}+1}(x), α, α′ ≤ a_{0}.

Then the required function φ_{α_{0}+1,β_{0}+1}(x) can be defined in a neighborhood of the subset Ū′_{γ} ∩ Ū′_{α_{0}+1} ∩ Ū′_{β_{0}+1} for any γ ≤ α_{0} by formula

φ_{α_{0}+1,β_{0}+1}(x) = φ_{γ,α_{0}+1}^{−1}(x)φ_{γ,β_{0}+1}(x).

Hence the required function φ_{α_{0}+1,β_{0}+1}(x) is defined in a neighborhood of the union

V = ∪_{γ≤α_{0}}(Ū′_{γ} ∩ Ū′_{α_{0}+1} ∩ Ū′_{β_{0}+1})

and satisfies the condition

∥ψ_{α_{0}+1,β_{0}+1}(x) − φ_{α_{0}+1,β_{0}+1}(x)∥ < 2Cε.  (1.71)

By the lemma (2) the function can be extended φ_{α_{0}+1,β_{0}+1}(x) from the closure of a neighborhood of the set V to a neighborhood of the set Ū′_{α_{0}+1} ∩ Ū′_{β_{0}+1} satisfying the same condition (1.71).

By induction, a family of functions φ_{αβ} with property

∥ψ_{α,β}(x) − φ_{α,β}(x)∥ < (2C)^{N^{2}}ε = ε′.

can be constructed. Now we pass to the construction of functions h_{α} satisfying the conditions

φ_{αβ}(x) = h_{α}(x)ψ_{αβ}(x)h_{β}^{−1}(x)

or

h_{α}(x) = φ_{αβ}(x)h_{β}(x)ψ_{βα}(x).  (1.72)

We construct the functions h_{α} by induction. Put

h_{1}(x) ≡ 1.

Assume that the functions h_{α}(x), 1 ≤ α ≤ α_{0} have already been constructed satisfying the conditions (1.72) and

∥1 − h_{α}(x)∥ < ε, α < α_{0}.

Then the function h_{α_{0}+1}(x) is defined by the formula (1.72)in a neighborhood of the set

V = ∪_{α<α_{0}} Ū′_{α_{0}+1} ∩ Ū′_{α}.

On the set V, the inequality

∥1 − h_{α_{0}+1}(x)∥ = ∥1 − φ_{α_{0}+1,β}(x)h_{β}(x)ψ_{β,α_{0}+1}(x)∥ ≤
≤ C^{2}∥h_{β} − φ_{β,α_{0}+1}ψ_{β,α_{0}+1}^{−1}∥ ≤ 2C^{2}ε  (1.73)

is satisfied. If the number ε is sufficient small then the function h_{α_{0}+1}(x) can be extended from a neighborhood V on the whole chart U_{α_{0}+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 bundle ν(X ⊂ Y).

Proof.

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 : ν(X)⟶Y.

Consider a normal vector ξ ∈ ν(X) at the point x ∈ X ⊂ Y. Notice that the vector ξ is orthogonal to the subspace T_{x}(X) ⊂ T_{x}(Y). Let γ(t) be the geodesic curve such that

γ(0) = x, \frac{dγ}{dt} (0) = ξ.

Put f(ξ) = γ(1). The mapping f has nondegenerate Jacobian for each point of the zero section of the bundle ν(X). Indeed, notice that

1. if ξ = 0, ξ ∈ T_{x}(Y) then f(ξ) = x,

2. f(λξ) = γ(λ).

Therefore the Jacobian matrix of the mapping f at a point of the zero section of ν(X) that maps the tangent spaces

Df : T_{x}(ν(X)) = T_{x}(X) ⊕ ν_{x}(X)⟶T_{x}(Y) = T_{x}(X) ⊕ ν_{x}(X)

is the identity. By the implicit function theorem there is a neighborhood V of zero section o f the bundle ν(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 ν(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 Δ ⊂ X × 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 locally trivial bundle with the fiber a smooth manifold.

Without loss of generality one can consider a chart U ⊂ Y diffeomorphic to R^{n} and part of the manifold X, namely, f^{−1}(U). Then the mapping f gives a vector valued function

f : X⟶R^{n}.

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 γ(x_{0}, 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

(ξ, η)_{1} = (ξ, η)(grad f, grad f).

Then

(grad f, ξ) = ξ(f) = (grad _{1}f, ξ)_{1} = (grad _{1}f, ξ)(grad _{1}f, grad _{1}f).

Hence

grad _{1}f = \frac{grad f}{(grad f, grad f)}.

Then

(grad _{1}f, grad _{1}f)_{1} = (grad _{1}f, grad _{1}f)(grad f, grad f) =
= \frac{(grad f, grad f)}{(grad f, grad f)^{2}} (grad f, grad f) = 1.

Thus the integral curves

\frac{d}{dt}f(γ(t)) = (grad f, grad f) ≡ 1.

Hence the function f(γ(t)) is linear. This means that if

f(x_{0}) = f(x_{1}),

then

f(γ(x_{0}, t)) = f(γ(x_{1}, t)) = f(x_{0}) + t.

Put

g : Z × R^{1}⟶X, g(x, t) = γ(x, t).

The mapping g is a fiberwise smooth homeomorphism. Hence the mapping

f : ⟶R^{1}

gives a locally trivial bundle. Further, the proof will follow by induction with respect to n. Consider a vector valued function

f(x) = {f_{1}(x), …, f_{n}(x)}

which satisfies the condition of the theorem. Choose a Riemannian metric on the manifold X such that gradients

grad f_{1}, …, grad f_{n}

are orthonormal. Such a metric exists. Indeed, consider firstly an arbitrary metric. The using the linear independence of the differentials {df_{i}} we know that the gradients are also independent. Hence the matrix

a_{ij} = ⟨grad f_{i}(x), grad f_{j}(x)⟩

is nondegenerate in each point. Let ∥b_{ij}(x)∥ be the matrix inverse to the matrix ∥a_{ij}(x)∥, that is,

∑_{α} b_{iα}(x)a_{jα}(x) ≡ δ_{ij}.

Put

ξ_{k} = ∑_{i} b_{kj}(x)grad f_{i}(x).

Then

ξ_{k}(f_{j}) = ∑_{i} b_{ki}(x)grad f_{i}(f_{j}) =

= ∑_{i} b_{ki}(x)⟨grad f_{i}, grad g_{j}⟩ = ∑_{i} b_{ki}(x)a_{ij}(x) ≡ δ_{kj}.

Let U_{α} be a sufficiently small neighborhood of a point of the manifold X. The system of vector fields {ξ_{1}, …, ξ_{n}} can be supplemented by vector fields η_{n+1}, …, η_{N} to form a basis such that

η_{k}(f_{i}) ≡ 0.

Consider the new metric in the chart U_{α} given by

⟨ξ_{i}, ξ_{j}⟩_{α} ≡ δ_{ij},
⟨ξ_{k}, η_{j}⟩_{α} ≡ 0.

Let φ_{α} be a partition of unity subordinate to the covering {U_{α}} and put

⟨ξ, η⟩_{0} = ∑_{α} φ_{α}(x)⟨ξ, η⟩_{α}.

Then

⟨ξ_{i}, ξ_{j}⟩_{0} ≡ δ_{ij},
⟨ξ_{k}, η⟩_{0} ≡ 0

for any vector η for which η(f_{i}) = 0.Let grad _{0}f_{i} be the gradients of the functions f_{i} with respect to the metric ⟨ξ, η⟩_{0}. This means that

⟨grad f_{i}, ξ⟩_{0} = ξ(f_{i})

for any vector ξ. In particular one has

⟨grad f_{i}, ξ_{j}⟩_{0} ≡ δ_{ij},
⟨grad f_{k}, η⟩_{0} ≡ 0

for any vector η for which η(f_{i}) = 0. Similar relations hold for the vector field ξ_{i}. Therefore

ξ_{i} = grad f_{i},

that is,

⟨grad f_{i}, grad f_{j}⟩δ_{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) = {f_{1}(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 × R^{n−1} and the functions f_{i} are the coordinate functions for the second factor. Then grad f_{n} is tangent to the first factor Z and hence the function f_{n}(x, t) does not depend on t ∈ R^{n−1}. Therefore one can apply the first step of the induction to the manifold Z, that is,

Z = Z_{1} × R^{1}.

Thus

X = Z_{1} × R^{n}.

7. An invariant formulation of the implicit function theorem Let f : X⟶Y be a smooth mapping of smooth manifolds. The point y_{0} ∈ Y is called a regular value of the mapping f if for any point x ∈ f^{−1}(y_{0}) the differential Df : T_{x}(X)⟶T_{y_{0}}(Y) is surjective. The implicit function theorem says that the inverse image

Z = f^{−1}(y_{0})

is a smooth manifold and that

T X_{|Z} = T Z ⊕ R^{m}, 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

T_{f(x)}Y = T_{f(x)}W ⊕ Df(T_{x}X).

In particular, the mapping f is transversal ’along’ each regular point y_{0} ∈ Y. The implicit function theorem says that the inverse image

Z = f^{−1}(W)

is a submanifold, the normal bundle ν(Z ⊂ X) is isomorphic to the bundle f^{∗}(ν(W ⊂ Y)) and the differential Df is the fiberwise isomorphism

ν(Z ⊂ X)⟶ν(W ⊂ Y).

8. Morse functions on manifolds Consider a smooth function f on a manifold X. A point p_{0} is called a critical point if

df(x_{0}) = 0.

A critical point x_{0} 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⟶R^{1}

there is a mapping

df : X⟶T^{∗}X  (1.74)

adjoint to Df which to each point x ∈ X associates the linear form on T_{x}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 X_{0} 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 X_{0} and df(X) at that point is transversal. If all the critical points of the function f are nondegenerate then 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 X_{0} ⊂ T^{∗}X.

9. Oriented manifolds. A manifold X is said to be oriented if there is an atlas {U_{α}} such that all the transition functions φ_{αβ} 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_{α}} be an atlas,

φ_{αβ} = ∥\frac{∂x_{α}^{k}}{∂x_{β}^{j}} ∥

be the transition functions of the tangent bundle and

ψ_{αβ} = h_{α}φ_{αβ}h_{β}^{−1}

be the new transition functions such that

det ψ_{αβ} > 0.

Of course, if the matrix valued functions

h_{α} : U_{α}⟶ GL (n, R)

had the form of the Jacobian matrix, that is, if

hα(x) = ∥\frac{∂y_{α}^{k}}{∂x_{α}^{j}} ∥

then the functions

y_{α}^{k} = y_{α}^{k}(x_{α}^{1}, …, x_{α}^{n})

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_{α}. 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_{α}. Hence, select new functions as follows:

h̄_{α}(x) = ∥\begin{matrix}±1 & 0 & … & 0 \\ 0 & 1 & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & 1 \end{matrix}∥ ,

where the sign of the first entry should coincide with the sign of det h_{α}(x) on each connected component of the chart U_{α}. Then the new transition functions

ψ̄_{αβ}(x) = h̄_{α}(x)φ_{αβ}(x)h̄_{β}^{−1}(x)

satisfy the same condition

det ψ_{αβ}(x) > 0.

On the other hand, the functions h̄_{α}(x) are the Jacobians of a change of the coordinates:

y_{α}^{1} = ±x_{α}^{1},
y_{α}^{2} = x_{α}^{2},
. . . . . . . . .
y_{α}^{n} = x_{α}^{n}.

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

We define an atlas {U_{k}}_{k=0}^{n} on RP^{n} as follows. Put

U_{k} = {[x_{0} : x_{1} : …: x_{n}] : x_{k} ≠ 0},

and define coordinates on U_{k} by the following functions:

y_{k}^{α} = \frac{x_{α}}{x_{k}} , 0 ≤ α ≤ n, α ≠ k.

where in the numbering of the coordinates by index α there is a gap when α = k. The change of variables on the intersection U_{k} ∩ U_{j}, (k ≠ j) of two charts has the following form:

y_{k}^{α} = \begin{cases}\frac{y_{j}^{α}}{y_{j}^{k}} & when & α ≠ j, \\ \frac{1}{y_{j}^{k}} & when & α = j. \end{cases}  (1.75)

The formula (1.75) is well defined because k ≠ j and on U_{k} ∩ U_{j}

y_{j}^{k} = \frac{x_{k}}{x_{j}} ≠ 0.

All the functions in (1.75) are smooth functions making RP^{n} 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 R^{n+1} and x is a point on l. The space E differs from the space R^{n+1} in that instead of zero vector of R^{n+1} in the space E there are many points of the type (l, 0). The mapping

p : E⟶RP^{n},  (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 RP^{n} × R^{n+1} defined by the following system of equations in each local coordinate system:

rank ∥\begin{matrix}x_{0} & x_{1} & … & x_{n} \\ y_{0} & y_{1} & … & y_{n} \end{matrix}∥ = 1,

where (x_{0}, x_{1}, …, x_{n}) are projective coordinates of a point of RP^{n}, and (y_{0}, y_{1}, …, y_{n}) are coordinates of the point of R^{n+1}. For example, in the case when U_{0} = {x_{0} ≠ 0} we put x_{0} = 1. Then

rank ∥\begin{matrix}1 & x_{1} & … & x_{n} \\ y_{0} & y_{1} & … & y_{n} \end{matrix}∥ = 1,

that is,

\begin{cases}det ∥\begin{matrix}1 & x_{k} \\ y_{0} & y_{k} \end{matrix} ∥ = 0, \\ det ∥\begin{matrix}x_{k} & x_{j} \\ y_{k} & y_{j} \end{matrix}∥ = 0, \end{cases}

Hence the set E is defined in the RP^{n} × R^{n+1} by the following system of n equations:

f_{k}(x_{1}, …, x_{n}, y_{0}, …, y_{n}) = y_{k} − x_{k}y_{0} = 0, k = 1, …, n.

The Jacobian matrix of the functions f_{k} is

∥\begin{matrix}−y_{0} & 0 & … & 0 & −x_{1} & 1 & 0 & … & 0 \\ 0 & −y_{0} & … & 0 & −x_{2} & 0 & 1 & … & 0 \\ … & … & … & … & … & … & … & … & … \\ 0 & 0 & … & −y_{0} & −x_{n} & 0 & 0 & … & 1 \end{matrix}∥  (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 (y_{0}, x_{1}, …, x_{n}). The projection (1.76) consists of forgetting the first coordinate y_{0}. So the inverse image p^{−1}(U_{0}) is homeomorphic to the Cartesian product U_{0} × R^{1}. Changing to another chart U_{k}, one takes another coordinate y_{k} as the coordinate in the fiber which depends linearly on y_{0}. 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 CP^{n}. In the both cases, real and complex, the corresponding principal bundles with the structure groups O(1) = Z_{2} and U(1) = S^{1} can be identified with subbundles of the Hopf bundle. The point is that the structure group O(1) can be included in the fiber R^{1}, O(1) = {−1, 1} ⊂ R^{1}, in a such way that the linear action of O(1) on R^{1} coincides on the subset {−1, 1} with the left multiplication. Similarly, the group U(1) = S^{1} can be included in C^{1}:

S^{1} = {z : |z| = 1} ⊂ C^{1},

such that linear action of U(1) on C^{1} coincides on S^{1} with left multiplication. Hence the Hopf bundle has the principal subbundle consisting of vectors of unit length. Let

p : E_{S}⟶RP^{n}

be principal bundle associated with the Hopf vector bundle. The points of the total space E_{S} are pairs (l, x), where l is a line in R_{n+1} and x ∈ l, |x| = 1. Since x ≠ 0, the pair (l, x) is uniquely determined by the vector x. Hence the total space E_{S} is homeomorphic to the sphere S^{n} of unit radius a nd the principal bundle

p : S^{n}⟶RP^{n}

is the two-sheeted covering. In the case of the complex Hopf bundle, the associated principal bundle

p : E_{S}⟶CP^{n}

is

E_{S} = S^{2n+1}

and the fiber is a circle S^{1}.

Both these two principal bundles are also called Hopf bundles. When n = 1, we have S^{3}⟶P^{1}, that is, the classical Hopf bundle

S^{3}⟶S^{2}.  (1.78)

In this last case it is useful to describe the transition functions for the intersection of charts. Let us consider the sphere S^{3} as defined by the equation

|z_{0}|^{2} + |z_{1}|^{2} = 1,

in the two dimensional complex vector space C^{2}. The map (1.78) associates the point (z_{0}, z_{1}) to the point in the CP^{1} with the projective coordinates [z_{0} : z_{1}]. So over the base CP^{1} we have the atlas consisting of two charts:

U_{0} = {[z_{0}, z_{1}] : z_{0} ≠ 0},
U_{1} = {[z_{0}, z_{1}] : z_{1} ≠ 0},

The points of the chart U_{0} are parametrized by the complex parameter

w_{0} = \frac{z_{1}}{z_{0}} ∈ C^{1}

whereas points of the chart U_{1} are parametrized by the complex parameter

w_{1} = \frac{z_{0}}{z_{1}} ∈ C^{1}.

The homeomorphisms

φ_{0} : p^{−1}(U_{0}) ⟶ S^{1} × C^{1} = S^{1} × U_{0},
φ_{1} : p^{−1}(U_{1}) ⟶ S^{1} × C^{1} = S^{1} × U_{1},

68 Chapter 1

have the form

φ_{0}(z_{0}, z_{1}) = (\frac{z_{0}}{|z_{0}|}, \frac{z_{1}}{z_{0}} ) = (\frac{z_{0}}{|z_{0}|}, [z_{0} : z_{1}]) ,
φ_{1}(z_{0}, z_{1}) = (\frac{z_{1}}{|z_{1}|}, \frac{z_{0}}{z_{1}} ) = (\frac{z_{1}}{|z_{1}|}, [z_{0} : z_{1}]) .

The mappings φ_{0} and φ_{1} clearly are invertible:

φ_{0}^{−1}(λ, w_{0}) = (\frac{λ}{\sqrt{1 + |w_{0}|^{2}}} , \frac{λw_{0}}{\sqrt{1 + |w_{0}|^{2}}} ) ,
φ_{1}^{−1}(λ, w_{1}) = (\frac{λ}{\sqrt{1 + |w_{1}|^{2}}} , \frac{λw_{1}}{\sqrt{1 + |w_{1}|^{2}}} ) .

Hence the transition function

φ_{01} : S_{1} × (U_{0} ∩ U_{1})⟶S_{1} × (U_{0} ∩ U_{1})

is defined by the formula:

φ_{01}(λ, [z_{0:z_{1}}]) = (\frac{z_{1}|z_{0}|}{z_{0}|z_{1}|}, [z_{0} : z_{1}], ) ,

that is, multiplication by the number

\frac{z_{1}|z_{0}|}{z_{0}|z_{1}|} = \frac{w_{1}}{|w_{1}|}.

1.6.3 The tangent bundle of the Hopf bundle

Let ξ be the complex Hopf bundle over CP^{n} and let T CP^{n} be the tangent bundle. Then

T CP^{n} ⊕ \overline{1} = ξ^{∗} ⊕ ξ^{∗} ⊕ … ⊕ ξ^{∗} = (n + 1)ξ^{∗}.  (1.79)

When n = 1 this gives

TCP^{1} ⊕ \overline{1} = ξ^{∗} ⊕ ξ^{∗}.

Indeed, using the coordinate transition function

w_{2} = \frac{1}{w_{1}} ,

the transition function for the tangent bundle TCP^{1} has the following form:

ψ_{01}(w_{1}) = −\frac{1}{w_{1}^{2}} = −\frac{w̄_{1}^{2}}{|w_{1}|^{4}} .  (1.80)

On the other hand, the transition function for the Hopf bundle is

φ_{01}(w_{1}) = \frac{w_{1}}{|w_{1}|}.  (1.81)

Using homotopies the formulas (1.80) and (1.81) can be simplified to

ψ_{01}(w_{1}) = w̄_{1}^{2},
φ_{01}(w_{1}) = w_{1}.

This shows that the matrix functions

∥\begin{matrix}w̄_{1}^{2} & 0 \\ 0 & 1 \end{matrix}∥ and ∥\begin{matrix}w̄_{1} & 0 \\ 0 & w̄_{1} \end{matrix}∥

are homotopic in the class of invertible matrices when w_{1} ≠ 0, w_{1} ∈ C^{1}. In general, let us consider the manifold CP^{n} as the quotient space of the unit sphere S^{2n+1} in C^{n+1} by the action of the group S^{1} ⊂ C^{1} of complex numbers with unit norm. The total space of the tangent bundle T CP^{n} 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 S^{1}. In other words, one has

T CP^{n} = \frac{\{(u, v) : u, v ∈ C^{n+1}, |u| = 1, ⟨u, v⟩ = 0\}}{\{(λu, λv) ∼ (u, v), λ ∈ S^{1}\}},

where ⟨u, v⟩ is the Hermitian inner product in C^{n+1} Consider the quotient space

A = \frac{\{(ū, s) : u ∈ C^{n+1}, |ū| = 1, s ∈ C^{1}\}}{\{(λū, λs) ∼ (ū, s), λ ∈ S^{1}\}}.

Associate to each pair (ū, s) ∈ A the line l which passes through vector ū ∈ C^{n+1} and the vector sū ∈ l. If (λū, λs) is an equivalent pair (which passes through vector λū) then it corresponds the same line l and the same vector sū = (λs)(\overline{λu}). This means that the space A is homeomorphic to the total space of vector bundle ξ^{∗}. Hence the total space of the vector bundle (n + 1)ξ^{∗} is homeomorphic to the space

B = \frac{\{(u, s) : u, v ∈ C^{n+1}, |u| = 1, \}}{\{(λu, λv) ∼ (u, v), λ ∈ S^{1}\}}.

The space T CP^{n} = is clearly a subspace of B. A complementary subbundle can be defined as a quotient space

D = \frac{\{(u, v) : u, v ∈ C^{n+1}, |u| = 1, v = su, s ∈ C^{1}\}}{\{(λu, λv) ∼ (u, v), λ ∈ S^{1}\}}.

The latter is homeomorphic to the space

D = \frac{\{(u, s) : u ∈ C^{n+1}, |u| = 1, s ∈ C^{1}\}}{\{λu ∼ u, λ ∈ S^{1}\}},

which is homeomorphic to the Cartesian product T CP^{n} × C^{1}. Thus one has the isomorphism (1.79).

1.6.4 Bundles of classical manifolds

Denote by V_{n,k} the space where the points are orthonormal sets of k vectors of n–dimensional euclidean space (R^{n} or C^{n}). If we need we will write V_{n,k}^{R} or V_{n,k}^{C}. Correspondingly, let us denote by G_{n,k} the space in which points are k–dimensional subspaces of n–dimensional euclidean space (R^{n} or C^{n}). By expanding a k–frame to a basis in C^{n} we obtain

V_{n,k}^{C} = U(n)/U(n − k),

where U(n − k) ⊂ U(n) is a natural inclusion of unitary matrices:

A_{n−k}⟶ ∥\begin{matrix}1_{k} & 0 \\ 0 & A_{n−k} \end{matrix}∥ .

Similarly, the space G_{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

(A_{k}, B_{n−k})⟶ ∥\begin{matrix}A_{k} & 0 \\ 0 & B_{n−k} \end{matrix}∥ .

Generally speaking, if G is a Lie group, and H ⊂ G is a subgroup then the projection

p : G⟶G/H

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

V_{n,k}\underrightarrow{V_{n−k_{1},k−k_{1}}}V_{n,k_{1}},

V_{n,k}^{C} ⟶^{U(k)}G_{n,k}^{C}.

(The fibers are shown over the arrows.) In particular,

V_{n,n}^{R} = O(n),
V_{n,n}^{C} = U(n),
V_{n,1}^{R} = S^{n−1},
V_{n,1}^{C} = S^{2n−1}.

Hence we have the following locally trivial bundles

U(n) ⟶^{U(n−1)}S^{2n−1},

O(n) ⟶^{O(n−1)}S^{n−1},

All the mappings above are defined by forgetting some of the vectors from the frame. The manifolds V_{n,k} are called the Stieffel manifolds, and the G_{n,k} called Grassmann manifolds.

72 Chapter 1

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 × I then the restrictions of the bundle to B ×{0} and B ×{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

Assume that the base B of a locally trivial bundle

p : E⟶B

is a cellular space, that is, the space B is a direct limit of its compact subspaces [B]^{n},

B = lim_{n⟶∞}[B]^{n},

and each space [B]^{n} is constructed from [B]^{(n−1)} by attaching to it a finite family of discs D_{i}^{n} by using continuous mappings

S_{i}^{n−1}⟶[B]^{(n−1)}.

The subspace [B]^{n} is called the n–dimensional skeleton of the space B.

Let us consider a principal G–bundle,

p_{G} : E_{G}⟶B_{G},  (2.1)

such that all homotopy groups of the total space E_{G} are trivial:

π_{i}(E_{G}) =), 0 ≤ i < ∞.  (2.2)

Theorem 13 Let B be a cellular space. Then any principal G–bundle

p : E⟶B  (2.3)

is isomorphic to the inverse image of the bundle (2.1) with respect to a continuous mapping

f : B⟶B_{G}.  (2.4)

Two inverse images of the bundle (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 all isomorphism classes of principal G–bundles over the base B is in one to one correspondence with the family of homotopy classes of continuous mappings from B to B_{G}.

Corollary 3 If two cellular spaces B and B′ are homotopy equivalent then the families of all isomorphism classes of principal G–bundles 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 B_{G} is a cellular space then it is defined by the condition (2.2) up to homotopy equivalence.

Proof of Theorem 13.

Homotopy Invariants 75

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 space, B_{0} be a cellular subspace, (that is, B_{0} is union of some cells of the space B and is closed subspace). Consider a principal bundle (2.3) and let

E_{0} = p^{−1}(B_{0}).

Let

F_{0} : E_{0}⟶E_{G}  (2.5)

be an equivariant continuous mapping. Let us show that F_{0} can be extended to a continuous equivariant mapping

F : E⟶E_{G}.  (2.6)

Lemma 3 Any principal G–bundle p over a disc D^{n} is isomorphic to a trivial bundle.

Proof.

In fact, put

h : D^{n} × I ⟶ D^{n},
h(x, t) = tx, 0 ≤ x ≤ 1, x ∈ D^{n}.

Then

h(x, 0) ≡ x
h(x, 1) ≡ 0.

Then by Theorem 3 of the section 1.2 of the chapter 1, inverse images of the bundle p with respect to the mappings

h_{0}(x) = h(x, 0),
h_{1}(x) = h(x, 1)

are isomorphic. On the other hand, the inverse image of the bundle p with respect to the identity mapping h_{1} is isomorphic to the bundle p and the inverse image with respect to the mapping h_{0} is trivial.

Returning to the proof of Theorem 13, let the (n − 1)–dimensional skeleton [B]^{(n−1)} lie in the subspace B_{0} with a cell D_{i}^{n} outside B_{0}. Let

φ_{i} : D_{i}^{n}⟶B

be the natural inclusion. Then

φ_{i}(S_{i}^{(n−1)}) ⊂ [B]^{(n−1)} ⊂ B_{0}.  (2.7)

Consider the restriction of the bundle p to the cell D_{i}^{n}. By Lemma 3 this restriction is a trivial bundle, that is, the total space is isomorphic to D_{i}^{n}×G.By (2.7) the mapping (2.5) induces the equivariant mapping

(F_{0})_{|(S_{i}^{(n−1)}×G)} : (S_{i}^{(n−1)} × G) ⟶ E_{G}
F_{0}(x, g) = F_{0}(x, 1)g,  x ∈ S_{i}^{(n−1)}, g ∈ G.

Consider the mapping

F_{0}(x, 1) : S_{i}^{(n−1)}⟶E_{G}.  (2.8)

Since

π_{(n−1)}(E_{G}) = 0,

the mapping (2.8) can be extended to a continuous mapping

F (x, 1) : D_{i}^{n}⟶E_{G}.

Put

F (x, g) = F (x, 1)g, x ∈ D_{i}^{n}, g ∈ G.  (2.9)

It is clear that the mapping (2.9) is equivariant and extends the mapping F_{0} to the set p^{−1}(B_{0} ∪ D_{i}^{n}) ⊃ E_{0}. 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 B_{0} = ∅. The second statement of the theorem follows if we substitute B for B × I and put

B_{0} = (B × {0}) ∪ (B × {1}).

Corollaries 2 and 3 follow directly from Theorem 13.

Proof of Corollary 4.

Let

p : E_{G} ⟶ B_{G},
p′ : E′_{G} ⟶ B′_{G}

be two principal G– bundles such that

π_{i}(E_{G}) = π_{i}(E′_{G}) = 0, 0 ≤ i ≤ ∞,

and let B_{G} and B′_{G} be cellular spaces. Then by Theorem 13 there are two continuous mappings

f : B_{G} ⟶ B′_{G},
g : B′_{G} ⟶ B_{G}

such that

f^{∗}(p′) = p,
g^{∗}(p) = p′.

Hence

(gf)^{∗}(p) = p = (Id)^{∗}(p),
(gf)^{∗}(p′) = p′ = (Id)^{∗}(p′).

By the second part of Theorem 13,

gf ∼ Id.
fg ∼ Id.

This means that the spaces B_{G} and B′_{G} are homotopy equivalent.

The bundle (2.1) is called the universal principal G–bundle and the base B_{G} is called the classifying space for bundles with structure group 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 G–bundle.

Definition 4 A mapping

p : X⟶Y

is said to satisfy the lifting homotopy axiom if for any cellular space P, any homotopy

f : P × I⟶Y

and any mapping

g_{0} : P × {0}⟶Y

satisfying the condition

pg_{0}(x, 0) = f(x, 0),

there is an extension

g : P × 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 the mapping f) with respect to the mapping p if the two mappings f and g satisfy the condition (2.10). So the lifting homotopy axiom states that if the initial mapping f(x, 0) has a lifting mapping then any homotopy f(x, t) has a lifting homotopy.

Theorem 14 Let

p : E⟶B  (2.11)

be a locally trivial bundle. Then p satisfies the lifting homotopy axiom.

Proof.

Assume firstly that (2.11) is a trivial bundle, that is,

E = B × F.

Then the lifting mapping g_{0} can be described as

g_{0}(x, 0) = (f(x, 0), h(x)).

Then let

g(x, t) = (f(x, t), h(x)).  (2.12)

Clearly the mapping (2.12) is a lifting mapping for the mapping f. Further, let P_{0} ⊂ P be a cellular subspace with a lifting homotopy

g_{0} : P_{0} × I⟶E = B × F.

In other words, the lifting mapping is defined on the subspace

(P_{0} × I) ∪ (P × {0}) ⊂ P × I.

We need to extend the mapping g_{0} to a lifting mapping

g : P × I⟶E.

Notice that it is sufficient to be able to construct an extension of the mapping g_{0} on a single cell

D_{i}^{k} × I ⊂ P × I.

The restriction of the mapping g_{0} defines the lifting mapping on the subset

(D_{i}^{k} × {0}) ∪ (S_{i}^{(k−1)} × I) ⊂ D_{i}^{k} × I.  (2.13)

This subset has the form of a ‘glass’ where the ‘bottom’ corresponds to D_{i}^{k}×{0} and the ‘sides’ to S_{i}^{(k−1)} ×I. So the Cartesian product D_{i}^{k} ×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 g_{0} from (2.13) to the whole Cartesian product is equivalent to the problem of an extension of the lifting mapping from the ‘bottom’ D_{i}^{k} × {0} to the D_{i}^{k} × I and the latter problem is already solved. To complete the proof, consider a sufficiently fine cell decomposition of the space

P = ∪D_{i}^{k}

and the interval

I = ∪I_{l}

such that image of each Cartesian product of a cell with an interval lies in a chart of the bundle p. Then the restriction of the mapping f onto D_{i}^{k} ×I_{l} 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 locally trivial bundle, x_{0} ∈ B, y_{0} ∈ p^{−1}(x_{0}) = 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

∂ : π_{k}(B, x_{0})⟶π_{k−1}(F, y_{0})

such that the sequence of homomorphisms

… ⟶π_{k}(F, y_{0})⟶^{j_{∗}}π_{k}(E, y_{0})⟶^{p_{∗}}π_{k}(B, x_{0})⟶^{∂}

⟶^{∂} π_{k−1}(F, y_{0})⟶^{j_{∗}}π_{k−1}(E, y_{0})⟶^{p_{∗}}π_{k−1}(B, x_{0})⟶^{∂} …
… ⟶π_{1}(F, y_{0})⟶^{j_{∗}}π_{1}(E, y_{0})⟶^{p_{∗}}π_{1}(B, x_{0})  (2.14)

is an exact sequence of groups.

Proof.

The elements of homotopy group π_{k}(X, x_{0}) can be represented as homotopy classes of continuous mappings

φ : I^{k}⟶X

such that

φ(∂I^{k}) = x_{0}.

To construct the mapping ∂ let

φ : I^{k}⟶B

be a representative of an element of the group π_{k}(B, x_{0}) and note that

I^{k} = I^{k−1} × I.

Hence

φ(I^{k−1} × {0}) = φ(I^{k−1} × {1}) = x_{0}.

The mapping φ can be considered as a homotopy of the cube I^{k−1}. Put

ψ(u, 0) ≡ y_{0}, u ∈ I^{k−1}.

Then the mapping ψ is a lifting of the mapping φ(u, 0). By the lifting homotopy axiom there is a lifting homotopy

ψ : I^{k−1} × T ⟶E,

such that

pψ(u, t) ≡ φ(u, t).

In particular,

pψ(u, 1) ≡ x_{0}.

This means that ψ(u, 1) maps the cube I^{k−1} into the fiber F. Moreover, since

φ(∂I^{k−1} × I) = x_{0}

the lifting ψ can be chosen such that

ψ(∂I^{k−1} × I) ≡ y_{0}.

Hence the mapping ψ(u, 1) maps ∂I^{k−1} to the point y_{0}. This means that the mapping ψ(u, 1) defines an element of the group π_{k−1}(F, y_{0}). This homomorphism ∂ is well defined. Indeed, if the mapping φ is homotopic to the mapping φ′ then the corresponding lifting mappings ψ and ψ′ are homotopic and a homotopy Ψ between ψ and ψ′ can be constructed as a lifting mapping for the homotopy Φ between φ and φ′. 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 π_{k}(E, y_{0}).

If

φ : S^{k}⟶F

represents an element of the group π_{k}(F, y_{0}), then the mapping

pjφ : S^{k}⟶B

represents the image of this element with respect to homomorphism p_{∗}j_{∗}. Since

p(F) = x_{0},

we have

pjφ(S^{k}) = x_{0}.

Hence

Im j_{∗} ⊂ Ker p_{∗}.

Conversely, let

φ : S^{k}⟶E

be a mapping such that pφ is homotopic to a constant, that is,

[φ] ∈ Ker p_{∗}.

Let Φ be a homotopy between pφ and a constant mapping. By the lifting homotopy axiom there is a lifting homotopy Ψ between φ and a mapping φ′ which is lifting of the constant mapping. Thus

pφ′(S^{k}) = x_{0},

that is,

φ′(S^{k}) ⊂ F.

82 Chapter 2

Hence

[φ′] = [φ] ∈ Im j_{∗},

that is,

Ker p_{∗} ⊂ Im j_{∗}.

2. Exactness in the term π_{k}(B, y_{0}). Let

φ : I^{k}⟶E

represent an element of the group π_{k}(B, x_{0}). The composition

pφ : I^{k} = I^{k−1} × I⟶B

is lifted by φ. According to definition, the restriction

φ(u, 1) : I^{k−1}⟶f ⊂ E

represents the element ∂[pφ]. Since

φ(u, 1) ≡ y_{0},

we have

∂p_{∗}[φ] = 0.

Now let

φ : I_{k} = I_{k−1} × I⟶B

represent an element of the group π_{k}(B, x_{0}) and

ψ : I^{k−1} × I⟶E

be a lifting of the mapping φ. Then the restriction ψ(u, 1) represents the element ∂[φ]. If the mapping

ψ(u, 1) : I^{k−1}⟶F

is homotopic to the constant mapping then there is a homotopy

ψ′ : I^{k−1} × I⟶F

between ψ(u, 1) and constant mapping. Let us now construct a new mapping ψ′′ : I^{k−1} × I⟶E,

ψ′′(u, 1) = \begin{cases}ψ(u, 2t), & 0 ≤ t ≤ \frac{1}{2} \\ ψ′(u, 2t − 1), & \frac{1}{2} ≤ t ≤ 1. \end{cases}

Then

pψ′′(u, 1) = \begin{cases}φ(u, 2t), & 0 ≤ t ≤ \frac{1}{2} \\ x_{0}, & \frac{1}{2} ≤ t ≤ 1. \end{cases}

Hence the mapping pψ′′ is homotopic to the mapping φ, that is,

p_{∗}[ψ′′] = [φ]

or

Ker ∂ ⊂ Im p_{∗}.

3. Exactness in the term π_{k}(F, y_{0}).

Let the mapping

φ : I^{k+1} = I^{k} × I⟶B

represent an element of the group π_{k+1}(B, x_{0}) and let

ψ : I^{k} × I⟶E

lift the mapping φ. Then the restriction ψ(u, 1) represents the element

∂([φ]) ∈ π_{k}(F, y_{0}).

Hence the element j_{∗}∂([φ]) is represented by the mapping ψ(u, 1). The latter is homotopic to constant mapping ψ(u, 0), that is,

j_{∗}∂([φ]) = 0.

Conversely, if

φ : I^{k} × I⟶E

is a homotopy between φ(u, 1) ⊂ F and constant mapping φ(u, 0), then according to the definition the mapping φ(u, 1) represents the element ∂([pφ]). Hence

Ker j_{∗} ⊂ Im ∂.

Remark

Let π_{0}(X, x_{0}) denote the set of connected components of the space X where the fixed element [x_{0}] is the component containing the point x_{0}. Then the exact sequence (2.14) can be extended as follows

… ⟶π_{1}(F, y_{0})⟶^{j_{∗}}π_{1}(E, y_{0})⟶^{p_{∗}}π_{1}(B, x_{0})⟶^{∂}

⟶π_{0}(F, y_{0})⟶^{j_{∗}}π_{0}(E, y_{0})⟶^{p_{∗}}π_{0}(B, x_{0}).

This sequence is exact in the sense that the image coincides with the inverse image of the fixed element.

Examples

1. Consider the covering

R^{1}⟶S^{1}

in which the fiber Z is the discrete space of integers. The exact homotopy sequence

… ⟶π_{k}(Z)⟶^{j_{∗}}π_{k}(R^{1})⟶^{p_{∗}}π_{k}(S^{1})⟶^{∂} …

… ⟶^{∂}π_{1}(Z)⟶^{j_{∗}}π_{1}(R^{1})⟶^{p_{∗}}π_{1}(S^{1})⟶^{∂}

⟶π_{0}(Z)⟶^{j_{∗}}π_{0}(R^{1})⟶^{p_{∗}}π_{0}(S^{1})

has the following form

… ⟶0⟶^{j_{∗}}0⟶^{p_{∗}}π_{k}(S^{1})⟶^{∂}0⟶ …

… ⟶^{∂}0⟶^{j_{∗}}0⟶^{p_{∗}}π_{1}(S^{1})⟶^{∂}

⟶Z⟶^{j_{∗}}0⟶^{p_{∗}}0,

since

π_{k}(R^{1}) = 0,  k ≥ 0,
π_{k}(Z) = 0,  k ≥ 1.

Hence

π_{1}(S^{1}) = Z,
π_{k}(S^{1}) = 0 when k ≥ 2.

2. Consider the Hopf bundle

p : S^{3}⟶S^{2}.

The corresponding exact homotopy sequence

… ⟶π_{k}(S^{1})⟶^{j_{∗}}π_{k}(S^{3})⟶^{p_{∗}}π_{k}(S^{2})⟶^{∂} …

… ⟶^{∂}π_{3}(S^{1})⟶^{j_{∗}}π_{3}(S^{3})⟶^{p_{∗}}π_{3}(S^{2})⟶^{∂}
… ⟶^{∂}π_{2}(S^{1})⟶^{j_{∗}}π_{2}(S^{3})⟶^{p_{∗}}π_{2}(S^{2})⟶^{∂}
… ⟶^{∂}π_{1}(S^{1})⟶^{j_{∗}}π_{1}(S^{3})⟶^{p_{∗}}π_{1}(S^{2})⟶^{∂}

has the following form:

… ⟶0⟶^{j_{∗}}π_{k}(S^{3})⟶^{p_{∗}}π_{k}(S^{2})⟶^{∂}0 dots

… ⟶^{∂}0⟶^{j_{∗}}Z⟶^{p_{∗}}π_{3}(S^{2})⟶^{∂}
… ⟶^{∂}0⟶^{j_{∗}}0⟶^{p_{∗}}π_{2}(S^{2})⟶^{∂}
… ⟶^{∂}Z⟶^{j_{∗}}0⟶^{p_{∗}}0,

since

π_{3}(S^{1}) = π_{2}(S^{1}) = π_{2}(S^{3}) =
= π_{1}(S^{3}) = π_{1}(S^{2}) = 0.

Hence

π_{2}(S^{2}) = π_{1}(S^{1}) = Z,
π_{3}(S^{2}) = Z.

The space S^{2} 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 projection p : S^{3}⟶S^{2} gives a generator of the group π_{3}(S^{2}) = Z. Indeed, from the exact homotopy sequence we see that a generator of the group π_{3}(S^{3}) is mapped to a generator of the group π_{3}(S^{2}) by the homomorphism p_{∗}. A generator of the group π_{3}(S^{3}) is represented by the identity φ : S^{3}⟶S^{3} and hence

p_{∗}[φ] = [pφ] = [p].

3. In the general case there are two bundles: the real Hopf bundle

S^{n}⟶RP^{n}

with fiber Z_{2}, and complex Hopf bundle

S^{2n−1}⟶CP^{n}

with fiber S^{1}. In the first case, the exact homotopy sequence shows that

π_{1}(S^{n}) = 0⟶π_{1}(RP^{n})⟶π_{0}(Z_{2}) = Z_{2}⟶0,  π_{1}(RP^{n}) = Z_{2},
π_{k}(S^{n}) = 0⟶π_{k}(RP^{n})⟶π_{k−1}(Z_{2}) = 0,  2 ≤ k ≤ n.

Hence

π_{k}(RP^{n}) = 0, 2 ≤ k ≤ n − 1.

When n = 1, we have RP^{1} = S^{1}. In the complex case, one has

0 = π_{2}(S^{2n−1})⟶π_{2}(CP^{n})⟶π_{1}(S^{1}) = Z  ⟶
⟶π_{1}(S^{2n−1}) = 0⟶π_{1}(CP^{n})⟶π_{0}(S^{1}) = 0
π_{k}(S^{2n−1}) = 0⟶π_{k}(CP^{n})⟶π_{k−1}(S^{1}) = 0

where 2n − 1 > k > 2. Hence

π_{1}(CP^{n}) = 0,
π_{2}(CP^{n}) = Z,
π_{k}(CP^{n}) = 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)⟶S^{n}.

So we have the exact homotopy sequence

… ⟶π_{k+1}(S^{n})⟶π_{k}(O(n))⟶^{j_{∗}}π_{k}(O(n + 1))⟶π_{k}(S^{n})⟶ …  (2.15)

If k + 1 < n that is k ≤ n − 2 then

π_{k}(O(n)) ≡ π_{k}(O(n + 1)).

Hence

π_{k}(O(k + 2)) ≡ π_{k}(O(k + 3)) ≡ π_{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

π_{1}(O(1)) = 0,
π_{1}(O(2)) = Z,
π_{1}(O(3)) = π_{1}(O(4)) = … = π_{1}(O(N)).

Homotopy Invariants 87

Also SO(3) = RP^{3}, and so

π_{1}(O(3) = π_{1}(RP^{3}) = Z_{2}.

Hence

π_{1}(O(n)) = Z_{2} for n ≥ 3.

Moreover from (2.15) one has the exact sequence

Z = π_{1}(O(2))⟶π_{1}(O(3)) = Z_{2}⟶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^{2n+1}.

The exact homotopy sequence

… ⟶π_{k+1}(S^{2n+1})⟶π_{k}(U(n))⟶^{j_{∗}}π_{k}(U(n + 1))⟶π_{k}(S^{2k+1})⟶ … (2.16)

shows that

π_{k}(U(n)) = π_{k}(U(n + 1)) for k < 2n.

and, in particular,

π_{1}(U(1)) = … = π_{1}(U(n)).

From U(1) = S^{1} it follows that

π_{1}(U(n)) = Z, n ≥ 1.

For k = 2,

π_{2}(U(2)) = … = π_{2}(U(n)).

From (2.16) it follows that

… ⟶p_{3}(S^{3})⟶π_{2}(U(1))⟶π_{2}(U(2))⟶π_{2}(S^{3})⟶ …,

and hence

0⟶π_{2}(U(2))⟶0,

or

π_{2}(U(2)) = 0.

Hence

π_{2}(U(n)) = 0 for any n.

Let us now calculate π_{3}(U(n)). Firstly, for n ≥ 2,

π_{3}(U(2)) = … = π_{3}(U(n)).

and so it is sufficient to calculate the group π_{3}(U(2)). Consider the locally trivial bundle

p : U(2)⟶S^{1},  (2.17)

where p(A) = det A. The fiber of (2.17) is the group SU(2). Thus we have the exact homotopy sequence

… ⟶π_{4}(S^{1})⟶π_{3}(SU(2))⟶π_{3}(U(2))⟶π_{3}(S^{1})⟶ ….

The end terms are trivial and hence

π_{3}(SU(2)) = π_{3}(U(2)).

Further, the group SU(2) is homeomorphic to the 3–dimensional sphere S^{3} ⊂ C^{2}, and therefore

π_{3}(SU(2)) = Z.

Thus,

π_{3}(U(1)) = 0,
π_{3}(U(n)) = Z, n ≥ 2.

6. In the same spirit as the two previous examples consider locally trivial bundle

p : Sp(n)⟶S^{4n−1}

with a fiber Sp(n − 1) where Sp(n) is the group of orthogonal quaternionic transformations of the quaternionic space K^{n} = R^{4n}. Each such transformation can be described as a matrix with entries which are quaternions. Hence each column is a vector in K^{n} = R^{4n}. The mapping p assigns the first column to each matrix. The exact homotopy sequence

… ⟶π_{k}(Sp(n − 1))⟶π_{k}(Sp(n))⟶π_{k}(S^{4n−1})⟶ …
… ⟶π_{1}(Sp(n − 1))⟶π_{1}(Sp(n))⟶π_{1}(S^{4n−1})⟶0

gives the following table of isomorphisms

π_{1}(Sp(n)) = π_{1}(Sp(1)) = π_{1}(S^{3}) = 0,
π_{2}(Sp(n)) = π_{2}(Sp(1)) = π_{2}(S^{3}) = 0,
pi_{3}(Sp(n)) = π_{3}(Sp(1)) = π_{3}(S^{3}) = Z,
π_{k}(Sp(n)) = π_{k}(Sp(n − 1))  for k < 4n − 2.

7. Let

p : E_{G}⟶B_{G}

be a universal principal G–bundle. By the exact homotopy sequence

… ⟶π_{k}(E_{G})⟶π_{k−1}(B_{G})⟶π_{k−1}(G)⟶π_{k−1}(E_{G})⟶ …

we have

π_{k}(B_{G}) ≡ π_{k−1}(G)

since π_{k}(E_{G}) = 0 for all k ≥ 0.

8. Let G = S^{1}. Then the space B_{S^{1}} can be constructed as a direct limit

CP^{∞} = lim CP^{n},

and total space ES^{1} the direct limit

S^{∞} = lim S^{2n+1}.

Therefore,

π_{2}(CP^{∞}) = Z
π_{k}(CP^{∞}) = 0 for k ≠ 2.  (2.18)

The table (2.18) means that the space CP^{∞} is an Eilenberg–McLane complex K(Z, 2). Thus the same space CP^{∞} classifies both the one–dimensional complex vector bundles and two–dimensional 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^{∗}(ξ),

where f : X⟶CP^{∞} is a continuous mapping, [f] ∈ [X, CP^{∞}], and ξ is universal one–dimensional vector bundle over CP^{∞}. Then the isomorphism

α : Line (X)⟶H^{2}(X, Z),

mentioned above is defined by the formula

α(η) = (l^{−1}(η))^{∗} (a) ∈ H^{2}(X, Z), η ∈ Line (X),

where a ∈ H^{2}(CP^{∞}, Z) is a generator.

Theorem 16 The additive structure in the group H^{2}(X, Z) corresponds with the tensor product in the family of one–dimensional complex vector bundles over X with respect to the isomorphism α.

Proof.

The structure group of a one–dimensional complex vector bundle is U(1) = S^{1}. The tensor product is induced by the homomorphism

φ : S^{1} × S^{1}⟶S^{1},  (2.19)

which is defined by the formula

φ(z_{1}, z_{2}) = z_{1}z_{2}.

The homomorphism (2.19) induces a mapping of the classifying spaces

ψ : CP^{∞} × CP^{∞}⟶CP^{∞}  (2.20)

such that universal bundle ξ over CP^{∞} goes to tensor product

ψ^{∗}(ξ) = ξ_{1} ⊗ ξ_{2}.

Hence the tensor product structure in the Line (X) is induced by the composition

X⟶^{Δ}X × X ⟶^{f_{1}×f_{2}}CP^{∞} × CP^{∞}⟶^{ψ}CP^{∞}.  (2.21)

On the other hand the mapping (2.21) induces the additive structure on the two–dimensional cohomology group. Indeed, the group structure (2.19) has the property that

φ(z, 1) = φ(1, z) = z.

Hence the same holds for (2.20):

ψ(x, x_{0}) = ψ(x_{0}, x) = x.

Therefore,

ψ^{∗}(a) = (a ⊗ 1) + (1 ⊗ a) ∈ H^{2}(CP^{∞}, Z).

Thus

α([f_{1}] ⊗ [f_{2}]) =
= (ψ(f_{1} × f_{2})Δ)^{∗}(a) = (Δ)^{∗}(f_{1} × f_{2})^{∗}ψ^{∗}(a) =
= (Δ)^{∗}(f_{1} × f_{2})^{∗}(a ⊗ 1 + 1 ⊗ a) =
= (Δ)^{∗} (f_{1}^{∗}(a) ⊗ 1 + 1 ⊗ f_{2}^{∗}(a)) = f_{1}^{∗}(a) + f_{2}^{∗}(a) =
= α([f_{1}]) + α([f_{2}]).

2.3 CONSTRUCTIONS OF THE CLASSIFYING SPACES

In this section we shall construct a universal principal G–bundle

p_{G} : E_{G}⟶B_{G}.

There are at least two different geometric constructions of a universal G–bundle. The problem is to construct a contractible space E_{G} with a free action of the group G.

2.3.1 Lie groups

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

U(n) ⊂ U(N + n)

and

U(N) ⊂ U(N + n)

are included as matrices

∥\begin{matrix}X & 0 \\ 0 & 1 \end{matrix}∥ , X ∈ U(n) and ∥\begin{matrix}1 & 0 \\ 0 & Y \end{matrix} ∥ , Y ∈ U(N)

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 V_{N+n,n} and the base is homeomorphic to complex Grassmann manifold G_{N+n,n}. Now consider the inclusion

j : U(N + n)⟶U(N + n + 1)

defined by

T ⟶ ∥\begin{matrix}T & 0 \\ 0 & 1 \end{matrix}∥ , T ∈ U(N + n).

The subgroup U(N) maps to the subgroup U(N + 1):

∥\begin{matrix}1 & 0 \\ 0 & Y \end{matrix} ∥ ⟶ ∥\begin{matrix}∥\begin{matrix}1 & 0 \\ 0 & Y \end{matrix} ∥ & {0 \atop 0} \\ 0 0 & 1 \end{matrix}∥ = ∥\begin{matrix}1 & 0 0 \\ {0 \atop 0} & ∥\begin{matrix}Y & 0 \\ 0 & 1 \end{matrix}∥ \end{matrix}∥ , 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 + n)/U(N)  ⟶^{j}  U(N + n + 1)/U(N + 1)
↓p  ↓p
U(N + n)/ (U(N) ⊕ U(n)) ⟶^{j} U(N + n + 1)/ (U(N + 1) ⊕ U(n))

or

\begin{matrix}V_{N+n,n} & ⟶^{j} & V_{N+n+1} \\ ↓p & & ↓p \\ G_{N+n,n} & ⟶^{j} & G_{N+n+1,n} \end{matrix}.  (2.22)

Let V_{∞,n} denote the direct limit

V_{∞,n} = lim V_{N+n,n},

and G_{∞,n} the direct limit

G_{∞,n} = lim G_{N+n,n}.

The commutative diagram (2.22) induces the mapping

p : V_{∞,n}⟶G_{∞,n}.  (2.23)

The mapping (2.23) gives a principal U(n)–bundle. To prove this it is sufficient to show that p gives a locally trivial bundle. Notice that by the definition of a direct limit, a set U ⊂ G_{∞,n} is open iff for any N the set U ∩ G_{N+n,n} is open in the space G_{N+n,n}. Hence charts for the bundle (2.23) can be constructed as a union

Ω = ∪Ω_{N}

where

Ω_{N} ⊂ G_{N+n,n}

is a open chart in G_{N+n,n} such that

Ω_{N} ⊂ Ω_{N+1}.

The existence of the sequence {Ω_{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

π_{k}(V_{∞,n}) = 0.  (2.24)

Consider the bundle

U(N + n)⟶V_{N+n,n}

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

⟶π_{k}(U(N))⟶π_{k}(U(N + n))⟶π_{k}(V_{N+n,n})⟶
⟶π_{k−1}(U(N))⟶π_{k−1}(U(N + n))⟶ …

it follows that

π_{k}(V_{N+n,n}) = 0

for k < 2N. Since

π_{k}(V_{∞,n}) = lim π_{k}(V_{N+n,n})

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}.

Proposition 3 The mapping

p_{G} : V_{∞,n}⟶BG = V_{∞,n}/G

gives a locally trivial principal G–bundle.

Proof.

Notice that the quotient mapping

q : U(n)⟶U(n)/G

gives a locally trivial G–bundle. 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) × W_{0}, W_{0} ⊂ G_{∞,n},

then the mapping

p_{G}|W : W = U(n) × W_{0}⟶U(n)/G × W_{0}

gives a locally trivial bundle and the set U(n)/G × W_{0} is an open set. Hence the mapping p_{G} gives a locally trivial bundle.

Proposition 4 For each finite dimensional matrix group G there is a universal principal G–bundle.

Proof.

Consider a subgroup

H ⊂ G

such that H is deformation retract of G, so

π_{k}(H) ≡ π_{k}(G).  (2.25)

Consider the commutative diagram

\begin{matrix}EH & ⟶ & EG \\ ↓ & & ↓ \\ BH & ⟶ & BG \end{matrix}  (2.26)

The corresponding homotopy sequences give the following diagram

⟶ π_{k}(H) ⟶ π_{k}(EH) ⟶ π_{k}(BH) ⟶ π_{k−1}(H) ⟶ ↓  ↓  ↓  ↓ ⟶ π_{k}(G) ⟶ π_{k}(EG) ⟶ π_{k}(BG) ⟶ π_{k−1}(G) ⟶ .

(2.27)

Since

π_{k}(EH) = π_{k}(EG) = 0

for all k, and because of (2.25),

π_{k}(BH) ≡ π_{k}(BG).

On the other hand if there is universal principal H–bundle 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

π_{k}(EG) = 0.

This means that for the group G there is universal principal G–bundle.

To finish the proof, it is sufficient to notice that any matrix group has a compact subgroup which is deformation retract of the group.

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 G–bundle. 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,

π_{0}(G) ≠ 0.

To kill π_{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 (α_{0}, g_{0}, α_{1}, g_{1}) where g_{0}, g_{1} ∈ G and

0 ≤ α_{0}, α_{1} ≤ 1, α_{0} + α_{1} = 1

are barycentric coordinates of a point on the unit interval, with the convention that

(0, g_{0}, 1, g_{1}) = (0, g′_{0}, 1, g_{1}),  g_{0}, g′_{0}, g_{1} ∈ G
(1, g_{0}, 0, g_{1}) = (1, g_{0}, 0, g′_{1}),  g_{0}, g_{1}, g′_{1} ∈ G

The action of G on G ∗ G is defined by

h (α_{0}, g_{0}, α_{1}, g_{1}) = (α_{0}, hg_{0}, α_{1}, hg_{1}), h ∈ G.

Similarly, the join of (k + 1) copies of the group G is denoted by

G^{∗k} = G ∗ G ∗ ⋯ ∗ G (k+1 times).

A point of the join has the form

x = (α_{0}, g_{0}, α_{1}, g_{1}, …, α_{k}, g_{k})  (2.29)

where

g_{0}, g_{1}, … g_{k} ∈ G

are arbitrary elements and

(α_{0}, α_{1}, … α_{k}) : α_{i} ≥ 0, α_{0} + α_{1} + ⋯ + α_{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

(α_{0}, g_{0}, α_{1}, g_{1}, …, α_{k}, g_{k}) ≡ (α′_{0}, g′_{0}, α′_{1}, g′_{1}, …, α′_{k}, g′_{k})

iff

α_{i} = α′_{i} for all i

and

g_{i} = g′_{i} for each i when α_{i} = 0.

The group G acts on the G^{∗k} by formula

h(α_{0}, g_{0}, α_{1}, g_{1}, …, α_{k}, g_{k}) = (α_{0}, hg_{0}, α_{1}, hg_{1}, …, α_{k}, hg_{k}), h ∈ G.

Proposition 5 When ≤ k − 1,

π_{j}(G^{∗(k+1)}) = 0.

Proof.

The join X ∗ Y is the union

X ∗ Y = ((CX × Y)) ∪ ((X × CY))  (2.30)

where CX, CY are cones over X and Y. The union (2.30) can represented as the union

X ∗ Y = ((CX × Y)) ∪ CY) ∪ ((X × CY)) ∪ CX) = P_{1} ∪ P_{2}

where

P_{1} = ((CX × Y) ∪ CY) ,
P_{2} = ((X × CY) ∪ CX) .

Both spaces P_{1} and P_{2} are contractible and intersection is

P_{3} = P_{1} ∩ P_{2} = (X × Y) ∪ CX ∪ CY.

Hence both P_{1} and P_{2} are homotopy equivalent to the cone CP_{3}. So the union P_{1} ∪ P_{2} is homotopy equivalent to the suspension SP_{3}. By induction, the space G^{∗(k+1)} is homotopy equivalent to the S^{k}Z for some Z. It is easy to check that

π_{0}(G^{∗(k+1)}) = H_{0}(G^{∗(k+1)}) = 0.

Hence, using the Hurewich theorem

π_{j}(G^{∗(k+1)}) = H_{j}(G^{∗(k+1)}) = 0 for j ≤ k.

Hence

G_{∞} = lim G^{∗k}

has trivial homotopy groups.

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 G_{N+n,n}^{R} = G_{∞,n}^{R}

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 cξ.

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)

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 ξ for which the complexification is trivial and gives concrete trivialization of the bundle cξ.

Proposition 6 Each element of the family [X, U(n)/O(n)] defines a real vector ξ bundle with a homotopy class of trivializations of the bundle cξ.

Proof.

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

\begin{matrix}ξ_{O} & ⟶^{g} & U(n) \\ ↓ & & ↓ \\ X & ⟶^{f} & U(n)/O(n) \end{matrix}.

Let ξ_{U} be the total space of the principal U(n)–bundle associated with the complexification of ξ_{O}. Then the space ξ_{O} lies in the space ξ_{U} and this inclusion ξ_{O} ⊂ ξ_{U} is equivariant with respect to the actions of the group O(n) on ξ_{O} and the group U(n) on ξ_{U}. Therefore, the map g extends uniquely to an equivariant map

ḡ : ξ_{U}⟶U(n).

The latter defines a trivialization of the bundle ξ_{U}. If

f_{1}, f_{2} : X⟶U(n)/O(n)

are homotopic then there is a bundle ξ_{O} over X × I with a trivialization of the complexification. Since the restrictions of the bundle ξ_{O} to the X × {t} are all isomorphic, the trivializations of ξ_{U}|_{X×{0}} and ξ_{U}|_{X×{1}} are homotopic. Conversely, let ξ_{O} be a principal O(n)–bundle and fix a trivialization of the complexification ξ_{U}. This means that there is map ḡ giving a commutative diagram

\begin{aligned}ξ_{U} & ⟶^{ḡ} & U(n) & ⊂ & EU(n) \\ ↓ & & ↓ & & ↓ \\ X & ⟶ & \{pt\} & ⊂ & BU(n) \end{aligned}.

Hence there is a commutative diagram

[figure: X; U(n)/O(n); BO(n); ξ_{O}; U(n); EU(n); f; g; X; pt; BU(n); ξ_{U}; U(n); EU(n); ḡ]

The mapping

f : X⟶U(n)/O(n)

is induced by the mapping ḡ and generates the bundle ξ_{O} and the trivialization ḡ of the bundle cξ_{O}. 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 (ξ, φ) where ξ is real bundle over X and φ is an isomorphism of cξ with a trivial bundle. Two pairs (ξ_{1}, φ_{1}) and (ξ_{2}, φ_{2}) are considered equivalent if the bundles ξ_{1} and ξ_{2} are isomorphic and the isomorphisms φ_{1} and φ_{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 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 α which associates to each n-dimensional vector bundle ξ over X a cohomology class α(ξ) ∈ H^{∗}(X) with some fixed coefficient group for the cohomology groups. In addition, we require functoriality : if

f : X⟶Y

is a continuous mapping, η an n–dimensional vector bundle over Y, and ξ = f^{∗}(η) the pull-back vector bundle over X, then

α(ξ) = f^{∗}(α(η)),  (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 ξ, then might hope to identify the bundle ξ, 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 all characteristic classes of n–dimensional real (complex) vector bundles is in one–to–one correspondence with the cohomology ring H^{∗} (BO(n)) (respectively, with H^{∗} (BU(n))).

Proof.

Let ξ_{n} be the universal bundle over the classifying space BO(n) and α a characteristic class. Then α(ξ_{n}) ∈ H^{∗} (BO(n)) is the associated cohomology class. Conversely, if x ∈ H^{∗} (BO(n)) is arbitrary cohomology class then a characteristic class α is defined by the following rule: if f : X⟶BO(n) is continuous map and ξ = f^{∗}(ξ_{n}) put

α(ξ) = f^{∗}(x) ∈ H^{∗}(X).  (2.33)

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

η = g^{∗}(ξ_{n}), ξ = g^{∗}(η),

then

α(ξ) = α((hg)^{∗}(ξ_{n})) = (hg)^{∗}(x) = g^{∗}(h^{∗}(x)) = g^{∗}(α(h^{∗}(ξ_{n}))) = g^{∗}(α(η)).

If

f : BO(n)⟶BO(n)

is the identity mapping then

α(ξ_{n}) = f^{∗}(x) = x.

Hence the class α 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

α = {α_{1}, α_{2}, …, α_{n}, …}  (2.34)

where each term α_{n} is a characteristic class defined on vector bundles of dimension n.

Definition 5 A class α of the form (2.34) is said to be stable if the following condition holds:

α_{n+1}(ξ ⊕ 1) = α_{n}(ξ),  (2.35)

for any n–dimensional vector bundle ξ.

In accordance with Theorem 17 one can think of α_{n} as a cohomology class

α_{n} ∈ H^{∗} (BO(n)) .

Let

φ : BO(n)⟶BO(n + 1)

be the natural mapping for which

φ^{∗}(ξ_{n+1}) = ξ_{n} ⊕ 1.

This mapping φ is induced by the natural inclusion of groups

O(n) ⊂ O(n + 1).

Then the condition (2.35) is equivalent to:

φ^{∗}(α_{n+1}) = α_{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[c_{1}, c_{2}, …, c_{n}], where

c_{k} ∈ H^{2k}(BU(n); Z).

The generators {c_{1}, c_{2}, …, c_{n}} can be chosen such that

1. the natural mapping

φ : BU(n)⟶BU(n + 1)  (2.37)

satisfies

φ^{∗}(c_{k}) = c_{k}, k = 1, 2, …, n,
φ^{∗}(c_{n+1}) = 0;  (2.38)

2. for a direct sum of vector bundles we have the relations

c_{k}(ξ ⊕ η) = c_{k}(ξ) + c_{k−1}(ξ)c_{1}(η)+
+ c_{k−2}(ξ)c_{2}(η) + … + c_{1}(ξ)c_{k−1}(η) + c_{k}(η) =
= ∑_{α+β=k} c_{α}(ξ)c_{β}(η),  (2.39)

where c_{0}(ξ) = 1.

The condition (2.38) means that the sequence

{0, …, c_{k}, c_{k}, …, c_{k}, …}

is a stable characteristic class which will also be denoted by c_{k}. This notation was used in (2.39). If dim ξ < k then by (2.38) it follows that c_{k}(ξ) = 0.

Formula (2.39) can be written in a simpler way. Put

c = 1 + c_{1} + c_{2} + … + c_{k} + …  (2.40)

The formal series (2.40) has a well defined value on any vector bundle ξ since in the infinite sum (2.40) only a finite number of the summands will be nonzero:

c(ξ) = 1 + c_{1}(ξ) + c_{2}(ξ) + … + c_{k}(ξ), if dim ξ = k.

Hence from (2.39) we see that

c(ξ ⊕ η) = c(ξ)c(η).  (2.41)

Conversely, the relations (2.39) may be obtained from (2.41) by considering the homogeneous components in (2.41).

Proof.

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}(S^{n}) = Z,
H^{n}(S^{n}) = Z,
H^{k}(S^{n}) = 0, when k ≠ 0 and k ≠ n,

the cohomology ring

H^{∗}(S^{n}) =
= ⊕_{k}H^{k}(S^{n}) =
= H^{0}(S^{n}) ⊕ H^{n}(S^{n})

is a free exterior algebra over the ring of integers Z with a generator a_{n} ∈ H^{n}(S^{n}). The choice of the generator a_{n} is not unique: one can change a_{n} for (−a_{n}). We write

H^{∗}(S^{n}) = Λ(a_{n}).

Now consider the bundle U(2)⟶S^{3} with fibre S^{1}. The second term of spectral sequence for this bundle is

E_{2}^{∗,∗} = ∑_{p,q} E_{2}^{p,q} =
= H^{∗}(S^{3}, H^{∗}(S^{1})) = H^{∗} (S^{3}) ⊗ H^{∗}(S^{1}) =
= Λ(a_{3}) ⊗ Λ(a_{1}) = Λ(a_{1}, a_{3}).

The differential d_{2} vanishes except possibly on the generator

1 ⊗ a_{1} ∈ E_{2}^{0,1} = H^{0} (S^{3}, H^{1}(S^{1})) .

But then

d_{2}(1 ⊗ a_{1}) ∈ E_{2}^{2,0} = H^{2} (S^{3}, H^{0}(S^{1})) = 0.

Hence

d_{2}(1 ⊗ a_{1}) = 0,
d_{2}(a_{3} ⊗ 1) = 0,
d_{2}(a_{3} ⊗ a_{1}) = d_{2}(a_{3} ⊗ 1)a_{1} − a_{3}d_{2}(1 ⊗ a_{1}) = 0.

Hence d_{2} is trivial and therefore

E_{3}^{p,q} = E_{2}^{p,q}.

Similarly d_{3} = 0 and

E_{4}^{p,q} = E_{3}^{p,q} = E_{2}^{p,q}.

Continuing, d_{n} = 0 and

E_{n+1}^{∗,∗} = E_{n}^{∗,∗} = … = E_{2}^{∗,∗} = Λ(a_{1}, a_{3}),
E_{∞}^{∗,∗} = Λ(a_{1}, a_{3}).

The cohomology ring H^{∗}(U(2)) is associated to the ring Λ(a_{1}, a_{3}), that is, the ring H^{∗}(U(2)) has a filtration for which the resulting factors are isomorphic to the homogeneous summands of the ring Λ(a_{1}, a_{3}). In each dimension, n = p+q, the groups E_{∞}^{p,q} vanish except for a single value of p, q. Hence

H^{0} (U(2)) = Z,
H^{1} (U(2)) = E_{∞}^{1,0},
H^{3} (U(2)) = E_{∞}^{0,3},
H^{4} (U(2)) = E_{∞}^{1,3}.

Let

u_{1} ∈ H^{1} (U(2)) , u_{3} ∈ H^{3} (U(2))

be generators which correspond to a_{1} and a_{3}, respectively. As a_{1}a_{3} is a generator of the group E_{∞}^{1,3}, the element u_{1}u_{3} 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 cells show the positions of the generators the groups E_{s}^{p,q} for each fixed s–level of the spectral sequence.

[figure: 1; a_{1}; a_{3}; a_{1}a_{3}; q; p]

Figure 2.1

For brevity the tensor product sign ⊗ is omitted. The arrow denotes the action of the differential d_{s} for s = 2, 3. Empty cells denote trivial groups.

Thus we have shown

H^{∗}(U(2)) = Λ(u_{1}, u_{3}).

[figure: 1; u_{1}; u_{2n−3}; a_{2n−1}; u_{1}a_{3}; q; p]

Figure 2.2

Proceeding inductively, assume that

H^{∗} (U(n − 1)) = Λ(u_{1}, u_{3}, …, u_{2n−3}), u_{2k−1} ∈ H^{2k−1} (U(n − 1)) ,

1 ≤ k ≤ n − 1,

and consider the bundle

U(n)⟶S^{2n−1}

with fiber U(n − 1). The E_{2} member of the spectral sequence has the following form:

E_{2}^{∗,∗} = H^{∗} (S^{2n−1}; H^{∗} (U(n − 1))) =
= Λ(a_{2n−1}) ⊗ Λ(u_{1}, …, u_{2n−3}) = Λ(u_{1}, u_{3}, …, u_{2n−3}, a_{2n−1}).

(see fig.2.2).

The first possible nontrivial differential is d_{2n−1}. But

d_{2n−1}(u_{k}) = 0, k = 1, 3, …, 2n − 3,

and each element x ∈ E^{0,q} decomposes into a product of the elements u_{k}. Thus d_{2n−1}(x) = 0. Similarly, all subsequent differentials d_{s} are trivial. Thus

E_{∞}^{p,q} = … = E_{s}^{p,q} = … = E_{2}^{p,q} =
= H^{p} (S^{2n−1}, H^{q} (U(n − 1))) ,
E_{∞}^{∗,∗} = Λ(u_{1}, u_{3}, …, u_{2n−3}, u_{2n−1}).

Let now show that the ring H^{∗} (U(n)) is isomorphic to exterior algebra

Λ(u_{1}, u_{3}, …, u_{2n−3}, u_{2n−1}).

Since the group E_{∞}^{∗,∗} has no torsion, there are elements v_{1}, v_{3}, …, v_{2n−3} ∈ H^{∗} (U(n)) which go to u_{1}, u_{3}, …, u_{2n−3} under the inclusion U(n − 1) ⊂ U(n). All the v_{k} are odd dimensional. Hence elements of the form v_{1}^{ε_{1}}v_{3}^{ε_{3}} ⋯ v_{2n−3}^{ε_{2n−3}} where ε_{k} = 0, 1 generate a subgroup in the group H^{∗} (U(n)) mapping isomorphically onto the group H^{∗} (U(n − 1)). The element a_{2n−1} ∈ E_{∞}^{2n−1,0} has filtration 2n − 1. Hence the elements of the form

v_{1}^{ε_{1}}v_{3}^{ε_{3}} ⋯ v_{2n−3}^{ε_{2n−3}}a_{2n−1}

form a basis of the group E_{∞}^{2n−1,0}. Thus the group H^{∗} (U(n)) has a basis consisting of the elements

v_{1}^{ε_{1}}v_{3}^{ε_{3}} ⋯ v_{2n−3}^{ε_{2n−3}}a_{2n−1}^{ε_{2n−1}}, ε_{k} = 0, 1.

Thus

H^{∗} (U(n)) = Λ = (v_{1}, v_{3}, …, v_{2n−3}, v_{2n−1}).

Now consider the bundle

EU(1)⟶BU(1)

with the fibre U(1) = S^{1}. From the exact homotopy sequence

π_{1} (EU(1)) ⟶π_{1} (BU(1)) ⟶π_{0} (S^{1})

it follows that

π_{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^{∗} (S^{1}) = Λ(u_{1}),

[figure: 1; c; c^{2}; c^{n}; c^{n+1}; u_{1}; u_{1}c; u_{1}c^{2}; u_{1}c^{n}; u_{1}c^{n+1}; q; p]

Figure 2.3

and cohomology of the total space

H^{∗} (EU(1)) = 0.

This means that

E_{∞}^{p,q} = ∩_{s}E_{s}^{p,q} = 0.

We know that

E_{2}^{p,q} = H^{p} (BU(1)) , H^{q}(S^{1}),

and hence

E_{2}^{p,q} = 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,

E_{2}^{p,1} ∼ E_{2}^{p,0} ⊗ u_{1} ∼ E_{2}^{p,0}.

But

E_{2}^{p,q} = 0 for q ≥ 2,

and it follows that

E_{s}^{p,q} = 0 for q ≥ 2.

Hence all differentials from d_{3} on are trivial and so

E_{3}^{p,q} = … = E_{∞}^{p,q} = 0.

Also

E_{3}^{p,q} = H(E_{2}^{p,q}, d_{2})

and thus the differential

d_{2} : E_{2}^{p,1}⟶E^{p+2,0}

is an isomorphism. Putting

c = d_{2}(u_{1}),

we have

d_{2}(u_{1}c^{k}) = d_{2}(u_{1})c^{k} = c^{k+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[c_{1}, …, c_{n−1}]

and consider the bundle

BU(n − 1)⟶BU(n)

with the fiber U(n)/U(n − 1) = S^{2n−1}. The exact homotopy sequence gives us that

π_{1} (BU(n)) = 0.

We know the cohomology of the fiber S^{2n−1} and the cohomology of the total space BU(n − 1). The cohomology of the latter is not trivial but equals the ring Z[c_{1}, …, c_{n−1}]. Therefore, in the spectral sequence only the terms E_{s}^{p,0} and E_{s}^{p,2n−1} may be nontrivial and

E_{2}^{p,2n−1} = E_{2}^{p,0} ⊗ a_{2n−1} = H^{p} (BU(n)) ⊗ = a_{2n−1}.

(see figure 2.4).

Hence the only possible nontrivial differential is d_{2n} and therefore

E_{2}^{p,q} = … = E_{2n}^{p,q},
H(E_{2n}^{p,q}, d_{2n}) = E_{2n+1}^{p,q} = … = E_{∞}^{p,q}

It is clear that if n = p + q is odd then E_{∞}^{p,q} = 0. Hence the differential

d_{2n} : E_{2n}^{0,2n−1}⟶E_{2n}^{2n,0}

is a monomorphism. If p + q = k < 2n − 1 then E_{2}^{p,q} = E_{∞}^{p,q}. Hence for odd k ≤ 2n − 1, the groups E_{2n}^{k,0} are trivial. Hence the differential

d_{2n} : E_{2n}^{k,2n−1}⟶E_{2n}^{k+2n,0}

[figure: 1; a_{2n−1}; c_{n}; q; p]

Figure 2.4

is a monomorphism for k ≤ 2n. This differential makes some changes in the term E_{2n+1}^{k+2n,0} only for even k ≤ 2n. Hence, for odd k ≤ 2n, we have

E_{2n+1}^{k+2n,0} = 0.

Thus the differential

d_{2n} : E_{2n}^{k,2n−1}⟶E_{2n}^{k+2n,0}

is a monomorphism for k ≤ 4n. By induction one can show that

E_{2n}^{k,0} = 0

for arbitrary odd k, and the differential

d_{2n} : E_{2n}^{k,2n−1}⟶E_{2n}^{k+2n,0}

is a monomorphism. Hence

E_{2n+1}^{k,2n−1} = 0
E_{2n+1}^{k+2n,0} = E_{2n}^{k2n,0}/E_{2n}^{k,2n−1} = E_{∞}^{k+2n,0}.

The ring H^{∗} (BU(n − 1)) has no torsion and in the term E_{∞}^{p,q} only one row is nontrivial (q = 0). Hence the groups E_{∞}^{k+2n,0} have no torsion. This means that image of the differential d_{2n} is a direct summand.

Let

c_{n} = d_{2n}(a_{2n−1}).

and then

d_{2n}(xa_{2n−1}) = xc_{2n}.

It follows that the mapping

H^{∗} (BU(n)) ⟶H^{∗} (BU(n))

defined by the formula

x⟶c_{n}x

is a monomorphism onto a direct summand and the quotient ring is isomorphic to the ring H^{∗} (BU(n − 1)) = Z[c_{1}, …, c_{n−1}]. Thus

H^{∗} (BU(n)) = Z[c_{1}, …, c_{n−1}, c_{n}].

Consider now the subgroup

T^{n} = U(1) × … × U(1) (n times ) ⊂ U(n)  (2.42)

of diagonal matrices. The natural inclusion T^{n} ⊂ U(n) induces a mapping

j_{n} : BT^{n}⟶BU(n).  (2.43)

But

BT^{n} = BU(1) × … × BU(1),

and hence

H^{∗}(BT^{n}) = Z[t_{1}, …, t_{n}].

Lemma 4 The homomorphism

j_{n}^{∗} : Z[c_{1}, …, c_{n}]⟶Z[t_{1}, …, t_{n}]

induced by the mapping (2.43) is a monomorphism onto the direct summand of all symmetric polynomials in the variables (t_{1}, …, t_{n}).

Proof.

Let

α : U(n)⟶U(n)

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 α acts on diagonal matrices by permutation of the diagonal elements. In other words, α permutes the factors in the group (2.42). The same is true for the classifying spaces and the following diagram

BT^{n} ⟶^{j_{n}} BU(n)
↓α ↓α
BT^{n} ⟶^{j_{n}} BU(n)

is commutative. The inner automorphism α is homotopic to the identity since the group U(n) is connected. Hence, on the level of cohomology, the following diagram

H^{∗} (BT^{n}) ⟵^{j_{n}^{∗}} H^{∗} (BU(n))
↑α^{∗} ↑α^{∗}
H^{∗} (BT^{n}) ⟵^{j_{n}^{∗}} H^{∗} (BU(n))

or

\begin{matrix}Z[c_{1}, …, c_{n}] & ⟶^{j_{n}^{∗}} & Z[t_{1}, …, t_{n}] \\ ↓α^{∗} & & ↓α^{∗} \\ Z[c_{1}, …, c_{n}] & ⟶^{j_{n}^{∗}} & Z[t_{1}, …, t_{n}] \end{matrix}.

is commutative. The left homomorphism α^{∗} is the identity, whereas the right permutes the variables (t_{1}, …, t_{n}). Hence, the image of j_{n}^{∗} consists of symmetric polynomials.

Now let us prove that the image of j_{n}^{∗} is a direct summand. For this it is sufficient to show that the inclusion

U(k) × U(1) ⊂ U(k + 1)

induces a monomorphism in cohomology onto a direct summand. Consider the corresponding bundle

B (U(k) × U(1)) ⟶BU(k + 1)

with fiber

U(k + 1)/ (U(k) × U(1)) = CP^{k}.

The E_{2} term of the spectral sequence is

E_{2}^{∗,∗} = H^{∗} (BU(k + 1); H^{∗}(CP^{k})) .  (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 d_{s} are trivial and

E_{2}^{p,q} = E_{∞}^{p,q}.

None of these groups have any torsion. Hence the group

H^{p} (BU(k + 1); Z) = E_{∞}^{p,0} ⊂ H^{∗} (BU(k) × BU(1))

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 (t_{1}, …, t_{n}) are the same.

Using Lemma 4, choose generators

c_{1}, …, c_{n} ∈ H^{∗} (BU(n))

as inverse images of the elementary symmetric polynomials in the variables

t_{1}, …, t_{n} ∈ H^{∗} (BU(1) × … × BU(1)) .

Then the condition (2.38) follows from the fact that the element c_{n+1} is mapped by j_{n+1}^{∗} to the product t_{1} · . . . · t_{n+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:

σ_{k}(t_{1}, …, t_{n}, t_{n+1}, …, t_{n+m}) = ∑_{α+β=k} σ_{α}(t_{1}, …, t_{n})σ_{β}(t_{n+1}, …, t_{n+m}).

The proof of Theorem 18 is finished.

The generators c_{1}, …, c_{n} are not unique but only defined up to a choice of signs for the generators t_{1}, …, t_{n}. Usually the sign of the t_{k} is chosen in such way that for the Hopf bundle over CP^{1} the value of c_{1} on the fundamental circle is equal to 1.

The characteristic classes c_{k} 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

c_{k}(X) = c_{k}(T X).

Example

Consider the complex projective space CP^{n}. It was shown in the section 6 of the chapter 1 that the tangent bundle T CP^{n} satisfies the relation

T CP^{n} ⊕ ¯1 = (n + 1)ξ^{∗}

where ξ is the Hopf bundle. Hence

c_{k} (T CP^{n}) = c_{k} (T CP^{n} ⊕ ¯1) = c_{k} ((n + 1)ξ^{∗}) =
= σ_{k}(−t_{1}, −t_{2}, …, −t_{n})|_{t_{k}=t} = (−1)^{k} (\begin{matrix}n + 1 \\ k \end{matrix}) t^{k}.

From this we can see that the tangent bundle of CP^{n} 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); Z_{2}).

Theorem 19 1. The ring H^{∗} (BSO(2n); Q) is isomorphic to the ring of polynomials Q[p_{1}, p_{2}, …, p_{n−1}, χ], where

p_{k} ∈ H^{4k} (BSO(2n); Q) , χ ∈ H^{2n} (BSO(2n); Q) .

2. The ring H^{∗} (BS = O(2n + 1); Q) is isomorphic to the ring of polynomials Q[p_{1}, p_{2}, …, p_{n}], where

p_{k} ∈ H^{4k} (BSO(2n + 1); Q) .

3. Generators {p_{k}} can be chosen such that the natural mapping

BSO(2n)⟶BSO(2n + 1)

induces a homomorphism such that p_{k} goes to p_{k} if 1 ≤ k ≤ n − 1 and p_{n} goes to χ^{2}. The natural mapping

BSO(2n + 1)⟶BSO(2n + 2)

induces a homomorphism such that p_{k} goes to p_{k}, 1 ≤ k ≤ n, and χ goes to zero.

4. For a direct sum of bundles:

p_{k}(ξ ⊕ η) = ∑_{α+β=k} p_{α}(ξ)p_{β}(η),

where p_{0} = 1.

Proof.

The group SO(2) is isomorphic to the circle S^{1} = U(1). Therefore, the cohomology ring of BSO(2) is known:

H^{∗}(BSO(2); Q) = Q[χ], χ ∈ H^{2}(BSO(2); Q).

Consider the bundle

BSO(2k − 2)⟶BSO(2k)  (2.45)

with the fiber SO(2k)/SO(2k − 2) = V_{2k,2}. The Stieffel manifold V_{2k,2} is the total space of a bundle

V_{2k,2}⟶S^{2k−1}  (2.46)

with fiber S^{2k−2}. The manifold V_{2k,2} may be described as the family of pairs of orthogonal unit vectors (e_{1}, e_{2}). We think of the first vector e_{1} as parametrising the sphere S^{2k−1} and the second vector e_{2} as tangent to the sphere S^{2k−1} at the point e_{1}. Since the sphere S^{2k−1} is odd dimensional, there is a nontrivial vector field, that is, a section of the bundle (2.46) inducing a monomorphism

H^{∗}(S^{2k−1})⟶H^{∗}(V_{2k,2}).

In the spectral sequence for the bundle (2.46) the only one possible nontrivial differential d_{2k} vanishes. Hence

E_{2}^{∗,∗} = e_{∞}^{∗,∗} = H^{∗}(V_{2k,2}) = Λ(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[p_{1}, …, p_{k−2}, χ],  (2.47)

where χ ∈ 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}(a_{2k−2}) = v ≠ 0,

[figure: 1; a_{2k−1}; a_{2k−2}; a_{2k−2}·; ·a_{2k−1}; w; q; p]

Figure 2.5

then

d_{2k−1}(a_{2k−2}v) = v^{2} = 0.

Hence

E_{∞}^{2k−1,2k−2} ≠ 0.

The latter contradicts the assumption that

H^{4k−3}(BSO(2k − 2); Q) = 0.

Therefore, one has that

d_{2k−1}(a_{2k−2}) = 0.

and hence

d_{2k−1} = 0.

The dimension of the element a_{2k−1} is odd and hence

d_{2k}(a_{2k−1}) = w ≠ 0.

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 d_{2k} maps the subgroup a_{2k−1}E_{2k}^{∗,∗} monomorphically onto the subgroup wE_{2k}^{∗,∗}. Therefore,

E_{∞}^{∗,∗} = E_{∞}^{∗,0} ⊕ a_{2k−2}E_{∞}^{∗,0}

which is associated to the polynomial ring (2.47). This means that

p_{1}, …, p_{k−2} ∈ E_{∞}^{∗,0},
[χ] = a_{2k−2},
[χ^{2}] ∈ E_{∞}^{∗,0},
E_{∞}^{∗,0} = Q[p_{1}, …, p_{k−2}, χ^{2}],
E_{2}^{∗,0} = Q[p_{1}, …, p_{k−2}, χ^{2}, w].

Put p_{k−1} = χ^{2}. Then

H^{∗} (BSO(2k); Q) = Q[p_{1}, …, p_{k−1}, w].

Consider the bundle

BSO(2k − 2)⟶BSO(2k − 1)  (2.48)

with the fiber S^{2k−2}. In the spectral sequence for the bundle (2.48), the member E_{2}^{∗,∗} has only two nontrivial rows: E_{2}^{∗,0} and E_{2}^{∗,2k−2} = a_{2k−2}E_{2}^{∗,0}. If

d_{2k−1}(a_{2k−2}) = v ≠ 0,

then

d_{2k−1}(a_{2k−2}v) = v^{2} = 0.

Hence

E_{∞}^{2k−1,2k−2} ≠ 0.

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}(a_{2k−2}) = 0,
d_{2k−1} = 0,
E_{2}^{∗,∗} = E_{∞}^{∗,∗}.

Hence the odd dimensional cohomology of the space BSO(2k − 1) is trivial.

Consider the bundle

BSO(2k − 1)⟶BSO(2k)  (2.49)

with fiber S^{2k−1}. The member E_{2}^{∗,∗} of the spectral sequence of the bundle (2.49) has the form

E_{2}^{∗,∗} = H^{∗} (BSO(2k)) ⊗ Λ(a_{2k−1}).

The odd dimensional cohomology of the total space BSO(2k − 1) is trivial and so

d_{2k}(a_{2k−1}) = v ≠ 0.

Using the calculation for the bundle (2.45), we see that

v = d_{2k}(a_{2k}) = w.

Hence

d_{2k}(a_{2k−1}x) = wx.

Thus

E_{2k+1}^{∗,∗} = E_{∞}^{∗,∗} = Q[q_{1}…. , p_{k−1}] = H^{∗} (BSO(2k − 1); Q) .

To prove the last relation of Theorem 19 we study the bundle

j_{n} : BSO(2) × … × BSO(2)⟶BSO(2n).

We need to prove that the image of j_{n}^{∗} coincides with the ring of symmetric polynomials of variables {t_{1}^{2}, p_{2}^{2}, …, p_{n}^{2}} where

t_{k} ∈ H^{∗} (BSO(2) × … × BSO(2); Q)

are generators. We leave this proof for reader.

The characteristic classes

p_{k} ∈ H^{4k} (BSO(n); Q)

constructed in Theorem 19 are called the rational Pontryagin classes . The class

χ ∈ H^{2n} (BSO(2n); Q)

is called the Euler class .

Theorem 20 The ring H^{∗} (BO(n); Z_{2}) is isomorphic to the polynomial ring Z[w_{1}, …, w_{n}],

H^{∗} (BO(n); Z_{2}) = Z[w_{1}, …, w_{n}], w_{k} ∈ H^{k} (BO(n); Z_{2}) .

The generators w_{k} can be chosen such that

1.

w_{k}(ξ) = 0 for k > dim ξ;

2.

w_{k}(ξ ⊕ 1) = w_{k}(ξ);

3.

w_{k}(ξ ⊕ η) = ∑_{α+β=k} w_{α}(ξ)w_{β}(η).

The proof of this theorem is left to the reader. The class w_{k} is called Stieffel– Whitney characteristic class .

Exercise

Prove that the structure group O(n) of the vector bundle ξ can bes reduced to the subgroup SO(n) if and only if w_{1}(ξ) = 0. For this consider the bundle

BSO(n)⟶BO(n),

a two–sheeted covering. If w_{1}(ξ) = 0 then the vector bundle ξ is said to be an oriented bundle .

Example

Consider the space BO(1) = RP^{∞} and the universal one–dimensional Hopf vector bundle over it. It is clear that

w_{1}(ξ) = w_{1} ∈ H^{1} (RP^{∞}; Z_{2})

is a generator and

w_{1}(ξ ⊕ ξ) = w_{1}(ξ) + w_{1}(ξ) = 2w_{1} = 0.

Hence the bundle ξ ⊕ ξ is orientable. The bundle ξ ⊕ ξ is nontrivial since

w_{2}(ξ ⊕ ξ) = w_{1}(ξ)w_{1}(ξ) = w_{1}^{2} ≠ 0.

2.5 GEOMETRIC INTERPRETATION OF SOME CHARACTERISTIC CLASSES

2.5.1 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 SX 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 SX. The bundle

SX⟶X  (2.50)

with the fiber S^{2n−1} is classified by a mapping

X⟶BSO(2n)  (2.51)

and the universal bundle

SESO(2n)⟶BSO(2n)

with the fiber S^{2n−1}.

The classifying space BSO(2n) can be taken as the space of 2n–dimensional 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 2n–dimensional 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 S^{2n−1}.

Let E_{s}^{p,q} be the spectral sequence for the bundle (2.53) and E_{s}^{p,q}(X) be the spectral sequence for the bundle (2.50). Let

f : E_{s}^{p,q}⟶E_{s}^{p,q}(X)

be the homomorphism induced by the mapping (2.51). Then the element

f^{∗}(χ) ∈ H^{2n}(X) = E_{2}^{2n,0}(X)

is the Euler class of the manifold X. In the section 2.4 (Theorem 19), it was shown that projection

p : BO(2n − 1)⟶BO(2n)

sends the Euler class to zero. Hence the same holds for the projection

p′ : SX⟶X

where:

p^{′∗}(f^{∗}(χ)) = 0.

If

α : X⟶SX

is a section, then

0 = α^{∗}p^{′∗}(f^{∗}(ξ)) = f^{∗}(χ).  (2.54)

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 2n–dimensional manifold. On the other hand, the triviality of the rational Euler class implies the triviality of the integer Euler class since 2n–dimensional 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.

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 n–dimensional skeleton [X]^{n}. It is sufficient to extend the section s over each n–dimensional cell σ_{i} ⊂ X separately. Over a cell σ_{i} the bundle is trivial, that is,

p^{−1}(σ_{i}) ≡ σ_{i} × F,

the restriction of the section s to the boundary ∂σ_{i} defines a mapping to the fiber F:

s : ∂σ_{i} : ⟶F,

and which defines an element of the homotopy group of the fiber:

[s]_{i} ∈ π_{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 ∈ π_{n−1}(F).

In other words, the cochain

[s] ∈ C^{n}(X, π_{n−1}(F))

defined by the section s may be considered as a obstruction to the extension of the section s to the n–dimensional 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 Σ. Let

f : ∂Σ⟶[X]^{n}

be the mapping which defines the gluing of the cell to the skeleton [X]^{n}. Consider the composition

f̃: ∂Σ⟶[X]^{n}⟶[X]^{n}/[X]^{n−1} = ∨σ_{i}/∂σ_{i}.

Using a homotopy of the mapping f we can assume that f̃ is smooth everywhere except at the fixed point x_{0} of ∨σ_{i}/∂σ_{i} and the points x_{i} ∈ σ_{i}/∂σ_{i} are nonsingular. This means that

f̃^{−1}(x_{0}) = ∪_{i,j}D_{ij}

where the D_{i,j} ⊂ ∂Σ are n–dimensional disks which are pairwise disjoint and for which each

f̃_{|D_{i,j}} : D_{i,j}⟶σ_{i}/∂σ_{i}  (2.55)

is a diffeomorphism on the interior of D_{i,j}. Let t_{i,j} = ±1 be the degree of the mapping (2.55). Then for the cell chain generated by single cell Σ we have the boundary homomorphism

δΣ = ∑_{i} (∑_{j} t_{i,j}) σ_{i}.

with corresponding adjoint boundary homomorphism on cochains

(∂x)(Σ) = ∑_{i,j} t_{i,j}x(σ_{i}).  (2.56)

Now the section s is defined on the ∂Σ\ ∪_{i,j} ∫ D_{i,j}. The restriction of the section s to ∂D_{i,j} gives an element [s]_{i,j} of π_{n−1}(F) for which

[s]_{i,j} = t_{i,j}[s]_{i} ∈ π_{n−1}(F).

Thus from (2.56),

(∂[s])(Σ) = ∑_{i,j} t_{i,j}[s](σ_{i}) = ∑_{i,j} t_{i,j}[s]_{i} = ∑_{i,j}[s]_{i,j}.

Lemma 5 Let D_{k} ⊂ ∂Σ be a family of disjoint disks and let

g : ∂Σ\ ∪_{k} ∫ D_{k}⟶F

be a continuous mapping which on each ∂D_{k} gives an element [g]_{k} ∈ π_{n−1}(F). Then

∑_{k}[g]_{k} = 0 ∈ π_{k}(F).

Lemma 5 finishes proof that [s] is cocycle.

The obstruction theory says that the cocycle [s] ∈ C^{n}(X, π_{n−1}(F)) gives a cohomology class [s] ∈ H^{n}(X; π_{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; π_{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 locally trivial bundle with the fiber F and let X be a simply connected cellular space. Let

π_{0}(F) = … π_{n−2}(F) = 0
π_{n−1}(F) = Z.

Let a ∈ H^{n−1}(F; Z) be a generator and let E_{s}^{p,q} be the spectral sequence of the bundle p. Then the first nontrivial differential acting on the element a is d_{n}, E_{n}^{n,0} = E_{2}^{n,0} = H^{n}(X; Z), and the element w = d_{n}(a) ∈ H^{n}(X; Z) is the first obstruction to constructing a section of the bundle p.

Theorem 22 Let ξ be a complex n–dimensional vector bundle over a base X. Then the Chern class c_{k}(ξ) ∈ H^{2k}(X; Z) coincides with the first obstruction to constructing of n − k + 1 linear independent sections of the bundle ξ.

Proof.

Constructing n − k + 1 linearly independent sections of ξ is the same as representing the bundle ξ as a direct sum

ξ = η ⊕ \overline{n − k + 1}.  (2.57)

Consider the bundle

p : BU(k − 1)⟶BU(n)  (2.58)

with the fiber U(n)/U(k − 1). Let

f : X⟶BU(n)

be a continuous mapping inducing the bundle ξ, that is,

ξ = f^{∗}(ξ_{n}).

[figure: 1; a; w; q; 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 ξ 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 p^{∗}(c_{k}) = 0, the element c_{k} ∈ E_{2}^{2k,0} is the image of the differential d_{2k},

d_{2k}(u_{2k−1}) = c_{k}.

By Theorem 21, we have

f^{∗}(c_{k}) = c_{k}(ξ)

giving the first obstruction to constructing n−k+1 linearly independent sections of the bundle x.

There are similar interpretations for the Stieffel–Whitney classes and the Pontryagin classes.

[figure: 1; u_{2k−1}; c_{k}; q; p]

Figure 2.7

2.6 K–THEORY AND THE CHERN CHARACTER

2.6.1 Definitions and simple properties

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 bundles over the base X subject to the following relations:

[ξ] + [η] − [ξ ⊕ η] = 0  (2.59)

for vector bundles ξ and η, and where [ξ] denotes the element of the group K(X) defined by the vector bundle ξ.

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

To avoid confusion, the group generated by all real vector bundles will be denoted by K_{O}(X), the group generated by all complex vector bundles will be denoted by K_{U}(X) and the group generated by all quaternionic vector bundles will be denoted by K_{Sp}(X).

The relations (2.59) show that in the group K(X) the direct sum corresponds to the group operation.

Proposition 7 Each element x ∈ K(X) can be represented as a difference of two vector bundles:

x = [ζ_{1}] − [ζ_{2}].

Proof.

In the general case, the element x is a linear combination

x = ∑_{i=1}^{p} n_{i}[ξ_{i}]

with the integer coefficients n_{i}. Let us split this sum into two parts

x = ∑_{i=1}^{p_{1}} n_{i}[ξ_{i}] − ∑_{j=1}^{p_{2}} m_{j}[η_{j}]

with

n_{i} > 0, m_{j} > 0.

Using the relations (2.59) we have

x = [⊕_{i=1}^{p_{1}} \underbrace{(ξ_{i} ⊕ … ⊕ ξ_{i})}_{n_{i} times}] − [⊕_{j=1}^{p_{2}} \underbrace{(η_{j} ⊕ … ⊕ η_{j})}_{m_{j} times}] =

= [ζ_{1}] − [ζ_{2}].

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

ζ_{2} ⊕ ζ_{3} = N̄

is a trivial bundle. Then

x = [ζ_{1}] − [ζ_{2}] = ([ζ_{1}] − [ζ_{2}]) + ([ζ_{3}] − [ζ_{=}3]) =
= ([ζ_{1}] + [ζ_{3}]) − ([ζ_{2}] + [ζ_{3}]) = [ζ_{2} ⊕ ζ_{3}) − N̄.

Proposition 8 Two vector bundles ξ_{1} and ξ_{2} define the same element in the group K(X) if and only if there is a trivial vector bundle N̄ such that

ξ_{1} ⊕ N̄ = ξ_{2} ⊕ N̄.  (2.60)

Condition 2.60 is called stable isomorphism of vector bundles . Proof.

If the condition (2.60) holds then by (2.59)

[ξ_{1}] + [N̄] = [ξ_{2}] + [N̄],

that is, [ξ_{1}] = [ξ_{2}]. Conversely, assume that [ξ_{1}] = [ξ_{2}] or [ξ_{1}] − [ξ_{2}] = 0 . Then by definition, this relation can be represented as a linear combination of determining relations:

[ξ_{1}] − [ξ_{2}] = ∑_{j} λ_{j}(η_{j} + ζ_{j} − η_{j} ⊕ ζ_{j}).  (2.61)

Without loss of generality we can consider that λ_{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

ξ_{1} + ∑_{j}(η_{j} ⊕ ζ_{j}) + ∑_{k}(η_{k} + ζ_{k}) =

= ξ_{2} + ∑_{j}(η_{j} + ζ_{j}) + ∑_{k}(η_{k} ⊕ ζ_{k})  (2.62)

where j runs through all indices for which λ_{j} = 1 and k runs through all indices for which λ_{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 summands in the left hand side is equal to the direct sum of all summand in the right hand side. Thus

ξ_{1} ⊕ ∑_{j}(η_{j} ⊕ ζ_{j}) ⊕ ∑_{k}(η_{k} ⊕ ζ_{k}) =

= ξ_{2} ⊕ ∑_{j}(η_{j} ⊕ ζ_{j}) ⊕ ∑_{k}(η_{k} ⊕ ζ_{k}).

Put

θ = ∑_{j}(η_{j} ⊕ ζ_{j}) ⊕ ∑_{k}(η_{k} ⊕ ζ_{k}).

Then we have

ξ_{1} ⊕ θ = ξ_{2} ⊕ θ.

Finally, let χ be a vector bundle such that N̄ = χ ⊕ θ then

ξ_{1} ⊕ N̄ = ξ_{2} ⊕ N̄.

Proposition 9 The operation of tensor product of vector bundles 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(x_{0}) where x_{0} 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(x_{0}) is isomorphic to the ring Z of integers. In addition, the difference [ξ] − [η] corresponds the number dim ξ − dim η ∈ Z.

2. If X = {x_{1}, x_{2}} is a two point space then any vector bundle is defined by two integers — the dimension of the fiber over the point x_{1} and the dimension of the fiber over the point x_{2}. Hence

K(x_{0}) = Z ⊕ Z.

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 ξ associates the inverse image f^{∗}(ξ). 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 × 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^{∗}(ξ) to the subspaces X × {0} and X × {1} are isomorphic. Hence

f^{∗}(ξ) = g^{∗}(ξ).

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, x_{0}) be a cellular space with a base point. The natural inclusion x_{0}⟶X induces a homomorphism of rings:

K(X)⟶K(x_{0}) = Z.  (2.63)

This homomorphism is defined by a formula for a difference [ξ] − [η]:

[ξ] − [η]⟶ dim ξ − dim η ∈ Z

Note that if X has many connected components then we take the dimension of the bundle x over the component containing the base point x_{0}. Therefore, the homomorphism (2.63) will also be denoted by dim : K(X)⟶K(x_{0}).

Let K^{0}(X, x_{0}) denote the kernel of the homomorphism (2.63):

K^{0}(X, x_{0}) = Ker (K(X)⟶K(x_{0})) .

Elements of the subring K^{0}(X, x_{0}) are represented by differences [ξ] − [η] for which dim ξ = dim h. The elements of the ring K(X) are called virtual bundles and elements of the ring K^{0}(X, x_{0}) are virtual bundles of trivial dimension over the point x_{0}.

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 n–times suspension of the space X:

S^{n}X = (S^{n} × X)/(S^{n} ∨ X).

Theorem 23 The pair (X, Y) induces an exact sequence

K^{0}(Y, x_{0})⟵K^{0}(X, x_{0})⟵K^{0}(X, Y)⟵
⟵K^{−1}(Y, x_{0})⟵K^{−1}(X, x_{0})⟵K^{−1}(X, Y)⟵
⟵K^{−2}(Y, x_{0})⟵K^{−2}(X, x_{0})⟵K^{−2}(X, Y)⟵ …
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
… ⟵K^{−n}(Y, x_{0})⟵K^{−n}(X, x_{0})⟵K^{−n}(X, Y)⟵ …
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.64)

Proof.

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)⟶

⟶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, x_{0})⟵^{i^{∗}}K^{0}(X, x_{0})⟵^{j^{∗}}K^{0}(X, Y)

at the middle term. Let ξ be a vector bundle over X for which the restriction to the subspace Y is trivial. This means that the bundle ξ is trivial in some neighborhood U of the Y. Hence the neighborhood U can serve as a chart of an atlas for ξ. 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 η over X/Y such that

j^{∗}([η]) = [ξ].

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 c_{1}(ξ), …, c_{n}(ξ) 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

φ^{n}(t_{1}, …, t_{n}) = e^{e_{1}} + e^{t_{2}} + … + e^{t_{n}}

and its the Taylor series

φ^{n}(t_{1}, …, t_{n}) = ∑_{k=0}^{∞} \frac{t_{1}^{k} + … + t_{n}^{k}}{k!} .  (2.65)

Each term of the series (2.65) is a symmetric polynomial in the variables t_{1}, …, t_{n} and hence may be expressed as a polynomial in the elementary symmetric polynomials

\frac{t_{1}^{k} + … + t_{n}^{k}}{k!} = φ_{k}^{n}(σ_{1}, …, σ_{n}),
σ_{1} = σ_{1}(t_{1}, …, t_{n}) = t_{1} + t_{2} + … + t_{n},
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
σ_{n} = σ_{n}(t_{1}, …, t_{n}) = t_{1}t_{2} ⋯ t_{n},

In other words,

φ^{n}(t_{1}, …, t_{n}) = ∑_{k=0}^{∞} φ_{k}^{n}(σ_{1}, …, σ_{n}).

Definition 7 Let ξ be a n–dimensional complex vector bundle over a space X. The characteristic class

ch ξ = ∑_{k=0}^{∞} ∑_{k=0}^{∞} φ_{k}^{n}(c_{1}(ξ), …, c_{n}(ξ))  (2.66)

is called the Chern character of the bundle ξ.

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 ξ is a nonhomogeneous element of the group

H^{2∗}(X; Q) = ⊕H^{2k}(X; Q)

since from some number k_{0} 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 following conditions:

1. ch (ξ ⊕ η) = ch (ξ) + ch (η);

2. ch _{0}(ξ) = dim ξ;

3. ch (ξ ⊗ η) = ch (ξ)ch (η).

Proof.

As in the section 2.4, we can represent the Chern classes by elementary symmetric polynomial in certain variables, that is,

c_{k}(ξ) = σ_{k}(t_{1}, …, t_{n}),
c_{k}(η) = σ_{k}(t_{n+1}, …, t_{n+m}),
c_{k}(ξ ⊕ η =) = σ_{k}(t_{1}, …, t_{n}, t_{n+1}, …, t_{n+m}).  (2.67)

Using (2.66),

ch _{k}(ξ) = φ_{k}^{n}(c_{1}(ξ), …, c_{n}(ξ)) = \frac{1}{k!} (t_{1}^{k} + ⋯ + t_{n}^{k}) ,
ch _{k}(η) = φ_{k}^{m}(c_{1}(η), …, c_{n}(η)) = \frac{1}{k!} (t_{n+1}^{k} + ⋯ + t_{n+m}^{k}) ,
ch _{k}(ξ ⊕ η) = φ_{k}^{n+m}(c_{1}(ξ ⊕ η =), …, c_{n+m}(ξ ⊕ η)) =
= \frac{1}{k!} (t_{1}^{k} + ⋯ + t_{n}^{k} + t_{n+1}^{k} + ⋯ + t_{n+m}^{k}) =
= ch _{k}(ξ) + ch _{k}(η).  (2.68)

and from (2.68) property 1 follows. Property 2 follows from

ch _{0}(ξ) = φ_{0}^{n}(c_{1}(ξ), …, c_{n}(ξ)) = n = dim ξ.

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) × BU(m),

where ξ = ξ_{n} is the universal vector bundle over the factor BU(n) and η = ξ_{m} is the universal vector bundle over the factor BU(m). Let

f : BU(n) × BU(m)⟶BU(n + m)

be a mapping such that

f^{∗}(ξ_{n+m}) = ξ_{n} ⊕ ξ_{m}.

Then we must prove that

f^{∗}(ch ξ_{n+m}) = ch ξ_{n} + ch ξ_{m}.  (2.69)

Consider a commutative diagram

[figure: BU(n) × BU(m); f; BU(n + m); p; q; \underbrace{BU(1) × … × BU(1)}_{n times} × \underbrace{BU(1) × … × BU(1)}_{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 = j_{n} × j_{m}, q = j_{n+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 ξ_{n+m}) = p^{∗}ch ξ_{n} + p^{∗}ch ξ_{m},

that is, from

q^{∗}(ch ξ_{n+m}) = p^{∗}ch ξ_{n} + p^{∗}ch ξ_{m}

or from

ch q^{∗}(ξ_{n+m}) = ch p^{∗}ξ_{n} + ch p^{∗}ξ_{m}.

Let the η_{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^{∗}ξ_{n+m} = η_{1} ⊕ … ⊕ η_{n+m},
p^{∗}ξ_{n} = η_{1} ⊕ … ⊕ η_{n},
p^{∗}ξ_{m} = η_{n+1} ⊕ … ⊕ η_{n+m}.

Therefore, we can put t_{i} = c_{1}(η_{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

ξ = η_{1} ⊕ … ⊕ η_{n}
η = η_{n+1} ⊕ … ⊕ η_{n+m}.

Then

ξ ⊗ η = ⨁_{i,j}(η_{i} ⊗ η_{n+j}).

So we need to calculate the first Chern class c_{1}(ξ′ ⊗ ξ′′) 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) × BU(1)⟶BU(1)

such that f^{∗}(ξ) = ξ′ ⊗ ξ′′. Then for some integer α and β

c_{1}(ξ′ ⊗ ξ′′) = αc_{1}(ξ′) + βc_{1}(ξ′′).

When ξ′ = 1 we have ξ′ ⊗ ξ′′ = ξ′′ and hence

c_{1}(ξ′′) = αc_{1}(1) + βc_{1}(ξ′′) = βc_{1}(ξ′′),

that is, β = 1. Similarly, α = 1. Finally,

ch (ξ ⊗ η) = ∑_{i,j} e^{c_{1}(η_{i}⊗η_{n+j})} =
= ∑_{i,j} e^{t_{i}+t_{n+j}} = ∑_{i,j} e^{t_{i}}e^{t_{n+j}} =

= (∑_{i} e^{t_{i}}) (∑_{j} e^{t_{n+j}} ) = ch ξch η.

Theorem 24 shows that the Chern character can be extended to a homomorphism of rings

ch : K(X)⟶H^{2∗}(X; Q).

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 Atiyah–Singer 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 ξ over the base X. Assume that the bundle ξ is trivial over a closed subspace Y ⊂ X. This means that there is a neighborhood U ⊃ Y such that the restriction of the bundle ξ to U is isomorphic to a Cartesian product:

ξ_{|U} ∼ U × R^{n}.

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 φ_{αβ} are defined at points of the complement X\Y. Therefore, the same transition functions can be used for the definition of a bundle η over the new base — the quotient space X/Y. The bundle η can be considered as a bundle obtained from ξ by identification of fibers over Y with respect to the isomorphism ξ_{|Y} ∼ Y × R^{n}.

This important construction can be generalized in the following way. Consider a pair (X, Y) of cellular spaces and a triple (ξ_{1}, d, ξ_{2}) where ξ_{1} and ξ_{2} are vector bundles over X and d is an isomorphism

d : ξ_{1}|_{Y} ⟶ξ_{2}|_{Y} .

In the case when ξ_{2} is a trivial bundle we have the situation described above.

Consider the family of all such triples. The triple (ξ_{1}, d, ξ_{2}) is said to be trivial if d can be extended to an isomorphism over the whole base X. If two isomorphisms d_{1}, d_{2} : ξ_{1}|_{Y} ⟶ξ_{2}|_{Y} are homotopic in the class of isomorphisms then the two triples (ξ_{1}, d_{1}, ξ_{2}) and (ξ_{1}, d_{2}, ξ_{2}) are said to be equivalent. If the triple (η_{1}, h, η_{2}) is trivial then the triples (ξ_{1}, d, ξ_{2}) and (ξ_{1} ⊕ η_{1}, d ⊕ h, ξ_{2} ⊕ η_{2}) are also said to be equivalent. We denote by K^{2}(X, Y) the family of classes of equivalent triples.

Theorem 25 The family K^{2}(X, Y) is an Abelian group with respect to the direct sum of triples. The group K^{2}(X, Y) is isomorphic to K^{0}(X, Y) by the correspondence

K^{0}(X, Y) ∋ ([ξ] − [N̄])⟶(p^{∗}ξ, d, N̄)

where p : X⟶X/Y is natural projection and d : p^{∗}ξ|_{Y} ⟶N̄ is natural isomorphism generated by trivialization on a chart which contains the point [Y] ∈ X/Y, (dim ξ = N).

Proof.

The operation of summation is defined by

(ξ_{1}, d_{1}, ξ_{2}) + (η_{1}, d_{2}, η_{2}) = (ξ_{1} ⊕ η_{1}, d_{1} ⊕ d_{2}, ξ_{2} ⊕ η_{2}).

Hence associativity is clear. It is clear that a trivial triple gives a neutral element. If (ξ_{1}, d_{1}, ξ_{2}) is trivial triple and d is homotopic to d′ in the class of isomorphisms then the triple (ξ_{1}, d′_{1}, ξ_{2}) also is trivial. Indeed, assume firstly that both vector bundles ξ_{1} and ξ_{2} are trivial. Then triviality means that the mapping d : Y ⟶GL(n) extends to the mapping d̄: X⟶GL(n). Hence there is a continuous mapping

D : (Y × I) ∪ (X × {0})⟶GL(n).

The space (Y × I) ∪ (X × {0}) is a retract in the space X × I. Hence D extends to a continuous mapping

D̄ : X × I⟶GL(n).

Geometric constructions of bundles 139

For nontrivial vector bundles the theorem can be proved by induction with respect to the number of charts. Hence the triple (ξ_{1}, d_{1}, ξ_{2}) is equivalent to a trivial triple if and only there is a trivial triple (η_{1}, h, η_{2}) such that the direct sum (ξ_{1} ⊕ η_{1}, d ⊕ h, ξ_{2} ⊕ η_{2}) is trivial.

What are the inverse elements? We show that the triple (ξ_{1}⊕ξ_{2}, d⊕d^{−1}, ξ_{2}⊕ξ_{1}) is equivalent to trivial triple. For this consider the homotopy

D_{t} = ∥\begin{matrix}d sin t & cos \\ − cos t & d^{−1} sin t \end{matrix}∥ : (ξ_{1} ⊕ ξ_{2})|_{Y} ⟶(ξ_{1} ⊕ ξ_{2})|_{Y}

When t = 0,

D_{0} = ∥\begin{matrix}d & 0 \\ 0 & d^{−1} \end{matrix}∥

and when t = 1,

D_{1} = ∥\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}∥ .

The latter matrix defines an isomorphism which can be extended to an isomorphism over the whole of X. Hence the family K^{2}(X, Y) is a group.

Consider a mapping

α : K^{0}(X, Y)⟶K^{0}(X, Y)

defined as follows. Let x ∈ K^{0}(X, Y) be represented as a difference x = [ξ]−[N̄], dim ξ = N, let p : X⟶X/Y be the projection, η = p^{∗}(ξ), and let f : η⟶ξ be the canonical mapping. Then let

d : η|_{Y} ⟶ N̄|_{Y} = Y × R^{N}
d(h) = (y, f(h)), h ∈ η_{y}, y ∈ Y.

The triple (η, d, N̄), which we denote by α(x), defines an element of the group K^{0}(X, Y). This element is well defined, that is, it does not depend on the representation of the element x.

We now prove that α is a monomorphism. Assume that the triple (η, d, N̄) is equivalent to a trivial triple. This means that there is a trivial triple (η_{1}, h, η_{2}) such that direct sum (η ⊕ η_{1}, d ⊕ h, N̄ ⊕ η_{2}) also is trivial. In particular, the bundles η_{1} and η_{2} are isomorphic over X. Without loss of generality we can assume that both η_{1} and η_{2} are trivial and h ≡ 1. Hence the triple (η ⊕ N̄′, d ⊕ 1, N̄ ⊕ N̄′) is trivial and so the bundle η ⊕ N̄′ is isomorphic to trivial bundle N̄ ⊕ N̄′. Hence x = 0.

Next we prove that α is an epimorphism. Consider a triple (ξ_{1}, d, ξ_{2}). If η is a bundle such that ξ_{2} ⊕ η = N̄. The triple (η, 1, η) is trivial and the triple (ξ_{1} ⊕ η, d ⊕ 1, N̄) is equivalent to the triple (ξ_{1}, d, ξ_{2}). Hence the bundle ξ_{1} ⊕ η is trivial over Y. This means that ξ_{1} ⊕ η = p^{∗}(ζ) for a bundle ζ over X/Y. Hence

α([ζ] − [N̄]) = (ξ_{1}, d, ξ_{2}).

The definition of the groups K^{0}(X, Y) has the following generalization. If (ξ_{1}, d, ξ_{2}) is a triple, it can be represented as a complex of vector bundles

¯0⟶ξ_{1}⟶^{d̄}ξ_{2}⟶¯0  (3.1)

which is exact over Y. For this we need to extend the isomorphism d (over Y) to a morphism d̄ 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

¯0⟶ξ_{1}⟶^{d_{1}}ξ_{2}⟶^{d_{2}}ξ_{3}⟶ ⋯ ⟶^{d_{n−1}}ξ_{n}⟶¯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 K^{n}(X, Y). There is a natural mapping

β : K^{2}(X, Y)⟶K^{n}(X, Y).

A bundle complex of the form

¯0⟶¯0⟶ ⋯ ⟶ξ_{k}⟶^{d_{k}}ξ_{k+1}⟶ ⋯ ⟶¯0

is said to be an elementary complex.

Lemma 6 Every trivial bundle complex has a splitting into a sum of elementary trivial complexes.

Proof.

follows from Theorem 7 of the chapter 1.

Lemma 7 Every bundle complex of type (3.2) is equivalent to an elementary complex, that is, the mapping β is epimorphism.

Proof.

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:

¯0⟶ξ_{1}⟶ ⋯ ⟶ξ_{n−3} ⟶^{D_{n−3}}ξ_{n−2} ⊕ η ⟶^{D_{n−2}}

⟶^{D_{n−2}} ξ_{n−1} ⊕ η ⟶^{D_{n−1}}ξ_{n}⟶¯0,  (3.3)

where

D_{k} = d_{k}, 1 ≤ k ≤ n − 4,

D_{n−3} = ∥\begin{matrix}d_{n−3} \\ 0 \end{matrix}∥ , D_{n−2} = ∥\begin{matrix}d_{n−2} & 0 \\ 0 & 1 \end{matrix}∥ , D_{n−1} = ∥d_{n−1} 0∥ .

The morphism d_{n−1} is an epimorphism over Y. Hence there is a neighborhood U ⊃ Y such that d_{n−1} is epimorphism over U. Let φ be a function which equals 1 on X\U and 0 on Y, and φ′ be a function which equals 1 on Y and is such that

φφ′ ≡ 0.

Consider the new bundle complex

¯0⟶ξ_{1}⟶ ⋯ ⟶ξ_{n−3} ⟶^{D′_{n−3}}ξ_{n−2} ⊕ η ⟶^{D′_{n−2}}

⟶^{D′_{n−2}} ξ_{n−1} ⊕ η ⟶^{D′_{n−1}}ξ_{n}⟶¯0,  (3.4)

where

D′_{k} = d_{k}, 1 ≤ k ≤ n − 4,

D′_{n−3} = ∥\begin{matrix}d_{n−3} \\ 0 \end{matrix}∥ , D′_{n−2} = ∥\begin{matrix}d_{n−2} & 0 \\ 0 & φ′ \end{matrix}∥ , D′_{n−1} = ∥d_{n−1} φf∥ .

and f : η⟶ξ_{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 D_{n−1} is an epimorphism over X. Hence we can split the bundle complex (3.4) into two direct summands:

¯0⟶ξ_{1}⟶ ⋯ ⟶ξ_{n−3} ⟶^{D′_{n−3}}ξ_{n−2} ⊕ η ⟶^{D′_{n−2}}Ker (D′_{n−1})⟶¯0,

and

¯0⟶ (Ker (D′_{n−2}))^{⊥} ⟶^{D′_{n−1}}ξ_{n}⟶¯0,

the latter being a trivial elementary complex. Further, we can apply induction on the length of the bundle complex.

Lemma 8 The mapping β is a monomorphism.

The proof is similar to the proof of Lemma 7.

Thus we have shown

Theorem 26 The natural mapping

β : K^{2}(X, Y)⟶K^{n}(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 (ξ_{1}, d, ξ_{2}) to X equals [ξ_{1}]−[ξ_{2}] and the restriction of the bundle complex (3.2) to the space X equals the alternating sum

[ξ_{1}] − [ξ_{2}] + … + (−1)^{n+1}[ξ_{n}].

Examples

1. Let us describe the group K^{0}(S^{n}). We represent the sphere S^{n} as a quotient space of the disc D^{n} by its boundary ∂D^{n} = S^{n−1}, S^{n} = D^{n}/S^{n−1}. Therefore,

K^{0}(S^{n}) = K^{2}(D^{n}, ∂D^{n}).

Since every bundle over D^{n} is trivial, any element of the group K^{2}(D^{n}, ∂D^{n}) may be defined as the homotopy class of a continuous mapping

d : ∂D^{n} = S^{n−1}⟶GL(N, C).

Hence

K^{0}(S^{2}) = π_{1}(U(N)) = π_{1}(U(1)) = π_{1}(S^{1}) = Z.

Then

K(S^{2}) = Z ⊕ Z.

Similarly,

K^{0}(S^{3}) = π_{2}(U(N)) = π_{2}(U(1)) = 0.

Hence

K(S^{3}) = Z.

2. Let SX denote the suspension of the space X, that is, the quotient space of the cone CX by its base, SX = CX/X. Then

K^{0}(SX) = K^{2}(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 (ξ_{1}, d, ξ_{2}) for two complexes of type (3.2) and a homomorphism giving a commutative diagram

\begin{aligned}0 & ⟶ & ξ_{1} & ⟶^{d_{11}} & ξ_{2} & ⟶^{d_{12}} & … & ⟶^{d_{1,n−1}} & ξ_{n} & ⟶ & 0 \\ & & ↓d′_{1} & & ↓d′_{2} & & & & ↓d′_{n} \\ 0 & ⟶ & η_{1} & ⟶^{d_{21}} & η_{2} & ⟶^{d_{22}} & … & ⟶^{d_{2,n−1}} & η_{n} & ⟶ & 0 \end{aligned}  (3.5)

Assume that horizontal homomorphisms d_{ij} are exact over Y and vertical homomorphisms d′_{k} 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

0⟶ξ_{1}⟶^{D_{1}}(ξ_{2} ⊗ η_{1})⟶^{D_{2}}(ξ_{3} ⊗ η_{2})⟶ … ⟶(ξ_{n} ⊗ η_{n−1})⟶η_{n}⟶0, (3.6)

in which the horizontal homomorphisms are defined by the matrices

D_{k} = ∥\begin{matrix}d_{1k} & 0 \\ d′_{k} & d_{2,k−1} \end{matrix}∥ .

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

More generally, consider a diagram

0 ⟶ ξ_{11} ⟶^{d_{11}} ξ_{12} ⟶^{d_{12}} … ⟶^{d_{1,n−1}} ξ_{1n} ⟶ 0
↓d′_{11} ↓d′_{12} ↓d′_{1n}
0 ⟶ ξ_{21} ⟶^{d_{21}} ξ_{22} ⟶^{d_{22}} … ⟶^{d_{2,n−1}} ξ_{2n} ⟶ 0
↓  ↓  ↓
⋮  ⋮  ⋮
↓  ↓  ↓
0 ⟶ ξ_{m1} ⟶^{d_{m1}} ξ_{m2} ⟶^{d_{m2}} … ⟶^{d_{m,n−1}} ξ_{mn} ⟶ 0
↓  ↓  ↓
0  0  0

with anticommutative squares:

d′_{k,l+1}d_{kl} + d_{k+1,l}d′_{kl} = 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) × K^{0}(X′, Y′)⟶K^{0}(X × X′, X × Y′ ∪ Y × Y′).  (3.7)

Consider a complex on X,

0⟶ξ_{1}⟶^{d_{1}}ξ_{2}⟶^{d_{2}} … ⟶^{d_{n−1}}ξ_{n}⟶0  (3.8)

which is exact over Y and a complex on X′

0⟶η_{1}⟶^{d_{1}}η_{2}⟶^{d_{2}} … ⟶^{d_{n−1}}η_{n}⟶0  (3.9)

which is exact over Y′. Consider the tensor product of the two complexes (3.8) and (3.9)

0  0  0
↓  ↓  ↓
0 ⟶ ξ_{1} ⊗ η_{1} ⟶^{d_{1}⊗1} ξ_{2} ⊗ η_{1} ⟶^{d_{2}⊗1} … ⟶^{d_{n−1}⊗1} ξ_{n} ⊗ η_{1} ⟶ 0
↓−1 ⊗ d′_{1} ↓1 ⊗ d′_{1} ↓(−1)^{n} ⊗ d′_{1}
0 ⟶ ξ_{1} ⊗ η_{2} ⟶^{d_{1}⊗1} ξ_{2} ⊗ η_{2} ⟶^{d_{2}⊗1} … ⟶^{d_{n−1}⊗1} ξ_{n} ⊗ η_{2} ⟶ 0
↓  ↓  ↓
⋮  ⋮  ⋮
↓  ↓  ↓
0 ⟶ ξ_{1} ⊗ η_{m} ⟶^{d_{1}⊗1} ξ_{2} ⊗ η_{m} ⟶^{d_{2}⊗1} … ⟶^{d_{n−1}⊗1} ξ_{n} ⊗ η_{m} ⟶ 0
↓  ↓  ↓
0  0  0

(3.10)

It is easy to check that this diagram has rows exact over X × Y′ and columns exact over Y × 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 ×X we have the tensor product as a bilinear operation

K(X, Y) × K(X, Y′)⟶K(X, Y ∪ Y′).

5. Consider a complex n–dimensional vector bundle ξ over the base X and let p : E⟶X be the projection 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⟶Λ_{0}η⟶^{d_{0}}Λ_{1}η⟶^{d_{1}}Λ_{2}η⟶^{d_{2}} … ⟶^{d_{n−1}}Λ_{n}η⟶0,  (3.11)

where η = p^{∗}ξ is inverse image of the bundle ξ and the homomorphism

d_{k} : Λ_{k}η⟶Λ_{k+1}η

is defined as the exterior multiplication by the vector y ∈ E, y ∈ ξ_{x}, x = p(y). It is known that if the vector y ∈ ξ_{x} is nonzero, y ≠ 0, then the complex (3.11) is exact. Consider the subspace D(ξ) ⊂ E which consists of all vectors y ∈ E such that |y| ≤ 1. Then the subspace S(ξ) ⊂ D(ξ) of all unit vectors gives the pair (Dξ), S(ξ)) for which the complex (3.11) is exact on S(ξ). Denote the element defined by (3.11) by β(ξ) ∈ K^{0}(D(ξ), S(ξ)). Then one has homomorphism given by multiplication by the element β(ξ)

β : K(X)⟶K^{0} (D(ξ), S(ξ)) .  (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

η_{0} = ξ_{1} ⊕ ξ_{3} ⊕ … = ⨁_{k=1}^{m} ξ_{2k−1},

η_{1} = ξ_{2} ⊕ ξ_{4} ⊕ … = ⨁_{k=1}^{m} ξ_{2k},

D = ∥\begin{matrix}d_{1} & d_{2}^{∗} & 0 & ⋯ & 0 \\ 0 & d_{3} & d_{4}^{∗} & ⋯ & 0 \\ 0 & 0 & d_{5} & ⋯ & 0 \\ & & ⋯ \\ 0 & 0 & 0 & ⋯ & d_{2m−2}^{∗} \\ 0 & 0 & 0 & ⋯ & d_{2m−1} \end{matrix}∥ .

First of all notice that there is a sequence

⋯ ⟶η_{0}⟶^{D_{0}}η_{1}⟶^{D_{1}}η_{0}⟶^{D_{0}} ⋯,

where

D_{0} = ∥\begin{matrix}d_{1} & 0 & ⋯ & 0 \\ 0 & d_{3} & ⋯ & 0 \\ & & ⋯ \\ 0 & 0 & ⋯ & d_{2m−1} \end{matrix}∥ , D_{1} = ∥\begin{matrix}0 & 0 & ⋯ & 0 & 0 \\ d_{2} & 0 & ⋯ & 0 & 0 \\ & & ⋯ \\ 0 & 0 & ⋯ & d_{2m−2} & 0 \end{matrix}∥ .

This sequence is exact over Y. On the other hand, D = D_{0} + D_{1}^{∗}. Hence D is isomorphism over Y.

7. Using the previous construction we can define the tensor product of two triples

0⟶ξ_{1}⟶^{d}ξ_{2}⟶0, 0⟶η_{1}⟶^{d′}η_{2}⟶0

as a triple

0⟶(ξ_{1} ⊗ η_{1}) ⊕ (ξ_{2} ⊗ η_{2})⟶^{D}(ξ_{1} ⊗ η_{2}) ⊕ (ξ_{2} ⊗ η_{1})⟶0,

where

D = ∥\begin{matrix}d ⊗ 1 & −1 ⊗ d^{′∗} \\ 1 ⊗ d′ & d^{∗} ⊗ 1 \end{matrix}∥ .

3.2 BOTT PERIODICITY

Consider the Bott element β = β(1) ∈ K^{0} (S^{2}) which was defined in the section 3.1. The two natural projections from X×S^{2} to the factors X and S^{2} generate a natural homomorphisms from K(X) and K (S^{∗}) to the ring K (X × S^{∗}). Then since K (S^{2}) ∼ Z[β]/{β^{2} = 0}, there is a homomorphism

h : K(X)[t]/{t^{2} = 0}⟶K (X × S^{2}) ,  (3.13)

defined by the formula

h(xt + y) = xβ + y.

Theorem 27 (Bott periodicity) The homomorphism (3.13) is an isomorphism.

Proof.

We need to construct an inverse homomorphism to (3.13). Consider the pair (X × S^{2}, X × {s_{0}}), where s_{0} is a fixed point. From the exact sequence in K–theory:

K (S (X × S^{2})) ⟶^{Si^{∗}}K(SX)⟶^{δ}K (X × S^{2}, X × {s_{0}}) ⟶^{j^{∗}}K (X × S^{2}) ⟶^{i^{∗}}K(X)

we see that δ is trivial since i^{∗} and Si^{∗} are epimorphisms. Hence the homomorphism h sends the element ty ∈ K(X)[t]/{t^{2} = 0} to K (X × S^{2}, X × {s_{0}}). Denote the restriction of this homomorphism by h̃:

h̃ : K(X)⟶K (X × S^{2}, X × {s_{0}}) .  (3.14)

h̃(y) = βy, y ∈ K(X).

The pair (X × S^{2}, X × {s_{0}}) is equivalent to the pair (X × D^{2}, X × S^{1}). Hence each element of the group K^{0} (X × D^{2}, X × S^{1}) is represented by a triple (ξ_{1}, d, ξ_{2}) where d is an isomorphism over X × S^{1}. Moreover, we can assume that both vector bundles ξ_{1} and ξ_{2} are trivial. Thus the isomorphism d is a continuous function

d : X × S^{1}⟶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 × S^{1}⟶GL(C, n).

Thus we have a continuous mapping

d(x, z) ∈ U(n), x ∈ X, z ∈ S^{1} ⊂ C, |z| = 1.

The first step

We produce a smooth ε–approximation of the function d(x, z), for example, by the formula

d_{1}(x, z) = \frac{q}{2ε} ∫_{φ−ε}^{φ+ε}d(x, e^{iφ})dφ.

The second step

Consider the Fourier series:

d_{1}(x, z) = ∑_{k=−∞}^{∞} a_{k}(x)z^{k},  (3.15)

where

a_{k}(x) = \frac{1}{2π} ∫_{0}^{2π}d_{1}(x, e^{−ikφ})dφ.

The series (3.15) converges uniformly with respect to the variables x and z and equipotentially in the class of continuous functions d_{1}(x, z) ∈ U(n).

The third step

Restrict to a finite part of the Fourier series:

f_{N}(x, z) = ∑_{k=−N}^{N} a_{k}(x)z^{k}.

The number N can be chosen sufficiently large so that f_{N}(x, z) ∈ GL(C, n).

The fourth step

Put

p_{N}(x, z) = z^{N}f_{N}(x, z).

The fifth step

Let

p_{N}(x, z) = ∑_{k=0}^{2N} b_{k}(x)z^{k} ∈ GL(C, n).

Put

L_{N}(p)(x, z) = ∥\begin{matrix}b_{0}(x) & b_{1}(x) & … & b_{N−1}(x) & b_{N}(x) \\ −z & 1 & … & 0 & 0 \\ 0 & −z & … & 0 & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & 0 & … & −z & 1 \end{matrix}∥ ∈ GL(C, nN). (3.16)

The sixth step

The function (3.16) is linear with respect to the variable z,

L_{N}(p)(x, z) = A(x)z + B(x) ∈ GL(C, nN).

Hence there is a projection

Q(x) : C^{nN}⟶C^{nN}

which commutes with L_{N}(p)(x, z) and satisfies the following condition: the space C^{nN} is split into two summands

C^{nN} = V_{+}(x) ⊕ V_{−}(x)

such that the operator L_{N}(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 ×C^{nN}. Put

ν_{N}(d) = ([V_{+}(x)] − \overline{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 n, negative or positive. First of all notice that the direct sum ⨁_{n≤0} K^{n}(X, Y)

can be given a ring structure. In fact,

K^{−n}(X, Y) = K^{0} (X × D^{n}, ((Y × D^{n}) ∪ (X × 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) × K^{−m}(X′, Y′) =

= K^{0} (X × D^{n}, ((Y × D^{n}) ∪ (X × S^{n−1}))) ×
×K^{0} (X′ × D^{m}, ((Y′ × D^{m}) ∪ (X′ × S^{m−1}))) ⟶
⟶ K^{0}(X × X′ × D^{n} × D^{m}, (X × Y′ × D^{n} × D^{m}) ∪
∪(X × X′ × D^{n} × S^{m−1}) ∪
∪(Y × X′ × D^{n} × D^{m}) ∪ (X × X′ × S^{n−1} × D^{m})) =
= K^{0}(X × X′ × D^{n} × D^{m}, (((X × Y′) ∪ (Y × X′)) × D^{n} × D^{m}) ∪

∪(X × X′ × D^{n} × S^{m−1}) ∪ (X × X′ × S^{n−1} × D^{m})) =
= K^{0}(X × X′ × D^{n+m},

((Y × X′) ∪ (X × Y′) × D^{n+m}) ∪ (X × X′ × S^{n+m−1})) =

= K^{−(n+m)}(X × X′, (X × Y′) ∪ (Y × X′)).

In particular, when X = X′ and Y = Y′ it gives the usual inner multiplication

K^{−n}(X, Y) × K^{−m}(X, Y)⟶K^{−(n+m)}(X, Y).

When X′ = S^{0}, Y′ = {s_{0}}, where S^{0} is a 0–dimensional sphere, we have the pairing

K^{−n}(X, Y) × K^{−m}(S^{0}, s_{0})⟶
⟶K^{−(n+m)}(X × S^{0}, (X × s_{0}) ∪ (Y × S^{0)}) =
= K^{−(n+m)}(X, Y).

The Bott element β ∈ K^{0}(S^{2}, s_{0}) = K^{−2}(S^{0}, s_{0}) = Z generates a homomorphism h̃:

K^{−n}(X, Y)⟶^{⊗β}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, x_{0})⟶K^{0}(Y, x_{0})⟶
⟶K^{1}(X, Y)⟶K^{1}(X, x_{0})⟶K^{1}(Y, x_{0})⟶
⟶K^{0}(X, Y)⟶  …  (3.19)

Let

K^{∗}(X, Y) = K^{0}(X, Y) ⊕ K^{1}(X, Y).

Then the sequence (3.19) can be written as

…  ⟶K^{∗}(X, Y)⟶^{j^{∗}}K^{∗}(X, x_{0})⟶^{i^{∗}}K^{∗}(Y, x_{0})⟶^{∂}

⟶^{∂}K^{∗}(X, Y)⟶^{j^{∗}}K^{∗}(X, x_{0})⟶^{i^{∗}}K^{∗}(Y, x_{0})⟶^{∂}

⟶^{∂}K^{∗}(X, Y)⟶ …

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 Z_{2}–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 cellular pair (X, Y).

Proof. 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^{∗}(D^{n}, S^{n−1}) ⊗ Q⟶H^{∗∗}(D^{n}, S^{n−1}; Q)

is an isomorphism. Let n = 2. Then the

K^{0}(D^{2}, S^{1}) ∼ Z

with generator

β = [ξ] − 1,

where ξ is the Hopf bundle. Then

ch β = ch ξ − 1 = e^{c_{1}} − 1,

where

c_{1} ∈ H^{2}(S^{2}, s_{0}; Q)

is the integer generator. Hence the homomorphism

ch : K^{0}(D^{2}, S^{1}) ⊗ Q⟶H^{2}(D^{2}, S^{1}; Q)

is an isomorphism. For the odd case, one has

K^{1}(D^{2}, S^{1}) = Hodd(D^{2}, S^{1}) = 0.

Thus we have proved initial step. This finishes the proof of theorem.

Corollary 5 For any element x ∈ H^{2k}(X) there is a bundle ξ and a number λ such that

ch ξ = dim ξ + λx + terms of higher dimension.

3.4 LINEAR REPRESENTATIONS AND BUNDLES

Consider a linear representation of a group G on a (complex) n–dimensional vector space

ρ : G⟶GL(n, C).

If ξ is a principal G–bundle over a base space X with transition functions

φ_{αβ} : U_{αβ}⟶G

then using the new transition functions

ψ_{αβ} = ρ ∘ φ_{αβ} : U_{αβ}⟶GL(n, R)

we obtains a new vector bundle ξ^{ρ}.

We apply this construction to the universal principal G–bundle ξ_{G} over the classifying space B_{G}. 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(BG).  (3.20)

Interest in the homomorphism (3.20) comes from the fact that it has applications to a number of different problems. The virtual bundle b(ρ) = [ξ_{G}^{ρ}] is an invariant of the virtual representation ρ. This bundle can be used to interpret algebraic properties of the representation ρ 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 α 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).

Consider some simple examples.

The group G = S^{1}.

Each linear complex representation of the group S^{1} = U(1) can be split into a direct sum of one–dimensional representations which in turn are described by the following formula

ρ_{k}(z) = z^{k},

where z ∈ G = S^{1} is a complex number with unit modulus. So the irreducible representations are parametrized by the integers. It is clear that

ρ_{k} ⊗ ρ_{s} = ρ_{k+s}.

Hence the ring R(S^{1}) is generated by two elements ρ_{1} and ρ_{−1} with the unique relation ρ_{1}ρ_{−1} = 1. Thus,

R(S^{1}) = Z[ρ_{1}, ρ_{−1}]/{ρ_{1}ρ_{−1} = 1},

the so called ring of Laurent polynomials.

On the other hand, the classifying space BG is homeomorphic to the direct limit

BG = lim_{n⟶∞} CP^{n} = CP^{∞}.

The representation ρ_{1} is the identity representation. Hence

b(ρ_{1}) = ξ_{1}

and

b(ρ_{k}) = \underbrace{ξ_{1} ⊗ … ⊗ ξ_{1}}_{k times}

for k ≥ 0. Notice that the representation ρ_{−1} is the complex conjugate of ρ_{1} since z^{−1} − z̄ for |z| = 1. Hence

b(ρ_{−k}) = \underbrace{ξ̄_{1} ⊗ … ⊗ ξ̄_{1}}_{k times}

Thus

Theorem 29 The image of the homomorphism

b : R(S^{1}) ⊗ Q⟶K(BS^{1}) ⊗ Q,

where K(BS^{1}) ⊗ Q is the completion with respect to the topology generated by the ideal of zero dimensional virtual bundles.

Proof.

Consider the homomorphism

R(S^{1})⟶^{b}K(BS^{1})\underrightarrow{ch}H^{∗∗}(BS^{1}; Q)

After tensoring with Q we have

R(S^{1}) ⊗ Q⟶^{b}K(BS^{1}) ⊗ Q\underrightarrow{ch}H^{∗∗}(BS^{1}; Q)  (3.21)

The group K(BS^{1}) is defined to be the inverse limit

K(BS^{1}) = lim_{⟵} K(CP^{n}).

Since

H^{∗∗}(BS^{1}; Q) = lim_{⟵} H^{∗∗}(CP^{n}; Q),

using theorem 28 we have that the homomorphism ch in (3.21) is a topological isomorphism. On the other hand, for u = ρ_{1} − 1,

ch b(u) = a + \frac{a^{2}}{2} + ⋯ + \frac{a^{k}}{k!} + · · · .

Hence this element can serve as a generator in the ring of formal series H^{∗∗}(BS^{1}; Q).

Consider the ideal of zero–dimensional virtual representations in R(G). The corresponding topology generates the completion R̂(G) of the ring R(G). It is easy to prove that homomorphism

b : R̂(S^{1}) ⊗ Q⟶K(BS^{1}) ⊗ Q

is an isomorphism.

The group G = Z.

The classifying space BZ can be taken to be homeomorphic to the circle S^{1} and K(S^{1}) = Z. Hence linear representations of the group Z are not distinguished by the homomorphism b.

The group G = Z × Z.

In this case the classifying space can be taken to be homeomorphic to T^{2} = S^{1} × S^{1}. The group K(T^{2}) can be calculated using the exact sequence of the pair (T^{2}, S^{1} ∨ S^{1}). Finally,

K^{0}(T^{2}, s_{0}) = Z,
K^{1}(T^{2}, s_{0}) = Z ⊕ Z.

Each irreducible representation ρ of the commutative group Z × Z is one– dimensional and homotopic to the trivial representation. Thus b(ρ) is trivial.

The group G = Z^{n} and the case of a family of representations.

The case of G = Z^{n} is similar to the case of G = Z^{2}. But one can obtain a nontrivial result by considering a continuous family of representations. Namely, let ρ_{y}, y ∈ Y be a continuous family of representations. Let ξ_{G} be the universal principal G–bundle over X = BG with the transition functions φ_{αβ}. We can consider the same transition functions over the base X × Y by taking them independent of the argument y ∈ Y. The compositions ψ_{αβ} = ρ_{y} ∘ φ_{αβ} depend on both arguments x ∈ X, y ∈ Y. The latter defines a vector bundle ξ_{G}^{ρ} over the base X × Y. Then in spite of the fact that the restriction of vector bundle ξ_{G}^{ρ_{y_{0}}} over each X × y_{0} may be trivial the whole vector bundle ξ_{G}^{ρ} over X × Y may be nontrivial.

For example, consider as the space Y, the character group of the group G,

Y = G^{∗},

and consider as representation ρ_{y} the corresponding character of the group G,

ρ_{y} : G⟶S^{1} = U(1).

When G = Z^{n} = ⨁ G_{j}, G_{j} = Z,

G^{∗} = T^{n} and BG = T^{n}.

Then

X × Y = BG × G^{∗} = T^{n} × T^{n}.

Theorem 30 Let ρ be the family of all characters of the group G = Z^{n}. Then

c_{1}(ξ_{G}^{ρ}) = x_{1}y_{1} + … + x_{n}y_{n}

where x_{k} ∈ H^{1}(BG; Z), y_{k} ∈ H^{1}(G^{∗}; Z) are corresponding generators.

Proof.

The two dimensional cohomology of the space X ×Y is generated by monomials of degree 2 in the variables x_{k}, y_{k}. Hence

c_{1}(ξ_{G}^{ρ}) = ∑ λ_{ij}x_{i}y_{j} + ∑ μ_{ij}x_{i}x_{j} + ∑ ν_{ij}y_{i}y_{j}

for some integers λ_{ij}, μ_{ij}, ν_{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 λ_{ij} can be interpreted as characteristic classes of restrictions to the subspaces BG_{i} × G_{j}^{∗} Therefore, it is clear that λ_{ij} = 0 when i ≠ j. Hence the proof reduces to the case where G = Z. In this case X = S^{1} and Y = S^{1}. There is an atlas consisting of two charts: the upper and lower semicircles, U_{1} and U_{2}. The intersection U_{12} = U_{1} ∩ U_{2} consists of two points: {1, −1}. The principal bundle ξ_{G} has one transition function φ_{12} where

φ_{12}(1) = 0 ∈ Z, φ_{12}(−1) = 1 ∈ Z.

Let y ∈ Y = S^{1} be a character of the group G = Z. Then the corresponding transition function ψ_{12} = ρ_{y} ∘ φ_{12} is defined by the formula

ψ_{12}(1, y) = 1, ψ_{12}(−1, y) = y.

It is easy to see that the bundle ξ_{G}^{ρ} is trivial over the wedge of a parallel and a meridian S^{1} ∨ S^{1} and on the quotient space T^{2}/(S^{1} ∨ S^{1}) = S^{2} it is isomorphic to the Hopf bundle. Hence

c_{1}(ξ_{G}^{ρ}) = xy ∈ H^{2}(T^{2}; Z).

3.5 EQUIVARIANT BUNDLES

Let G be a compact Lie group. A G–space 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(gx) = gf(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 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 projection π onto the orbit space Y = X/G gives a locally trivial bundle.

Theorem 31 The family of G-equivariant vector bundles over a space X with a free action of a group G is in one–to–one correspondence with the family of vector bundles over the orbit space Y = X/G. This correspondence is given by the projection π:

η⟶π^{∗}η

with the natural action of the group G on the total space of vector bundle ψ.

Proof.

In fact, in the total space of the bundle π^{∗}(η), there is natural action of the group G. By definition the charts of the bundle π^{∗}(η) are inverse images of the charts U on Y. Then

π^{−1}(U) ∼ U × G

with a left action along the second summand. The corresponding trivialization of the vector bundle π^{∗}(η) is homeomorphic to U × G × F where F is the fiber. Then, by definition, the action of the group G on the total space is the left action along the second summand.

Conversely, let ξ⟶X be arbitrary G-equivariant vector bundle with a free G action. Consider a small chart U ⊂ Y and put V = π^{−1}(U) = U × G. For U sufficiently small that the restriction of ξ to U × g_{0} is trivial

ξ_{U×g_{0}} ∼ U × g_{0} × F.

Then the restriction of the bundle ξ_{U×G} also is trivial:

U × g_{0} × G × F ⟶ ξ_{U×G},
(u, g_{0}, g, f)⟶g(u, g_{0}, 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.

Theorem 32 Suppose the group G acts trivially on the base X. Then for each G–equivariant vector bundle x there exist irreducible representations of the group G in vector spaces V_{i} and vector bundles ξ_{i} such that

ξ = ⨁_{i}(ξ_{i} ⊗ V_{i})

with the natural action of the group G on the second factors.

Proof.

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 ξ_{x} gives a direct sum

ξ_{x} = ⊕_{i}(ξ_{x,i} ⊗ V_{i}).  (3.22)

Assume that locally the presentation (3.22) does not depend on the point x, that is, in some small neighborhood U_{α} we have

ξ_{|U_{α}} = ξ_{x_{0}} × U_{α} = [⊕_{i}(ξ_{x_{0},i} ⊗ V_{i})] × U_{α}.  (3.23)

Then using Schur’s lemma the equivariant transition function ψ_{αβ} decomposes into a direct sum

ψ_{αβ} = ⊕_{i}(ψ_{αβ,i} ⊗ V_{i}).

The functions ψ_{αβ,i} define transition functions for some vector bundles ξ_{i}. Thus

ξ = ⊕_{i}(ξ_{i} ⊗ V_{i}).

Our problem is to prove the decomposition (3.23). We give here an elegant proof due to M.F. Atiyah. Consider a G–equivariant 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 projection 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 projection since

P (P (v)) = ∫_{G} h(P (v))dμ(h) = ∫ ∫_{G×G} hg(v)dμ(h)dμ(g) =

= ∫ ∫_{G×G} 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 (ξ_{1}, ξ_{2}) (for the definition see section 1.3) . If both bundles ξ_{1} and ξ_{2} are G–equivariant then the bundle HOM (ξ_{1}, ξ_{2}) also is G–equivariant with the action defined as follows:

φ⟶gφg^{−1}.

Then the fixed point set HOM _{G}(ξ_{1}, ξ_{2}) gives a locally trivial bundle. Now we can construct the bundles ξ_{i} mentioned above. Let V_{i} be an irreducible G– vector space and \underline{V_{i}} be the corresponding trivial G–equivariant vector bundle. Put

ξ_{i} = HOM _{G}(\underline{V_{i}}, ξ).

Then there is an isomorphism

α : ⊕_{i} (\underline{V_{i}} ⊗ HOM _{G}(\underline{V_{i}}, ξ)) ⟶ξ

defined on each summand by formula

α(x, φ) = φ(x), x ∈ V_{i}.  (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.

Consider the group G = Z_{2} and a complex vector bundle ξ over the G–space X. This means that the group G acts on the space X. Let E be the total space of the bundle ξ and let

p : E⟶X

be the projection in the definition of the vector bundle ξ. 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 τ ∈ G = Z_{2} is the generator then

τ(λx) = λ̄τ(x), λ ∈ C, x ∈ E.  (3.25)

A vector bundle ξ with the action of the group G satisfying the condition (3.25) is called a KR–bundle . The operator τ is called the anticomplex involution. The corresponding Grothendieck group of KR–bundles is denoted by KR(X).

Below we describe some of the relations with classical real and complex vector bundles.

Proposition 10 Suppose that the G–space X has the form X = Y ×Z_{2} and the involution τ transposes the second factor. Then the group KR(X) is naturally isomorphic to the group K_{U}(Y) and this isomorphism coincides with restriction of a vector bundle to the summand Y ×{1}, 1 ∈ G = Z_{2}, ignoring the involution τ.

Proof.

The complex bundle ξ over X consists of a union of two bundles: ξ_{1} — restriction of ξ on Y × {1} and ξ_{τ} — restriction of ξ on Y × {τ}. Since τ interchanges the two summands — Y × {1} and Y × {τ}, the involution τ gives an isomorphism

τ : ξ_{1}⟶ξ_{τ}

which is anticomplex. Hence

ξ_{τ} = τ^{∗}(\overline{ξ_{1}}).

Hence given a complex vector bundle ξ_{1} over Y × {1}, then we can take the same vector bundle over Y × {τ} with the conjugate complex structure. Then the identity involution τ is anticomplex and induces a KR– bundle over X.

Proposition 11 Suppose the involution τ on X is trivial. Then

KR(X) ≈ K_{O}(X).  (3.26)

The isomorphism (3.26) associates to any KR–bundle the fixed points of the involution τ.

Proposition 12 The operation of forgetting the involution induces a homomorphism

KR(X)⟶K_{U}(X)

and when the involution is trivial on the base X this homomorphism coincides with complexification

c : K_{O}(X)⟶K_{U}(X).

Similar to the realification of vector bundles, we can construct, for any base X with involution τ, a homomorphism

K_{U}(X)⟶KR(X)  (3.27)

which coincides with realification

r : K_{U}(X)⟶K_{O}(X),

when the involution on the base X is trivial. The homomorphism (3.27) associates to any complex vector bundle ξ over X the bundle η = ξ ⊕ τ^{∗}(ξ̄) with an involution which transposes the summands.

As for complex vector bundles the operation of tensor product can be defined for KR–bundles making KR(X) into a ring. The extension of the groups KR(X) to a cohomology theory is more interesting. An equivariant analogue of suspension for Z_{2}–spaces must be defined.

Definition 8 Denote by D^{p,q} the disc D^{p+q} with involution τ which changes signs in the first p coordinates and denote by S^{p,q} the boundary of D^{p,q}. Then the suspension of pair (X, Y) with involution is given by the pair

Σ^{p,q}(X, Y) = (X × D^{p,q}, (X × S^{p,q}) ∪ (Y × D^{p,q}))

Hence in a natural way we can define a bigraded KR–theory

KR^{−p,−q}(X, Y) = KR (Σ^{p,q}(X, Y)) ,  (3.28)

where KR(X, Y) is defined as the kernel of the mapping

KR(X/Y)⟶KR(pt).

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

… ⟶KR^{p,q}(X, Y)⟶KR^{p,q}(X, Z)⟶KR^{p,q}(Y, Z)⟶ …

… ⟶KR^{p,q+1}(X, Y)⟶KR^{p,q+1}(X, Z)⟶KR^{p,q+1}(Y, Z)⟶ = ts

… ⟶KR^{p,0}(X, Y)⟶KR^{p,0}(X, Z)⟶KR^{p,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) ⊗ KR^{p′,q′}(X, Y′)⟶KR^{p+p′,q+q′}(X, Y ∪ Y′).

Moreover, the proof of Bott periodicity can be extended word by word to KR– theory:

Theorem 33 There is an element β ∈ KR(D^{1,1}, S^{1,1}) = KR^{−1,−1}(pt) such that the homomorphism given by multiplication by β

β : KR^{p,q}(X, Y)⟶KR^{p−1,q−1}(X.Y)  (3.29)

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 × S^{1,0} × D^{0,1} ≈ X × S^{1,0} × D^{1,0},
2)X × S^{2,0} × D^{0,2} ≈ X × S^{2,0} × D^{2,0},
3)X × S^{4,0} × D^{0,4} ≈ X × S^{4,0} × D^{4,0}.

In all three cases, the homeomorphism

μ : X × S^{p,0} × D^{0,p}⟶X × S^{p,0} × D^{p,0}

is defined by the formula

μ(x, s, u) = (x, s, su), x ∈ X, s ∈ S^{p,0}, u ∈ D^{0,p},

where we identify S^{p,0}, D^{p,0} and D^{0,p}, respectively, with the unit sphere or unit disk in the field of real, complex or quaternionic numbers, respectively.

Thus we have the following isomorphisms:

KR^{p,q}(X × S^{1,0}) ≈ KR^{p−1,q+1}(X × S^{1,0}),  (3.30)
KR^{p,q}(X × S^{2,0}) ≈ KR^{p−2,q+2}(X × S^{2,0}),  (3.31)
KR^{p,q}(X × S^{4,0}) ≈ KR^{p−4,q+4}(X × S^{4,0}).  (3.32)

In the case of (3.30) using the Bott periodicity (3.29)

KR^{p,q}(X × S^{1,0}) ≈ KR^{p,q+2}(X × 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 α ∈ KR^{0,−8}(pt) = KR(D^{0,8}, S^{0,8}) such that the homomorphism given by multiplication by α

α : KR^{p,q}(X, Y)⟶KR^{p,q−8}(X, Y)

is an isomorphism.

In the case of the trivial action of an involution on the base we get the classical 8–periodicity 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, IJ + JI = 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 ξ and satisfies

J^{4} = 1, IJ + JI = 0.  (3.33)

Then the vector bundle ξ can be split into two summands ξ = ξ_{1} ⊕ ξ_{2} both invariant under action of J, that is, J = J_{1} ⊕ J_{2} such that

J_{1}^{2} = 1, J_{2}^{2} = −1.  (3.34)

Hence the vector bundle ξ_{1} is the complexification of a real vector bundle and ξ_{2} is quaternionic vector bundle.

Consider a similar situation over a base X with an involution τ such that the operator (3.33) commutes with τ. Such vector bundle will be called a KRS– bundle .

Lemma 10 A KRS–bundle ξ is split into an equivariant direct sum ξ = ξ_{1}⊕ξ_{2} such that J^{2} = 1 on ξ_{1} and J^{2} = −1 on ξ_{2}.

Lemma 10 shows that the Grothendieck group KRS(X) generated by KRS– bundles has a Z_{2}–grading, that is,

KRS(X) = KRS_{0}(X) ⊕ KRS_{1}(X).

It is clear that KRS_{0}(X) = KR(X). In the case when the involution τ acts trivially, KRS_{1}(X) = K_{Q}(X), that is,

KRS(X) = K_{O}(X) ⊕ K_{Q}(X)

where K_{Q}(X) is the group generated by quaternionic bundles.

The Z_{2}–graded group KRS(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 τ on the base and action of the operator J on the total space. Moreover, this action is compatible with the grading, that is, if ξ_{1} ∈ KRS_{i}(X), ξ_{2} ∈ KRS_{j}(X) then ξ_{1} ⊗ ξ_{2} ∈ KRS_{i+j}(X). As in KR–theory, the groups KRS^{p,q}(X, Y) = KRS_{0}^{p,q}(X, Y) ⊕ KRS_{1}^{p,q}(X, Y) may be defined:

KRS^{−p,−q}(X, Y) = KRS (D^{p,q} × X, (S^{p,q} × X) ∪ (D^{p,q} × Y)) ,

where

KRS(X, Y) = Ker (KRS(X/Y)⟶KRS(pt)) .

Bott periodicity generalizes: if β ∈ KR (D^{1,1}, S^{1,1}) = KRS_{0}^{−1,−1}(pt) is the Bott element then Theorem 35 The homomorphism

β : KRS^{p,q}(X, Y)⟶KRS^{p−1,q−1}(X, Y)

given by multiplication by the element β is an isomorphism. As in Theorem 34 we have the following Theorem 36 There is an element

γ ∈ KRS_{1}^{0,−4}(pt) = KRS_{1}(D^{0,4}, S^{0,4})

such that the homomorphism of given by multiplication by γ

γ : KRS^{p,q}(X, Y)⟶KRS^{p,q−4}(X, Y)

or

γ : KRS_{0}^{p,q}(X, Y)⟶KRS_{1}^{p,q−4}(X, Y)

or

γ : KRS_{1}^{p,q}(X, Y)⟶KRS_{0}^{p,q−4}(X, Y)

are isomorphisms. Moreover,

γ^{2} = α ∈ KRS^{0,−8}(pt) = KR^{0,−8}(pt) = K_{O}^{−8}(pt).

Theorem 36 says that γ ∈ K_{Q}^{−4}(pt). In particular, we have the following isomorphisms:

K_{O}^{q}(X) ≈ K_{Q}^{q−4}(X)
K_{Q}^{q}(X) ≈ K_{O}^{q−4}(X).

K_{Q}  Z uγ^{2}  0 0  Z_{2} h^{2}γ Z_{2} hγ  Z γ  0  0  0  Z u Z α = γ^{2} 0  0  0  Z uγ  0  Z_{2} h^{2}  Z_{2} h  Z u^{2} = 4 K_{O} q  -8  -7  -6  -5  -4  -3  -2  -1  0

Figure 3.1

A list of the groups K_{O} and K_{Q} are given in the figure 3.1.

168 Chapter 3

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

X_{0} ⊂ X_{1} ⊂ … ⊂ X_{N} = X  (4.1)

be a increasing filtration of the space X. Put

D = ⨁_{p,q} D^{p,q} = ⨁_{p,q} K^{p+q}(X_{p})
E = ⨁_{p,q} E^{p,q} = ⨁_{p,q} K^{p+q}(X_{p}, X_{p−1}).  (4.2)

Consider the exact sequence of a pair (X_{p}, X_{p−1}):

…  ⟶K^{p−1+q}(X_{p−1})⟶^{∂}K^{p+q}(X_{p}, X_{p−1})⟶^{j^{∗}}
⟶K^{p+q}(X_{p})⟶^{i^{∗}}K^{p+q}(X_{p−1})⟶ …  (4.3)

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⟶^{j^{∗}}D⟶^{i^{∗}}D⟶^{∂}E⟶ …,  (4.4)

where the bigradings of the homomorphisms ∂, j^{∗}, i^{∗} are as follows:

deg i^{∗} = (−1, 1)
deg j^{∗} = (0, 0)
deg ∂ = (1, 0).

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

[figure: D; i^{∗}; D; j^{∗}; ∂; E]

(4.5)

Put

d = ∂j^{∗} : E⟶E.

Then the bigrading of d is deg d = (1, 0) and it is clear that d^{2} = 0. Put

D_{1} = D,
E_{1} = E,
i_{1} = i^{∗},
j_{1} = j^{∗},
∂_{1} = ∂,
d_{1} = d.

Now put

D_{2} = i_{1}(D_{1}) ⊂ D_{1},
E_{2} = H(E_{1}, d_{1}).

The calculation methods in K–theory 171

The grading of D_{2} is inherited from the grading as an image, that is,

D_{2} = ⨁ D_{2}^{p,q},
D_{2}^{p,q} = i_{1}(D_{1}^{p+1,q−1}).

The grading of E_{2} is inherited from E_{1}. Then we put

i_{2} = i_{1}|_{D_{2}} : D_{2}⟶D_{2},
∂_{2} = ∂_{1}i_{1}^{−1},
j_{2} = j_{1}.

It is clear that i_{2}, j_{2} and ∂_{2} are well defined. In fact, if x ∈ D_{2} then i_{1}(x) ∈ D_{2}. If x ∈ D_{1} then x = i_{1}(y) and

∂_{2}(x) = [∂_{1}(y)] ∈ H(E_{1}, d_{1}).

The latter inclusion follows from the identity

d_{1}∂_{1}(y) = ∂_{1}j_{1}∂_{1}(y) = 0.

If i_{1}(y) = 0, the exactness of the triangle (4.5) gives y = j_{1}(x) and then

∂_{2}(x) = [∂_{1}j_{1}(z)] = [d_{1}(z)] = 0.

Hence ∂_{2} is well defined. Finally, if x ∈ E_{1}, d_{1}(x) = 0 then ∂_{1}j_{1}(x) = 0. Hence j_{1}(x) = i_{1}(z) ∈ D_{2}. If x = d_{1}(y), x = ∂_{1}j_{1}(z) then j_{1}(x) = j_{1}∂_{1}j_{1}(z) = 0 Hence j_{2} is well defined. Thus there is a new exact triangle

[figure: D_{2}; i_{2}^{∗}; D_{2}; j_{2}^{∗}; ∂_{2}; E_{2}]

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

Repeating the process one can construct a series of exact triangles

[figure: D_{n}; i_{n}^{∗}; D_{n}; j_{n}^{∗}; ∂_{n}; E_{n}]

The bigradings of homomorphisms i_{n}, j_{n}, ∂_{n} and d_{n} = ∂_{n}j_{n} are as follows:

deg i_{n} = (−1, 1),
deg j_{n} = (0, 0),
deg ∂_{n} = (n, −n + 1),
deg d_{n} = (n, −n + 1).

The sequence

(E_{n}, d_{n})  (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). Proof. According to the definition (4.2), E_{1}^{p,q} = 0 for p > N. Hence for n > N,

d_{n} = 0,

that is,

E_{n}^{p,q} = E_{n+1}^{p,q} = … = E_{∞}^{p,q}.

By definition, we have D_{n}^{p,q} = Im (K^{p+q}(X_{p+n})⟶K^{p+q}(X_{p})) .

Hence for n > N,

D_{n}^{p,q} = Im (K^{p+q}(X)⟶K^{p+q}(X_{p})) .

Hence the homomorphism i_{n} is an epimorphism. Thus we have the exact sequence

0⟶E_{n}^{p,q}⟶^{j_{n}}D_{n}^{p,q}⟶^{i_{n}}D_{n}^{p−1,q+1}⟶0.

Hence

E_{n}^{p,q} = Ker i_{n} = = Ker (K^{p+q}(X)⟶K^{p+q}(X_{p−1})) /Ker (K^{p+q}(X)⟶K^{p+q}(X_{p}))

Let p : Y ⟶X be a locally trivial bundle with fibre F. Then the fundamental group π_{1}(X, x_{0}) acts on the fiber F in the sense that there is natural homomorphism

π_{1}(X, x_{0})⟶[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 π_{1}(X, x_{0}) on the groups K^{∗}(F).

Theorem 38 Let p : Y ⟶X be a locally trivial bundle with fiber F and the trivial action of π_{1}(X, x_{0}) on K–groups of the fiber. Then the spectral sequence generated by filtration Y_{k} = p^{−1}([X]^{k}), where [X]^{k} is k–dimensional skeleton of X, converges to the groups associated to K^{∗}(Y) and the second term has the following form:

E_{2}^{p,q} = 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

E_{2}^{p,q} = \begin{cases}H^{p}(X; Z), & for even q \\ 0 & for odd q. \end{cases}  (4.8)

Proof.

The first term of the spectral sequence E_{1} is defined to be

E_{1}^{p,q} = K^{p+q}(Y_{p}, Y_{p−1}).

Since the locally trivial bundle is trivial over each cell, the pair (Y_{p}, Y_{p−1}) has the same K–groups as the union

⋃_{j}(σ_{j}^{p} × F, ∂σ_{j}^{p} × F),

that is,

E_{1}^{p,q} ≈ ⨁_{j} K^{p,q}(σ_{j}^{p} × F, ∂σ_{j}^{p} × F) = ⨁_{j} K^{q}(F).

Hence, we can identify the term E_{1} with the cochain group

E_{1}^{p,q} = C^{p} (X; K^{q}(F))  (4.9)

with coefficients in the group K_{q}(F). What we need to establish is that the differential d_{1} 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 E_{1} is isomorphic to the chain group with a local system of coefficients defined by the action of the fundamental group π_{1}(X, x_{0}) in the group K^{q}(F).

We say that the spectral sequence is multiplicative if all groups E_{s} = ⨁_{p,q} E_{s}^{p,q}

are bigraded rings, the differentials d_{s} are derivations, that is,

d_{s}(xy) = (d_{s}x)y + (−1)^{p+q}x(d_{s}y), x ∈ E_{s}^{p+q},

and the homology of d_{s} 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 = S^{n}. Then the E_{2} term of the spectral sequence is

E_{2}^{p,q} = H^{p} (S^{n}; K^{q}(∗)) = \begin{cases}Z & for p = 0, n and for even q, \\ 0 & in other cases. \end{cases}\} (see fig.4.1.)

[figure: Z; Z; Z; n; Z; q; p]

Figure 4.1

For a one point space all differentials are trivial. Hence

d_{s}|_{E_{s}^{o,q}} = 0.

Thus all differentials for a sphere are trivial and E_{2} = E_{∞}. Finally, we have for even n,

K^{0}(S^{n}) = Z ⊕ Z, K^{1}(S^{n}) = 0,

and for odd n,

K^{0}(S^{n}) = Z, K^{1}(S^{n}) = Z.

2. Let X = CP^{n}. Then

E_{2}^{∗∗} = H^{∗} (CP^{n}; K^{∗}(pt)) = Z[t]/{t^{n+1} = 0}.

Nontrivial terms E_{2}^{p,q} exist only when p + q is even. Hence all differentials are trivial and we have

E_{∞}^{∗∗} = Z[t]/{t^{n+1} = 0}.  (4.10)

Hence the additive structure of the group K(CP^{n}) is the same as in the term E_{∞}. To show that the multiplicative structure of the group K^{∗}(CP^{n}) coincides with (4.10), consider an element u ∈ K(CP^{n}) which corresponds to the element t ∈ E_{∞}^{2,0}. The element u should be trivial on the one dimensional skeleton of CP^{n}. Further, the element u^{k}, k ≤ n is trivial on the (2k − 1)–dimensional skeleton of CP^{n} and corresponds to the element t^{k} ∈ E_{∞}^{2k,0}. Consider the element u^{n+1} ∈ K(CP^{n}). This element is trivial on the (2n + 1)–dimensional skeleton, that is, u^{n+1} = 0. Hence

K^{∗}(CP^{n)} = Z[u]/{u^{n+1} = 0}.

It is easy to show that the element u can be chosen to be equal to [ξ] − 1 where ξ is the Hopf bundle.

3. Let us show that all differentials d_{s} 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 Z_{2}–graded cohomology with rational coefficients induced by the filtration with respect to skeletons of X:

ch : (E_{s}^{p,q}, d_{s})⟶(′E_{s}^{p,q},′d_{s}).

Since all differentials ′d_{s} are trivial and ch ⊗ QQ is an isomorphism, we have

d_{s} ⊗ Q = 0.

Thus the differential d_{s} has finite order on each group E_{s}^{p,q}.

For example, the differential

d_{2} : E_{2}^{p,q}⟶E_{s}^{p+2,q−1}

can be written as

d_{2} : H^{p} (X; K^{q}(∗)) ⟶H^{n+2} (X; K^{k−1}(∗)) .

Hence d_{2} = 0 and E_{2}^{p,q} = E_{3}^{p,q}. Thus the differential d_{3} has the form of a stable cohomology operation:

d_{3} : H^{p}(X; Z)⟶H^{p+3}(X; Z).

Therefore, d_{3} may take one of only two possible values: d_{3} = 0 or d_{3} = Sq^{3}. To check which possibility is realized it suffices to consider a special example for the space X where the differential d_{3} can be calculated.

4. As in cohomology theory we can obtain a formula for the K–groups of a Cartesian product X × Y. Assume firstly that the group K(Y) has no torsion. Let X_{1} ⊂ X be a subspace of X which differs from X in only one cell. Consider the exact sequences for pairs (X, X_{1}) and (X × Y, X_{1} × Y):

[figure: K^{∗}(X × Y, X_{1} × Y); K^{∗}(X × Y); K^{∗}(X_{1} × Y); K^{∗}(X, X_{1}) ⊗ K^{∗}(Y); K^{∗}(X) ⊗ K^{∗}(Y); K^{∗}(X_{1}) ⊗ K^{∗}(Y); f; g; h; …; …; …; …](4.12)

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 Y_{1} be a space such that K^{∗}(Y_{1}) has no torsion and suppose there is a map

f : Y ⟶Y_{1}

such that the homomorphism

f^{∗} : K^{∗}(Y_{1})⟶K^{∗}(Y)

is an epimorphism. Then the exact sequence for the pair (Y_{1}, Y) has the form of a free resolution

0⟶K^{∗}(Y_{1}, Y)⟶K^{∗}(Y_{1})⟶K^{∗}(Y)⟶0.

Consider the diagram

[figure: K^{∗}(Y_{1} × X, Y × X); K^{∗}(Y_{1} × X); K^{∗}(Y × X); K^{∗}(Y_{1}, Y) ⊗ K^{∗}(X); K^{∗}(Y_{1}) ⊗ K^{∗}(X); K^{∗}(Y) ⊗ K^{∗}(X); f; g; h; …; …; β′; α′; β; α; …; 0]

We will show that the homomorphism f is a monomorphism. If f(x) = 0 then there exists y such that α(y) = x. Then α′(g(y)) = 0. So there exists z such that g(y) = β′(z). Since h is an isomorphism, we have z = h(w) or g(y) = gβ(w). Since g is also an isomorphism, y = β(w). Hence x = α(w) = 0.

Now let us show that h(Ker β) = Ker β′. It is clear that h(Ker β) ⊂ Ker β′. If β′(x) = 0, x = h(y) then gβ(y) = β′h(y) = 0, that is, β(y) = 0 or y ∈ Ker β. Hence there is an exact sequence

0⟶K^{∗}(Y) ⊗ K^{∗}(X)⟶K^{∗}(Y × X)⟶Ker β⟶0.

The group Ker β is denoted by tor (K^{∗}(Y), K^{∗}(X)). So there is a Künneth formula

0⟶K^{∗}(Y) ⊗ K^{∗}(X)⟶K^{∗}(Y × X)⟶Tor (K^{∗}(Y), K^{∗}(X)) ⟶0.

To complete the proof it remains for us to construct the space Y_{1}. The space Y_{1} 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 bundles which split into a sum of one dimensional bundles then the same relation holds for all vector bundles.

This principle is based on the following theorem

Theorem 40 For any vector bundle ξ over the base X there is a continuous mapping f : Y ⟶X such that f^{∗}(ξ) splits into a sum of one dimensional bundles and the homomorphism

f^{∗} : K^{∗}(X)⟶K^{∗}(Y)

is a monomorphism.

Proof.

Let p : E⟶X give the vector bundle ξ. Denote by P (E) the space of all one dimensional subspaces of fibers of the ξ. Let π : P (E)⟶X be the natural mapping. Then the inverse image π^{∗}(ξ) split into sum π^{∗}(ξ) = η ⊕ ζ where η is the Hopf bundle over P (E). Hence dim ζ = dim ξ − 1. On the other hand, the homomorphism

π^{∗} : K^{∗}(X)⟶K^{∗} (P (E))

is a monomorphism. In fact, consider the continuous mapping f : X⟶BU(n) and commutative diagram

[figure: X; BU(n); P (E); P EU(n)]  (4.13)

The diagram (4.13) induces a commutative diagram of spectral sequences

[figure: ^{3}E_{s}^{p,q}; ^{4}E_{s}^{p,q}; ^{1}E_{s}^{p,q}; ^{2}E_{s}^{p,q}]

where

1E_{s}^{p,q} = H^{p} (X, K^{q}(CP^{n−1})) ,
2E_{s}^{p,q} = H^{p} (BU(n), K^{q}(CP^{n−1})) ,
3E_{s}^{p,q} = H^{p} (X, K^{q}(pt)) ,
4E_{s}^{p,q} = 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

1E_{s}^{∗,∗} = ^{3}E_{s}^{∗,∗} ⊗ K^{∗}(CP^{n−1}).

Hence the homomorphism

3E_{∞}^{∗,∗}⟶^{1}E_{∞}^{∗,∗}

is a monomorphism. Therefore, π^{∗} is also a monomorphism. Now one can complete the proof by induction with respect to the dimension of vector bundle ξ.

4.2 OPERATIONS IN K–THEORY

By definition, a cohomology operation is a transformation

α : K^{∗}(X)⟶K^{∗}(X)

which is functorial, that is, for any continuous mapping f : X⟶Y the following diagram

[figure: K^{∗}(X); K^{∗}(Y); K^{∗}(X); K^{∗}(Y); α; α; f^{∗}; f^{∗}]

is commutative. In particular, if f : X⟶BU(N), x = f^{∗}(ξ_{N}) then

α(x) = f^{∗}(α(ξ_{N})).

Here the element α(ξ_{N}) ∈ K^{∗}(BU(N)) may be an arbitrary element and it completely defines the cohomology operation α. 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

j_{n} : G(N + n, N)⟶G(N + n + 1, N).

Then the cohomology operation α is defined as an inverse sequence of elements α_{n} ∈ K^{∗}(G(N +n, N)) such that j_{n}^{∗}(α_{n+1}) = α_{n}. This means that the sequence {α_{n}} defines an element in the inverse limit

α ∈ lim_{n^{⟵}} K^{∗}(G(N + n, N)) = K^{∗}(BU(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 α_{N} ∈ K^{∗}(BU(N)) such that

α_{N+1}(ξ ⊕ 1) = α_{N}(ξ).

Then the natural mapping

k_{N} : BU(N)⟶BU(N + 1)

sends the element α_{N+1} to α_{N}. This means that the sequence {α_{N}} defines an element

α ∈ lim_{N^{⟵}} K^{∗} (BU(N)) .

Let us put BU = lim_{N^{⟶}} BU(N). Then we can show that

K^{∗}(BU) = lim_{N^{⟵}} K^{∗}(BU(N)).

Definition 9 The Adams operation Ψ^{k} : K(X)⟶K(X) is the operation characterized by the following property

Ψ^{k}(ξ_{1} ⊕ … ξ_{n}) = ξ_{1}^{k} ⊕ … ξ_{n}^{k}

for any one dimensional vector bundles ξ_{j}.

We show now that such an operation Ψ^{k} exists. Consider the symmetric polynomial s_{k}(t_{1}, …, t_{n}) = t_{1}^{k} + … + t_{n}^{k}. Then the polynomial s_{k} can be expressed as a polynomial

s_{k}(t_{1}, …, t_{n}) = P_{k}^{n}(σ_{1}, …, σ_{n}).

The polynomial P_{k}^{n} has integer coefficients and the σ_{k} are the elementary symmetric polynomials:

P_{k}^{n}(σ_{1}, …, σ_{n}) = ⨁ λ_{l_{1},l_{2},...,l_{n}}^{k}σ_{1}^{l_{1}}σ_{2}^{l_{2}} ⋯ σ_{n}^{l_{n}}.

Then

Ψ^{k}(ξ) = ⨁ λ_{l_{1},l_{2},...,l_{n}}^{k}ξ^{l_{1}} ⊗ (Λ_{2}ξ)^{l_{2}} ⊗ … ⊗ (Λ_{n}ξ)^{l_{n}}

It is easy to check that

Ψ^{k} (ξ_{1} ⊕ ξ_{2}) = Ψ^{k} (ξ_{1}) + Ψ^{k} (ξ_{2}) .

Theorem 41 The Adams operations have the following properties:

1. Ψ^{k}(xy) = Ψ^{k}(x)Ψ^{k}(y);

2. Ψ^{k}Ψ^{l} = Ψ^{kl};

3. Ψ^{p}(x) ≡ x^{p}( mod p) for prime p.

As was shown earlier, it is sufficient to check the identities on vector bundles which can be split into one–dimensional summands. Let ξ = ξ_{1} ⊕ … ⊕ ξ_{n}, η = η_{1} ⊕ … ⊕ η_{m} and then

ξ ⊗ η ⨁_{i,j} (ξ_{i} ⊗ η_{j}) .

Then we have

Ψ^{k}(ξ ⊗ η) = ⨁_{i,j} (ξ_{i} ⊗ η_{j})^{k} =

= ⨁_{i,j} (ξ_{i}^{k} ⊗ η_{j}^{k}) =

= (⨁_{i} ξ_{i}^{k}) ⊗ (⨁_{j} η_{j}^{k}) =

= Ψ^{k}(ξ)Ψ^{k}(η).

Further,

Ψ^{k}Ψ^{l}(ξ_{1} ⊕ … ⊕ ξ_{n}) = Ψ^{k}(ξ_{1}^{l} ⊕ … ⊕ ξ_{n}^{l}) =
= ξ_{1}^{kl} ⊕ … ⊕ ξ_{n}^{kl} = Ψ^{kl}(ξ_{1} ⊕ … ⊕ ξ_{n}).

Finally, if p is a prime integer then,

Ψ^{p}(ξ_{1} ⊕ … ⊕ ξ_{n}) = ξ_{1}^{p} ⊕ … ⊕ ξ_{n}^{p} ≡
≡ (ξ_{1} ⊕ … ⊕ ξ_{n})^{p} ( mod p) .

Examples

1. Consider the sphere S^{2n} and the generator β_{n} ∈ K^{0} (S^{2n}). The element β_{n} is the image of the element

β ⊗ … ⊗ β ∈ K^{0} (S^{2} × … × S^{2})

by the homomorphism induced by the continuous mapping

f : S^{2} × … × S^{2}⟶S^{2n}

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

Then

f^{∗}Ψ^{k} (β_{n}) Ψ^{k} (β ⊗ … ⊗ β) (Ψ^{k}β) ⊗ … ⊗ (Ψ^{k}β) .

Now β = η − 1 where η is the Hopf bundle. Hence

Ψ^{k}β = η^{k} − 1 = (1 + β)^{k} − 1 = kβ.

Thus

f^{∗}Ψ^{k}β_{n} = k^{n} (β ⊗ … ⊗ β) k^{n}f^{∗}(β_{n}),

that is,

Ψ^{k}β_{n} = k^{n}β_{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 (2–dimensional) projective 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

S^{3n−1}⟶P (V^{3})

with the fiber S^{n−1}. As for CP^{n} (see 18), we can use the spectral sequence to show that

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]/{u^{3} = 0}, u ∈ K^{0} (P (V^{3})) .

Moreover, the space P (V^{3}) is the union of the sphere S^{n} = P (V^{2}) and the disk D^{2n} glued on by a continuous mapping φ : ∂D^{2n} = S^{2n−1}⟶S^{n}. In other words, we have the pair

S^{n} ⊂ P (V^{3})⟶P (V^{3})/S^{n} = S^{2n}.  (4.15)

The exact sequence for (4.15) has the following form

0⟶K^{0} (S^{2n}) ⟶^{j^{∗}}K^{0} (P (V^{3})) ⟶^{i^{∗}}K^{0} (S^{n}) ⟶0.

Let

a ∈ K^{0} (S^{n}) ; b ∈ K^{0} (S^{2n})

be generators and let

i^{∗}(x) = a; j^{∗}(b) = y.

Let y = αu + βu^{2}. Then 0 = y^{2} = (αu + βu^{2})^{2}α^{2}u^{2}. Hence α = 0. Let x = γu + δu^{2}. The elements (x, y) forms the basis in the group K^{0} (P (V^{3})) and so β = ±1; γ = ±1. By choosing a and b appropriately we can assume that

x^{2} = y.

Now we can calculate the action of the Adams operations. Firstly,

j^{∗}Ψ^{k}(x) = Ψ^{k}(a) = k^{m}a.

Hence

Ψ^{k}(x) = k^{m}x + λy

and

Ψ^{k}(y) = Ψ^{k}j_{∗}(b) = j^{∗}Ψ^{k}(b) = k^{2m}y.

Moreover,

Ψ^{k}(x) ≡ x^{k}( mod k)

when k is prime. For example, when k = 2, the number λ must be odd; when k = 3, the number λ is divisible by 3.

Consider the operation Ψ^{6} = Ψ^{2}Ψ^{3} = Ψ^{3}Ψ^{2}. Then

Ψ^{2}Ψ^{3}(x) = Ψ^{2} (3^{m}x + μy) 6^{m}x + (3^{m}λ + 2^{2m}μ) y,
Ψ^{3}Ψ^{2}(x) = Ψ^{3} (2^{m}x + λy) 6^{m}x + (3^{2m}λ + 2^{m}μ) y,

or

2^{m} (2^{m} − 1) μ = 3^{m} (3^{m} − 1) λ,

where λ is odd. This means that the number (3^{m} − 1) is divisible by 2^{m}. Let m = 2^{k}l where l is odd. Then

3^{m} − 1 = (3^{2^{k}})^{l} − 1 (3^{2^{k}} − 1) ((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. Hence the number 3^{2^{k}} − 1 is divisible by 2^{m} and hence is divisible by 2^{2^{k}}. On the other hand,

3^{2^{k}} − 1 = (3^{2^{k−1}} + 1) (3^{2^{k−2}} + 1) · . . . · (3 + 1) (3 − 1).  (4.16)

It is easy to see that the number 3^{2^{k}} + 1 is divisible by 2 but not divisible by 4. Hence by the decomposition (4.16), the number 3^{2^{k}} − 1 is divisible by 2^{k+2} but not divisible by 2^{k+3}. Hence

2^{k}l ≤ k + 2.  (4.17)

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 8–dimensional (nonassociative ) algebra, the Cayley numbers. Even though the algebra of the Cayley numbers is not associative, the projective 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

Consider a complex n–dimensional vector bundle

p : E⟶X,

denoted by ξ. The total space E is not compact. Denote by ξ^{E} the one–point compactification of E and call it the Thom space of the bundle ξ. This space can be represented a quotient space D(ξ)/S(ξ) where D(ξ) is the subbundle of unit balls and S(ξ) is subbundle of unit spheres. Then we say that the group K^{0} (D(ξ)/S(ξ)) = K_{c}(E) is K–group with compact supports of this noncompact space. This notation can be extended to suspensions. Hence there is a bigraded group

K_{c}^{∗}(E) = K_{c}^{0}(E) ⊕ K_{c}^{1}(E).

Consider the element

β(ξ) ∈ K^{0} (D(ξ), S(ξ))  (4.18)

defined in the chapter 3 (3.11). Then the homomorphism given by multiplication (see (3.12 ))

β : K^{∗}(X)⟶K_{c}^{∗}(E)

is called the Thom homomorphism.

Theorem 42 The Thom homomorphism is an isomorphism.

Proof.

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 ξ 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 σ. Then the vector bundle ξ is trivial over σ and also over ∂σ. Therefore, we can apply Bott periodicity to the pair (X, Y) to obtain:

β : K^{∗}(X, Y)⟶K_{c}^{∗}(E, E_{1}),

where E_{1} = p^{−1}(Y). Consider the commutative diagram

[figure: K_{c}^{∗}(E, E_{1}); K_{c}^{∗}(E); K_{c}^{∗}(E_{1}); K^{∗}(X, Y); K^{∗}(X); K^{∗}(Y); β; β; β; …; …; …; …]

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 β can be extended to noncompact bases as a isomorphism of K–groups with compact supports:

β : K_{c}^{∗}(X)⟶K_{c}^{∗}(E).

The homomorphism β 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 β by i_{∗}. Consider another vector bundle ψ over the base X. Then there is a commutative diagram of vector bundles:

[figure: X; E; E_{1}; E_{2}; η; p^{−1}(η); j; ξ; i]

where i and j are inclusions as zero sections.

Theorem 43 The inclusions i and j satisfy the following conditions:

1. (ji)_{∗} = j_{∗}i_{∗},

2. i_{∗} (xi^{∗}(y)) = i_{∗}(x)y for any y ∈ K_{c}^{∗}(E) and x ∈ K_{c}^{∗}(X).

Proof.

Since y = i_{∗}(z) = βz, we have

i_{∗} (xi^{∗}(βz)) = βi_{∗}(β)xz,
i_{∗}(x)βz = β^{2}xz.

Hence it is suffices to prove that

β^{2} = βi^{∗}(β)  (4.19)

and it is suffices to prove this last equality for the special case when the vector bundle ξ is the one dimensional Hopf bundle η over CP^{n}. Then the Thom space η^{CP^{n}} is homeomorphic to CP^{n+1}. In this case, the element β is equal to η − 1. Let t ∈ H^{2} (CP^{n}; Q) and α ∈ H^{2} (CP^{n+1}; Q) be generators. Passing to cohomology via the Chern character in the formula (4.19), we get

(ch β)^{2} = (ch β) (ch i^{∗}(β)) .

Therefore, we need to prove

(α + \frac{α^{2}}{2!} + … + \frac{α^{n+1}}{(n + 1)!}) (α + \frac{α^{2}}{2!} + … + \frac{α^{n+1}}{(n + 1)!}) =

= (α + \frac{α^{2}}{2!} + … + \frac{α^{n+1}}{(n + 1)!}) (t + \frac{t^{2}}{2!} + … + \frac{t^{n}}{n!} ) .  (4.20)

The equality (4.20) follows because

i^{∗}(α) = t
α^{k}t^{l} = α^{k+l}.

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_{∗} : K_{c}(T X)⟶K_{c}(T Y)

such that

1. (fg)_{∗} = f_{∗}g_{∗},

2. f_{∗} (xf^{∗}(y)) = f_{∗}(x)y for any y ∈ K_{c}^{∗}(T Y) and x ∈ K_{c}^{∗}(T X).

Proof.

Let φ : X⟶V be an inclusion into a Euclidean space of large dimension. Then the mapping f splits into a composition

X⟶^{i}Y × V ⟶^{p}Y,  (4.21)

where i(x) = (f(x), φ(x)) and p is the projection onto the first summand. Let U ⊂ Y × 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 × T V

be the identity inclusion of an open domain. Let

k : T Y ⟶T Y × T V

be the zero section inclusion.

Definition 10 Define

f_{∗}(x)k_{∗}^{−1}j_{∗}i_{∗}(x), x ∈ K_{c}(T X),

where j_{∗} : K_{c}^{∗}(T U)⟶K_{c}^{∗}(T (Y × V)) is a natural extension of the difference construction from T U to T (Y × V).

It is clear that if dim V is sufficiently large then the definition 10 is independent of the choice of the inclusion φ.

The property 1 of Theorem 44 follows from the following commutative diagram of inclusions and projections:

[figure: X; Y × V; Y; Z × W; Z; Z × V × W; (gf, φ, ψf); (f, φ); (g, ψ)]

Let us prove the property 2 of Theorem 44. If x ∈ K_{c}^{∗}(T X), y ∈ K_{c}^{∗}(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

j_{∗}i_{∗}(x) = p^{∗} (k_{∗}^{−1}j_{∗}i_{∗}(x)) k_{∗}(1).

Hence

k_{∗}^{−1} (j_{∗}i_{∗}(x)p^{∗}(y)) (k_{∗}^{−1}j_{∗}i_{∗}(x)) y = f_{∗}(x)y.

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 RIEMANN–ROCH 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 element φ(1) ∈ H_{c}^{2n}(E; Z) where n = dim ξ is the dimension of the complex vector bundle ξ with total space E.

It is natural to compare these two Thom isomorphisms. Consider the diagram

[figure: H^{∗}(X; Q); H_{c}^{∗}(E; Q); K^{∗}(X); K_{c}^{∗}(E); ch; ch; φ; β]  (4.22)

The diagram (4.22) is not, in general, commutative. In fact,

ch β(x) = ch (β ⊗ x) = ch βch x,
φch x = φ(1)ch x.  (4.23)

Therefore, the left hand sides of (4.22) differ from each of other by the factor T (ξ) = φ^{−1} (ch β):

ch β(x) = T (ξ)φ (ch (x))

The class T (ξ) is a characteristic class and, therefore, can be expressed in terms of symmetric functions. Let ξ = ξ_{1}⊕…⊕ξ_{n} be a splitting into one–dimensional summands. Then it is clear that

β = β_{1} ⊗ … ⊗ β_{n},

where the β_{k} the Bott elements of the one dimensional vector bundles. Similarly,

φ(1) = φ_{1}(1) · . . . · φ_{n}(1),

where each φ_{k} : H^{∗}(X; Z)⟶H_{c}^{∗}(E_{k}; Z) is the Thom isomorphism for the bundle ξ_{k}. Then

T (ξ) = φ^{−1}(ch β) = φ_{1}^{−1}(ch β_{1}) · . . . · φ_{n}^{−1}(ch β_{n}) = T (ξ_{1}) · . . . · T (ξ_{n}).

Hence it is sufficient to calculate T (ξ) for a one dimensional vector bundle, particularly for the Hopf bundle over CP^{n}. Let t ∈ H^{2}(CP^{n}; Z) be the generator. Then

T (ξ) = φ^{−1}(ch β) = \frac{e^{t} − 1}{t}.

Thus for an n–dimensional vector bundle ξ we have

T (ξ) = ∏_{k=1}^{n} \frac{e^{t_{k}} − 1}{t_{k}}.

The class T (ξ) is a nonhomogeneous cohomology class which begins with 1. The inverse class T^{−1}(ξ) is called the Todd class of the vector bundle ξ.

Theorem 45 Let f : X⟶Y be a smooth mapping of smooth manifolds. Then

ch f_{∗}(x)T^{−1}(T Y) = f_{∗} (ch xT^{−1}(T X)) , x ∈ K_{c}^{∗}(T X).

Proof.

Let

T X⟶^{i}U⟶^{j}T Y × R^{2n}⟶^{p}T Y

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) =
= k_{∗}^{−1}j_{∗} (i_{∗}ch (x)T (ξ)) = f_{∗} (ch (x)T (ξ)) ,

where ξ is the normal bundle to the inclusion i. Then up to a trivial summand we have

ξ = −[T X] + f^{∗}[T Y].

Therefore,

T (ξ) = T^{−1}(T X)f^{∗}T (T Y)

and thus,

ch f_{∗}(x) = f_{∗} (\frac{f^{∗}T (T Y)}{T (T X)}) .  (4.24)

Examples

1. Let X be a compact almost complex 2n–dimensional 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

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 n–dimensional 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 ∈ K_{c}^{∗}(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 zero–dimensional 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(BG).

from the ring of virtual representations of the compact group G. Let T ⊂ G be a maximal torus. Then there is the commutative diagram

[figure: R(T); K(BT); R(G); K(BG); i_{1}; i_{2}; b; b]

The lower horizontal homomorphism b becomes an isomorphism after completion of the rings. The homomorphism i_{1} 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 i_{2} also maps the group K(BG) into subring of K(BT) of elements which are invariant with respect to the action of the Weyl group. It is known that the fiber of i_{2}, G/T, admits an almost complex structure with unit Todd genus (A.Borel, F.Hirzebruch [12]). Then Corollary 7 says that i_{2} 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]).

194 Chapter 4

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^{∞}(R^{n})⟶C^{∞}(R^{n}).

The operator A is a finite linear combination of partial derivatives

A = ∑_{α} a_{α}(x)\frac{∂^{|α|}}{∂x^{α}} ,  (5.1)

where the α = (α_{1}, …, α_{n}) is multi-index, a_{α}(x) are smooth functions and

\frac{∂^{|α|}}{∂x^{α}} = \frac{∂|α|}{(∂x^{1})^{α_{1}}(∂x^{2})^{α_{2}} … (∂x^{n})^{α_{n}}} ,

|α| = α_{1} + … + α_{n}

is the operator given by the partial derivatives.

The maximal value of |α| is called the order of the differential operator differential operator , so the formula (5.1) can be written

A = ∑_{|α|≤m} a_{α}(x)\frac{∂^{|α|}}{∂x^{α}} ,

Let us introduce a new set of variables ξ = (ξ_{1}, ξ_{2}, …, ξ_{n}). Put

a(x, ξ) = ∑_{|α|≤m} a_{α}(x)ξ^{α}i^{|α|},

where ξ^{α} = ξ_{1}^{α_{1}}ξ_{2}^{α_{2}} ⋯ ξ_{n}^{α_{n}}. The function a(x, ξ) is called the symbol of a differential operator A. The operator A can be reconstructed from its symbol by substitution of the operators \frac{1}{i} \frac{∂}{∂x^{k}} for the variables ξ_{k}, that is,

A = a (x, \frac{1}{i} \frac{∂}{∂x}) .

Since the symbol is polynomial with respect to variables ξ, it can be split into homogeneous summands

a(x, ξ) = a_{m}(x, ξ) + a_{m−1}(x, ξ) + … + a_{0}(x.ξ).

The highest term a_{m}(x, x) is called the principal symbol of the operator A while whole symbol sometimes is called the full symbol . The reason for singling out the principal symbol is as follows:

Proposition 13 Let

y^{k} = y^{k} (x^{1}, …, 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

(Bu)(y) = (Au (y(x)))_{x=x(y)}

is again a differential operator of order m for which the principal symbol is

b_{m}(y, η) = a_{m} (x(y), η \frac{∂y(x(y))}{∂x}) .  (5.2)

Elliptic operators and K–theory 197

The formula (5.2) shows that variables ξ 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_{x⟶ξ}(u)(ξ) = \frac{1}{(2π)^{n/2}} ∫_{R^{n}} e^{−i(x,ξ)}u(x)dx

and

F_{ξ⟶x}(v)(x) = \frac{1}{(2π)^{n/2}} ∫_{R^{n}} e^{i(x,ξ)}v(ξ)dξ

are the direct and inverse Fourier transformations then

(Au) (x) = F_{ξ⟶x} (a(x, ξ) (F_{x⟶ξ}(u)(ξ)))  (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

|\frac{∂^{|α|}}{∂ξ^{α}} \frac{∂^{|β|}}{∂x^{β}} a(x, ξ)| ≤ C_{α,β}(1 + |ξ|)^{m−|α|}  (5.4)

for some constants C_{α,β}. Denote by S the Schwartz space of functions on R^{n} which satisfy the condition

|x^{β} \frac{∂^{|α|}}{∂x^{α}} u(x)| ≤ C_{α,β}

for any multiindexes α and β. 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_{α}} be an atlas of charts with a local coordinate system x_{α} = (x_{α}^{1}, …, x_{α}^{n}). Without loss of generality we can assume that the local coordinate system x_{α} maps the chart U_{α} onto the space R^{n}. Let ξ_{α} = (ξ_{1α}, …, ξ_{nα}) be the corresponding components of a cotangent vector. Let {φ_{α}} be a partition of unity subordinate to the atlas of charts, that is,

0 ≤ φ_{α}(x) ≤ 1, ∑_{α} φ_{α}(x) ≡ 1, Supp φ_{α} ⊂ U_{α}.

Finally, let ψ_{α}(x) be functions such that

Supp ψ_{α} ⊂ U_{α}, φ_{α}(x)ψ_{α}(x) ≡ φ_{α}(x).

Then we can define an operator A by the formula

A(u)(x) = ∑_{α} ψ_{α}(x)A_{α} (φ_{α}(x)u(x)) ,  (5.5)

where A_{α} is a pseudodifferential operator on the chart U_{α} (which is diffeomorphic to R^{n}) with principal symbol

a_{α}(x_{α}, ξ_{α}) = a(x, ξ).

When the function a(x, ξ) is polynomial (of order m) the operator A defined by formula (5.5) is a differential operator not depending on the choice of functions ψ_{α}. In general, the operator A depends on the choice of functions ψ_{α}, φ_{α} and the local coordinate system x_{α}, 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 ξ_{1} and ξ_{2} be two vector bundles over the manifold M. Consider a linear mapping

a : π^{∗}(ξ_{1})⟶π^{∗}ξ_{2},  (5.6)

where π : T^{∗}M⟶M is the natural projection. Then in any local coordinate system (x_{α}, ξ_{α}) 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) : Γ^{∞}(ξ_{1})⟶Γ^{∞}(ξ_{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 Proposition 14 Let

a : π^{∗}(ξ_{1})⟶π^{∗}(ξ_{2}),; b : π^{∗}(ξ_{a})⟶π^{∗}(ξ_{3}),;

be two symbols of orders m_{1}, m_{2}. Let c = ba be the composition of the symbols. Then the operator

b(D)a(D) − c(D) : Γ^{∞}(ξ_{1})⟶Γ^{∞}(ξ_{3})

is a pseudodifferential operator of order m_{1} + m_{2} − 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, BA = 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, ξ) = 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, ξ)b(x, ξ) = 1 + symbol of order (−1).  (5.8)

In particular, the condition (5.8) holds if

Condition 1 a(x, ξ) is invertible for sufficiently large |ξ| ≥ C.

In fact, if the condition (5.8) holds then we could put

b(x, ξ) = a^{−1}(x, ξ)χ(x, ξ),

where χ(x, ξ) is a function such that

χ(x, ξ) ≡ 1 for |ξ| ≥ 2C,
χ(x, ξ) ≡ 0 for |ξ| ≤ C.

Then the pseudodifferential operator a(D) is called an elliptic if Condition 1 holds.

The final generalization for elliptic operators is the substitution of a sequence of pseudodifferential operators for a single elliptic operator. Let ξ_{1}, ξ_{2}, …, ξ_{k} be a sequence of vector bundles over the manifold M and let

0⟶π^{∗}(ξ_{1})⟶^{a_{1}}π^{∗}(ξ_{2})⟶^{a_{2}} … ⟶^{a_{k−1}}π^{∗}(ξ_{k})⟶0  (5.9)

be a sequence of symbols of order (m_{1}, …, m_{k−1}). Suppose the sequence (5.9) forms a complex, that is, a_{s}a_{s−1} = 0. Then the sequence of operators

0⟶Γ^{∞}(ξ_{1}) ⟶^{a_{1}(D)}Γ^{∞}(ξ_{2})⟶ … ⟶Γ^{∞}(ξ_{k})⟶0  (5.10)

in general, does not form a complex because we can only know that the composition a_{k}(D)a_{k−1}(D) is a pseudodifferential operator of the order less then m_{s} + m_{s−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 elliptic complex of pseudodifferential operators.

5.2 FREDHOLM OPERATORS

A bounded linear operator

F : H⟶H

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 \overline{F (X)} is compact.

Theorem 46 Let F be a Fredholm operator. Then

1. there exists ε > 0 such that if ∥F − G∥ < ε 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.

Proof.

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 + Kx = 0, that is, Kx = −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 x_{n} ∈ Im F, lim x_{n} = x. Then

x_{n} = F (y_{n}) = y_{n} + Ky_{n}.

Without loss of generality we can assume that

y_{n} ⊥ Ker F.

If the set of numbers {∥y_{n}∥} is unbounded then we can assume that

∥y_{n}∥⟶∞.

Then

lim_{n⟶∞} (\frac{y_{n}}{∥y_{n}∥} + K \frac{y_{n}}{∥y_{n}∥}) = lim_{n⟶∞} \frac{x}{∥y_{n}∥} = 0

The sequence y_{n}/∥y_{n}∥ is bounded and by passing to subsequence we can assume that there is a limit

lim_{n⟶∞} K (\frac{y_{n}}{∥y_{n}∥}) = z.

Hence

lim_{n⟶∞} \frac{y_{n}}{∥y_{n}∥} = −z,  (5.12)

that is,

F (z) = z + Kz = 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 {y_{n}} is bounded and hence there is a subsequence such that

lim_{n⟶∞} Ky_{n} = z.

Therefore,

lim_{n⟶∞} y_{n} = lim_{n⟶∞} (x_{n} − Ky_{n}) = x − z.

Hence

x = lim_{n⟶∞} x_{n} = lim_{n⟶∞} F (y_{n}) = F (x − z),

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 = Ker F ⊕ (Ker F)^{⊥}
H = (Im F)^{⊥} ⊕ Im F.  (5.13)

Then the operator F can be represented as a matrix with respect to the splittings (5.13):

F = ∥\begin{matrix}0 & 0 \\ 0 & F_{1} \end{matrix}∥ ,  (5.14)

where F_{1} is an isomorphism. We define a new operator G by the following matrix

G = ∥\begin{matrix}0 & 0 \\ 0 & F_{1}^{−1} \end{matrix}∥ ,

Then the operators

K = F G − 1 = ∥\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}∥ and K′ = GF − 1 = ∥\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}∥

are finite dimensional and hence compact.

Let ε > 0 be such that ∥G∥ < \frac{1}{ε}. Suppose that the operator α is such that ∥α∥ < ε and F_{1} = F + α. Notice that both operators 1 + αG and 1 + Gα are invertible. Then

(1 + Gα)^{−1}GF_{1} = (1 + Gα)^{−1}(1 + Dα + K) = 1 + (1 + Gα)^{−1}K,
F_{1}G(1 + αG)^{−1} = (1 + αG + K′)(1 + αG)^{−1} = 1 + K′(1 + αG)^{−1}.

Since operators (1 + Gα)^{−1}K and K′(1 + αG)^{−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 = ∥\begin{matrix}F_{1} & 0 \\ 0 & F_{2} \end{matrix}∥  (5.15)

then, evidently,

index F = index F_{1} + index F_{2}.

If both Hilbert spaces H_{1} and H_{2} are finite dimensional and F : H_{1}⟶H_{2} is bounded then F is Fredholm and

index F = dim H_{1} + dim H_{2}.

If F is invertible then

index F = 0.

If in the operator (5.15), F_{1} = 0 is finite dimensional and F_{2} is invertible then F is Fredholm and index F = 0.

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∥ < ε. Let

G = ∥\begin{matrix}α_{11} & α_{12} \\ α_{21} & F_{2} + α_{22} \end{matrix}∥ ,  (5.16)

where ∥α_{22}∥ < ε. For sufficiently small α the operator F_{2} + α_{22} is invertible. Then for X = −(F_{2} + α_{22})^{−1}α_{21} we have

G = ∥\begin{matrix}1 & 0 \\ X & 0 \end{matrix}∥ ∥\begin{matrix}β_{11} & β_{12} \\ 0 & F_{2} + β_{22} \end{matrix}∥ .

Similarly, for Y = −β_{12}(F_{2} + β_{22})^{−1},

G = ∥\begin{matrix}1 & 0 \\ X & 1 \end{matrix}∥ ∥\begin{matrix}γ_{11} & 0 \\ 0 & F_{2} + γ_{22} \end{matrix}∥ ∥\begin{matrix}1 & Y \\ 0 & 1 \end{matrix} ∥ .

Thus index G = index F.

Now let us prove (5.11). We can represent the operators F and K as a matrices

F = ∥\begin{matrix}F_{1} & 0 \\ 0 & F_{2} \end{matrix}∥; K = ∥\begin{matrix}K_{11} & K_{12} \\ K_{21} & K_{22} \end{matrix}∥

such that F_{1} is finite dimensional and K_{22} is sufficiently small. Then we are in situation of the condition (5.16). Hence

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⟶C_{0}⟶^{d_{0}}C_{1}⟶^{d_{1}} … ⟶^{d_{n−1}}C_{n}⟶0.  (5.17)

We say that the sequence (5.17) is Fredholm complex if d_{k}d_{k−1}0, Im d_{k} is a closed subspace and

dim (Ker d_{k}/Coker d_{k−1}) = dim H (C_{k}, d_{k}) < ∞.

Then the index of Fredholm complex (5.17) is defined by the following formula:

index (C, d) = ∑_{k}(−1)^{k} dim H(C_{k}, d_{k}).

Theorem 47 Let

0⟶C_{0}⟶^{d_{0}}C_{1}⟶^{d_{1}} … ⟶^{d_{n−1}}C_{n}⟶0  (5.18)

be a sequence satisfying the condition that each d_{k}d_{k−1} is compact. Then the following conditions are equivalent:

1. There exist operators f_{k} : C_{k}⟶C_{k−1} such that f_{k+1}d_{k} + d_{k−1}f_{k} = 1 + r_{k} where each r_{k} is compact.

2. There exist compact operators s_{k} such that the sequence of operators d′_{k} = d_{k} + s_{k} forms a Fredholm complex. The index of this Fredholm complex is independent of the operators s_{k}.

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_{ev} = ⨁_{k} C_{2k},
C_{odd} = ⨁_{k} C_{2k+1},

D = ∥\begin{matrix}d_{0} & d_{1}^{∗} & 0 & … & 0 \\ 0 & d_{2} & d_{3}^{∗} & … & 0 \\ 0 & 0 & d_{4} & … & 0 \\ … & & & ... \end{matrix}∥  (5.19)

Prove that the complex (5.19) is Fredholm with the same index.

5.3 THE SOBOLEV NORMS

Consider the Schwartz space S. Define the Sobolev norm by the formula

∥u∥_{s}^{2} = ∫_{R^{n}} ū(x)(1 + Δ)^{s}u(x)dx,

where

Δ = ∑_{k=1}^{n} (i\frac{∂}{∂x^{k}} )^{2}

is the Laplace operator. Using the Fourier transformation this becomes

∥u∥_{s}^{2} = ∫_{R^{n}} (1 + |ξ|^{2})^{s}|û(ξ)|^{2}dξ  (5.20)

where the index s need not be an integer.

The Sobolev space H_{s}(R^{n}) is the completion of S with respect to the Sobolev norm (5.20).

Proposition 15 The Sobolev norms (5.20) for different coordinate systems are all equivalent on any compact K ⊂ R^{n}, that is, there are two constants C_{1} > 0 and C_{2} > 0 such that for Supp u ⊂ K

∥u∥_{1,s} ≤ C_{1}∥u∥_{2,s} ≤ C_{2}∥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 ξ. Let {U_{α}} be an atlas of charts where the vector bundle is trivial over each U_{α}. Let {φ_{α}} be a partition of unity subordinate to the atlas. Let u ∈ Γ^{∞}(M, ξ) be a section. Put

∥u∥_{s}^{2} = ∑_{α} ∥φ_{α}u∥_{s}^{2}.  (5.21)

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 Γ^{∞}(M, ξ) does not depend on the choice of partition of unity or the choice of local coordinate system in each chart U_{α}. We shall denote this completion by H_{s}(M, ξ).

Theorem 48 Let M be a compact manifold, ξ be a vector bundle over M and s_{1} < s_{2}. Then the natural inclusion

H_{s_{2}}(M, ξ)⟶H_{s_{1}}(M, ξ)  (5.22)

is a compact operator.

Theorem 48 is called the Sobolev inclusion theorem.

Proof.

For the proof it is sufficient to work in a local chart since any section u ∈ Γ^{∞}(M, ξ) can be split into a sum

u = ∑_{α} φ_{α}u,

where each summand φ_{α}u has support in the chart U_{α}. So let u be a function defined on R^{n} with support in the unit cube I^{n}. These functions can be considered as functions on the torus T^{n}. Then a function u can be expanded in a convergent Fourier series

u(x) = ∑_{l} a_{l}e^{i(l,x)}.

Partial differentiation transforms to multiplication of each coefficient a_{l} by the number l_{k}, where l is a multiindex l = (l_{1}, …, l_{n}). Therefore,

∥u∥_{s}^{2} = ∑_{l} |a_{l}|^{2} (1 + |l_{1}|^{2} + … + |l_{n}|^{2})^{s} .  (5.23)

By formula (5.23), the space H_{s}(T^{n}) is isomorphic to the Hilbert space l_{2} of the square summable sequences by the correspondence

u(x) ↦ \{b_{l} = \frac{a_{l}}{(1 + |l_{1}|^{2} + … + |l_{n}|^{2})^{s/2}} \}

Then the inclusion (5.22) becomes an operator l_{2}⟶l_{2} defined by the correspondence

\{b_{l}\} ↦ \{\frac{b_{l}}{(1 + |l_{1}|^{2} + … + |l_{n}|^{2})^{(s_{2}−s_{1})/2}} \}

which is clearly compact.

Theorem 49 Let

a(D) : Γ^{∞}(M, ξ_{1})⟶Γ^{∞}(M, ξ_{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) : H_{s}(M, ξ_{1})⟶H_{s−m}(M, ξ_{2}).  (5.26)

Proof.

The theorem is clear for differential operators. Indeed, it is sufficient to obtain estimates in a chart, as the symbol σ has compact support. If a(D) = \frac{∂}{∂x^{j}} then

∥\frac{∂}{∂x^{j}} u(x)∥_{s}^{2} = ∫_{R^{n}} (1 + |ξ|^{2})^{s} | ˆ\frac{∂}{∂x^{j}} u|^{2} dξ =

= ∫_{R^{n}} (1 + |ξ|^{2})^{s} |ξ_{j}û(ξ)|^{2}dξ ≤ ∫_{R^{n}} (1 + |ξ|^{2})^{s+1} |û(ξ)|^{2}dξ =

= ∥u(x)∥_{s+1}^{2}.  (5.27)

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 elliptic 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.

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 special operator (1+Δ)^{k} with symbol (1 + |ξ|^{2})^{k}. Since the norm ∥(1+Δ)^{k}u∥_{s} is equivalent to the norm ∥u∥_{s+2k}, the operator

(1 + Δ)^{k} : H_{s+2k}(ξ)⟶H_{s}(ξ)

is an isomorphism. Then the operator

A = (1 + Δ)^{−k}σ(D)(1 + Δ)^{k} : H_{s+2k}(ξ_{1})⟶H_{s}(ξ_{1})⟶

⟶H_{s−m}(ξ_{2})⟶H_{s+2k−m}(ξ_{2})

differs from σ(D) by a compact operator and therefore has the same index.

Corollary 8 The kernel of an elliptic operator σ(D) consists of infinitely smooth sections.

Proof.

Indeed, by increasing the number s we have a commutative diagram

[figure: H_{s+1}(ξ_{1}); H_{s+1−m}(ξ_{2}); H_{s}(ξ_{1}); H_{s−m}(ξ_{2}); ∪; ∪; σ(D); σ(D)]  (5.28)

An increase in the number s can only decrease the dimension of the kernel, Ker σ(D). Similarly, the dimension of the cokernel may only decrease since the cokernel is isomorphic to the kernel of the adjoint operator σ(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

⋂_{s} H_{s}(ξ_{1}) = Γ^{∞}(ξ_{1}).

5.4 THE ATIYAH–SINGER FORMULA FOR THE INDEX OF AN ELLIPTIC OPERATOR

In the sense explained in previous sections, an elliptic operator σ(D) is defined by a symbol

σ : π^{∗}(ξ_{1})⟶π^{∗}(ξ_{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 (π^{∗}(ξ_{1}), σ, π^{∗}(ξ_{2})) which in turn defines an element

[σ] ∈ K_{c}(T^{∗}M).

Theorem 51 The index index σ(D) of the Fredholm operator σ(D) depends only on the element [σ] ∈ K_{c}(T^{∗}M).

The mapping

index : K_{c}(T^{∗}M)⟶Z

is an additive homomorphism.

Proof.

A homotopy of an elliptic symbol gives a homotopy of Fredholm operators and, under homotopy, the index does not change. Assume that the symbol σ is an isomorphism not only away from zero section but everywhere. The operator σ(D) can be decomposed into a composition of an invertible operator (1 + Δ)^{m/2} and a operator σ_{1}(D) of the order 0. The symbol σ_{1} is again invertible everywhere and is therefore homotopic to a symbol σ_{2} which is independent of the cotangent vector and also invertible everywhere. Then the operator σ_{2}(D) is multiplication by the invertible function σ_{2}. Therefore, σ_{2}(D) is invertible. Thus

index σ(D) = 0.

Finally, if σ = σ_{1} ⊕ σ_{2} then σ(D) = σ_{1}(D) ⊕ σ_{2}(D) and hence

index σ(D) = index σ_{1}(D) + index σ_{2}(D).

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_{∗} : K_{c}(T^{∗}M)⟶K_{c}(pt) = Z,

as described in Section 4.3

Theorem 52 Let σ(D) be an elliptic pseudodifferential operator. Then

index σ(D) = p_{∗}[σ].  (5.30)

Using the Riemann-Roch theorem 45, the formula (5.30) can be written as

index σ(D) = (ch [σ]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

φ : H^{∗}(M)⟶H_{c}^{∗}(T^{∗}M).  (5.31)

Therefore, the formula (5.31) has the form

index σ(D) = (φ^{−1}ch [σ]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 Atiyah–Singer formula.

Proof.

The proof of the Atiyah–Singer 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 K_{c}(T^{∗}M).

Let j : M_{1}⟶M_{2} be a smooth inclusion of compact manifolds and let σ_{1} be an elliptic symbol on the manifold M_{1}. Assume that σ_{2} is a symbol on the manifold M_{2} such that

[σ_{2}] = j_{∗}[σ_{1}].  (5.33)

Theorem 53 If two symbols σ_{1} and σ_{2} satisfy the condition (5.33) then the elliptic operators σ_{1}(D) and σ_{2}(D) have the same index,

index σ_{1}(D) = index σ_{2}(D).

Theorem 52 follows from Theorem 53.

In fact, Theorem 52 holds for M = pt. Then the inclusion

j : pt⟶S^{n}.

gives the direct image

j_{∗} : K_{c}(pt)⟶K_{c} (T^{∗}S^{n}, T_{s_{0}}^{∗})

which is an isomorphism.Therefore, for a symbol σ defining a class

[σ] ∈ K_{c} (T^{∗}S^{n}, T_{s_{0}}^{∗})

we have

[σ] = j_{∗}[σ′], [σ′] = p_{∗}[σ],
index σ(D) = index σ′(D) = [σ′] = p_{∗}([σ]).

Finally, let j : M⟶S^{n} an inclusion and q : S^{n}⟶pt be the natural projection. Then

p = qj, j_{∗}([σ]) = [σ′],

and by Theorem 53

index σ(D) = index σ′(D) = q_{∗}[σ′] = q_{∗}j_{∗}[σ]p_{∗}[σ].

Thus the Atiyah–Singer formula follows from Theorem 53.

Let us explain Theorem 53 for the example when the normal bundle of the inclusion M_{1}⟶M_{2} is trivial. The symbol [σ_{2}] = j_{∗}[σ_{1}] is invertible outside of a neighborhood of submanifold M_{1} ⊂ T^{∗}M_{2}. Suppose that the symbols are of order 0. Then the symbol σ_{2} can be chosen so that it is independent of the cotangent vector outside of a neighborhood U of submanifold M_{1}. Then we can chose the operator σ_{2}(D) such that if Supp ∩ U = ∅ then σ_{2}(D)(u) is multiplication by a function. Hence we can substitute for the manifold M_{2}, the Cartesian product M_{1} × T^{k}, where T^{k} is torus, equipped with an elliptic 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 τ(D) on the circle such that

index τ(D) = 1

and

[τ] ∈ K_{c} (T^{∗}S^{1}, T_{s_{0}}^{∗})

is a generator.

214 Chapter 5

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 2n–dimensional rational cohomology given by the formula

(x, y) = ⟨xy, [X]⟩ ,  (6.1)

where x, y ∈ H^{2n}(X; Q), [X] is fundamental 4n–dimensional cycle and ⟨, ⟩ is the natural value of a cocycle on a cycle.

By Poincaré duality we see that the quadratic form (6.1) is nondegenerate. This 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) = ⨁_{k=0}^{4n} H^{k}(X) using the same definition (6.1). This 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^{odd}(X) ⊕ (H^{0}(X) ⊕ H^{4n}(X)) ⊕

⊕ (H^{2}(X) ⊕ H^{4n−2}(X)) ⊕ … ⊕ (H^{2n−2}(X) ⊕ H^{2n+2}(X)) ⊕ H^{2n}(X).

The matrix of the bilinear form then has the form

∥\begin{matrix}0 & A \\ A^{t} & 0 \end{matrix}∥  (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 4n–dimensional manifold X is the boundary of an oriented compact 4n + 1–dimensional manifold Y then

sign X = 0.

Proof.

The homology group (with rational or real coefficients) can be thought of as

H_{k}(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] ∈ H_{4n}(X) inducing Poincaré duality. The manifold Y also has a Poincaré duality induced by ∩– product with the fundamental cycle [Y] ∈ H_{4k+1}(Y, ∂Y):

D′ = ∩[Y] :  H^{k}(Y)⟶H_{4n+1−k}(Y, X),
D′′ = ∩[Y] :  H^{k}(Y, X)⟶H_{4n+1−k}(Y).

Applications of vector bundle theory 217

The exact homology sequence for the pair (Y, X) fits into a commutative diagram

[figure: H_{4n+1−k}(Y); H_{4n+1−k}(Y, X); H_{4n−k}(X); H_{4n−k}(Y); H^{k}(Y, X); H^{k}(Y); H^{k}(X); H^{k+1}(Y, X); D′′; D′; D; D′′; …; …; j^{∗}; δ; i^{∗}; j^{∗}; i^{∗}; ∂; …; …] (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 δ = codim Ker 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) + …

Since dim H^{k}(X) = dim H_{k}(X),

dim Im i^{∗} = codim Im i^{∗} = \frac{1}{2} dim H^{2n}(X).

Then the group H^{2n}(X) decomposes into two summands:

H^{2n}(X) = A ⊕ B, A = Im i^{∗}, B = (Ker i_{∗})^{∗}

By commutativity of the diagram, D(A) ⊂ B^{∗} and hence

D = ∥\begin{matrix}0 & D_{2} \\ D_{3} & D_{4} \end{matrix}∥ , D_{3} = D_{2}^{t}.

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 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 Hirzebruch–Pontryagin class.

6.1.1 Calculation of the Hirzebruch–Pontryagin class

The Hirzebruch–Pontryagin 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⟶Ω_{0}⟶^{d}Ω_{1}⟶^{d} … ⟶^{d}Ω_{4n}⟶0  (6.4)

The space Ω_{k} is the space of sections of the vector bundle Λ_{k} over X, the exterior power of the cotangent bundle T^{∗}X = Λ_{1}. The complex (6.4) is an elliptic differential complex for which the symbols are given by exterior multiplication by the cotangent vector ξ ∈ T^{∗}X.

Notice that the signature of the manifold X is defined in terms of multiplication of differential forms

(ω_{1}, ω_{2}) = ∫_{X} ω_{1} ∧ ω_{2}.

Fix a Riemannian metric g on the manifold X. The metric g induces a scalar product on the vector bundles Λ_{k}. Let e_{1}, …, e_{4k} be an orthogonal basis in the fiber of T^{∗}X. Then the corresponding orthogonal basis in the fiber of Λ_{k} consists of the vectors e_{i_{1}} ∧ … ∧ e_{i_{k}}. Consider the mapping

∗ : Λ_{k}⟶Λ_{4n−k},

which to each basis vector e_{i_{1}} ∧ … ∧ e_{i_{k}} associates the vector (−1)_{σ}e_{j_{1}} ∧ … ∧ e_{j_{4n−k}} where the e_{j_{l}} complement the system of vectors e_{i_{1}}, …, e_{i_{k}} to form the basis and σ is the parity of the permutation (i_{1}, …, i_{k}, j_{1}, …, j_{4n−k}). Then ∗^{2} = (−1)^{k} on Λ_{k}. Thus we have a homomorphism

∗ : Γ (Λ_{k}) ⟶Γ (Λ_{4n−k}) .

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 = \sqrt{det g_{ij}}dx^{1} ∧ … ∧ dx^{4n}.

It is clear that μ = ∗(1) and for any function f we have ∗(fω) = f ∗ (ω).

Denote by ⟨ω_{1}, ω_{2}⟩ the function whose value is equal to the scalar product of the forms ω_{1} and ω_{2}. Then

⟨ω_{1}, ω_{2}⟩μ = ω_{1} ∧ ∗ω_{2} = (−1)^{k} ∗ ω_{1} ∧ ω_{2}.  (6.5)

In the space ⨁_{k} ΓΛ_{k} we have the positive scalar product

⟨θ_{1}, ω_{2}⟩ = ∫ ⟨ω_{1}, ω_{2}⟩μ.  (6.6)

Then taking into account (6.5),

(ω_{1}, ω_{2}) = ⟨ω_{1}, ∗ω_{2}⟩  (6.7)

for ω_{1} ∈ Γ(Λ_{k}), ω_{2} ∈ ΓΛ_{4n−k}).

Let d^{∗} denote the adjoint operator with respect to the scalar product (6.6). This means that

⟨ω_{1}, dω_{2}⟩ ≡ ⟨d^{∗}ω_{1}, ω_{2}⟩.

Using (6.7),

⟨d^{∗}ω_{1}, ω_{2}⟩ = ⟨ω_{1}, dω_{2}⟩ = ∫_{X} ω_{1} ∧ ∗dω_{2} =

(−1)^{k} ∫_{X} ∗ω_{1} ∧ ω_{2} = ∫_{X} d(∗ω_{1} ∧ ω_{2}) − ∫_{X} d ∗ ω_{1} ∧ ω_{2} =

− ∫_{X} d ∗ ω_{1} ∧ ω_{2} = − ∫_{X} ∗d ∗ ω_{1} ∧ ∗ω_{2} = −⟨∗d ∗ ω_{1}, ω_{2}⟩.

Hence

d^{∗} = − ∗ d ∗ .  (6.8)

Now we can identify the de Rham cohomology with the kernel of the Laplace operator

Δ = (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 elliptic differential operator and in the space Γ^{∞}(Λ)

Ker d = Ker D ⊕ Im d.

Proof.

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 ω ∈ Γ^{∞} (Λ_{k}). Then ω ∈ H_{s} (Λ_{∗}). Decompose the element ω into a sum

ω = ω_{1} + ω_{2},

where

ω_{1} ∈ Ker D ⊂ H_{s} (Λ_{∗}) , ω_{2} ∈ (Ker D)^{⊥} .

On the other hand,

(D (H_{s+1} (Λ_{∗})))^{⊥} = Ker D

Hence

(Ker D)^{⊥} = D (H_{s+1} (Λ_{∗})) ,

that is,

ω_{2} = D(Ω), Ω ∈ H_{s+1} (Λ_{∗}) .

Let s′ > s and

ω = ω′_{1} + ω′_{2}, ω′_{1} ∈ Ker D ⊂ H_{s′} (Λ_{∗}) , ω′_{2} ∈ (Ker D)^{⊥} .

Then

ω_{1} − ω′_{1} = ω_{2} − ω′_{2} ∈ (Ker D)^{⊥} .

Hence ω_{1} = ω′_{1}, ω_{2} = ω′_{2} and ω_{2} ∈ Γ^{∞} (Λ_{∗}). If ω ∈ Ker d then

ω = ω_{1} + Dω_{2}, ω_{1} ∈ Ker D,

that is,

0 = dξω = dω_{1} + dDω_{2}.

Hence

0 = dDω_{2} = Dd^{∗}ω_{2}, dd^{∗}ω_{2} = 0, d^{∗}ω_{2} = 0.

Thus

ω = ω_{1} + d^{∗}ω_{2}.

Theorem 55 says that cohomology group H^{k}(X) is isomorphic to Ker D in the space Γ^{∞} (Λ_{k}). Notice that by formula (6.8), the operator ∗ preserves the kernel Ker D.

Consider a new operator

α = i^{k(k−1)+2n}∗ : Λ_{k}⟶Λ_{4n−k}.

Then

α^{2} = 1, d^{∗} = −αdα, αD = −Dα.

Thus there is a splitting

Λ_{∗} = Λ^{+} ⊕ Λ^{−}

into the two eigen subspaces corresponding to the eigenvalues +1 and −1 of the operator α. Then

D (Γ^{∞} (Λ^{+})) ⊂ Γ^{∞} (Λ^{−}) ,
D (Γ^{∞} (Λ^{−})) ⊂ Γ^{∞} (Λ^{+}) .

Hence

Ker D = Ker D^{+} ⊕ Ker D^{−},

where D^{+} and D^{−} are restrictions of D to Γ^{∞} (Λ_{+}) and Γ^{∞} (Λ_{−}). Thus

sign X = dim Ker D^{+} − dim Ker D^{−}.

Since the operator D is selfadjoint, the operator D^{+} is adjoint to the operator D^{−}. Hence

sign X = index D^{+}.

The operator D^{+} is called the Hirzebruch operator. To calculate the index of the Hirzebruch operator, let Σ be the symbol of the Hirzebruch operator. The symbol Σ is a homomorphism

Σ : π^{∗} (cΛ^{+}) ⟶π^{∗} (cλ^{−}) ,  (6.9)

which is an isomorphism away from the zero section of the cotangent bundle. Hence

index D^{+} = ⟨ch [Σ]T (X), [T^{∗}X]⟩,

where T (X) is the Todd class of the complexification of the tangent bundle.

The construction of the element [Σ] 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 projection on the total space of the bundle ξ. Let

0⟶p^{∗}Λ_{0}(cξ)⟶^{σ}p^{∗}Λ_{1}(cξ)⟶^{σ} … ⟶^{σ}p^{∗}Λ_{n}(cξ)⟶0

be the complex of vector bundles where σ is the exterior multiplication by the vector ξ ∈ E. Similar to the tangent bundle, the bundle Λ_{∗}(ξ) = ⊕Λ_{k}(ξ) has an operation

∗ : Λ_{k}(σ)⟶Λ_{n−k}(ξ)

which induces a splitting

Λ_{∗}(cξ) = Λ^{+}(ξ) ⊕ Λ^{−}(ξ)

and a homomorphism

Σ = σ − ∗σ∗

mapping

Σ : Λ^{+}(ξ)⟶Λ^{−}(ξ)  (6.10)

isomorphically away from the zero section of the total space E. Hence the triple (6.10) defines an element [Σ] ∈ K_{c}(E). Notice that if X = X_{1} × X_{2} and ξ = ξ_{1} ⊕ ξ_{2}, where ξ_{k} are vector bundles over X_{k}, then

[Σ(ξ)] = [Σξ_{1}] ⊗ [Σξ_{2}].

Then the problem can be reduced to the universal two dimensional vector bundle ξ over BSO(2) = BU(1) = CP^{∞}. In this case, the bundle ξ is isomorphic to the complex Hopf bundle and cξ = ξ ⊕ ξ^{∗}. Hence

Λ^{+}(ξ) = \overline{1} ⊕ ξ^{∗}
Λ^{−}(ξ) = ξ ⊕ \overline{1}.

Then the symbol Σ : Λ^{+}(ξ)⟶Λ^{−}(ξ) has the following form

Σ = ∥\begin{matrix}z & 0 \\ 0 & z \end{matrix}∥ ,

where z is a vector in the fiber of the bundle ξ. Hence

[Σ] = ([η] − 1) ⊕ (1 − [η^{∗}]) = [η] − [η_{∗}],

where η is the Hopf bundle over the Thom complex ξ^{CP^{∞}} = CP^{∞}. Hence

ch [Σ] = e^{α} − e^{−α}, α ∈ H_{2} (CP^{∞}; Z) .

Thus if the bundle ξ is decomposed into a direct sum of two dimensional bundles

ξ = ξ_{1} ⊕ … ⊕ ξ_{n},

then

ch (Σ(ξ)) = ∏_{k=1}^{n} (e^{α_{k}} − e^{−α_{k}}) = φ (∏_{k=1}^{n} \frac{e^{α_{k}} − e^{−α_{k}}}{α_{k}} ) ,

where φ : H^{∗}(X)⟶H_{c}^{∗} (ξ^{X}) is the Thom isomorphism. Then

φ^{−1} (ch (Σ)T (X)) = ∏_{k=1}^{n} \frac{α_{k}}{e^{α_{k}} − 1} \frac{(−α_{k})}{e^{−α_{k}} − 1} \frac{e^{α_{k}} − a^{−α_{k}}}{α_{k}} =

= 2^{n} ∏_{k=1}^{n} \frac{α_{k}/2}{tanh(α_{k}/2)}.

Thus for a 4n–dimensional manifold X,

L(X) = 2^{2n} ∏_{k=1}^{n} \frac{t_{k}/2}{tanh(t_{k}/2)},

where the t_{k} are formal generators for the tangent bundle of the manifold X.

6.2 C^{∗}–ALGEBRAS AND K–THEORY

The space Γ(ξ) of sections of a vector bundle ξ 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 Γ(\overline{1}) of the one dimensional trivial vector bundle \overline{1}. Hence Γ(\overline{1}) is a one dimensional free module. Hence for an arbitrary vector bundle ξ, the module Γ(ξ) is a finitely generated projective module.

Theorem 56 The correspondence ξ ↦ Γ(ξ) is an equivalence between the category of complex vector bundles over a compact base X and the category of finitely generated projective modules over the algebra C(B).

Proof.

If two vector bundles ξ and η are isomorphic then the modules Γ(ξ) and Γ(η) of sections are isomorphic. Therefore, it is sufficient to prove that any finitely generated projective module P over the C^{∗}–algebra C(B) is isomorphic to a module Γ(ξ) for a vector bundle ξ, and any homomorphism

φ : Γ(ξ)⟶Γ(η)  (6.11)

is induced by a homomorphism of bundles

f : ξ⟶η.

We start by proving the second statement. Let (6.11) be a homomorphism of modules. Assume first that both vector bundles ξ and η are trivial, that is,

ξ = B × C^{n}, η = B × C^{m}.

Let

a_{1}, …, a_{n} ∈ C^{n}
b_{1}, …, b_{m} ∈ C^{m}

be bases. The vectors a_{k}, b_{k} induces constant sections

ā_{k} ∈ Γ(ξ), b̄_{k} ∈ Γ(η).

Hence any sections σ ∈ Γ(ξ), τ ∈ Γ(η) decompose into linear combinations

σ = ∑_{k} σ^{k}(x)ā_{k},  σ^{k}(x) ∈ C(B),
τ = ∑_{l} τ^{l}b̄_{l},  τ^{l}(x) ∈ C(B).

Therefore, the homomorphism φ has the following form

φ(σ) = ∑_{k} σ_{k}(x)φ(ā_{k}) = ∑_{k} σ^{k}(x) (∑_{l} φ_{k}^{l}(x)b̄_{l}) =

= ∑_{k,l} σ^{k}(x)φ_{k}^{l}(x)b̄_{l}.

The matrix valued function φ_{k}^{l}(x) defines a fiberwise homomorphism of bundles

f = ∥φ_{k}^{l}(x)∥ : ξ⟶η.  (6.12)

Clearly, the homomorphism (6.12) induces the initial homomorphism (6.11).

Now let ξ and η be arbitrary vector bundles. Complement them to form trivial bundles, that is,

ξ ⊕ ξ_{1} = n̄, η ⊕ η_{1} = m̄.

Let

p : n̄⟶n̄, q : m̄⟶m̄

be projections onto the first summands ξ and η, respectively. Define a new homomorphism

ψ : Γ(n̄)⟶Γ(m̄)

by the matrix

ψ = ∥\begin{matrix}φ & 0 \\ 0 & 0 \end{matrix}∥ .

It is clear that

ψπ = κψ,

where π : Γ(n̄) and κ : Γ(m̄) are homomorphisms induced be p and q. We see that ψ is induced by a homomorphism of trivial vector bundles

g : n̄⟶m̄.

Hence the homomorphism g has the matrix form

g = ∥\begin{matrix}f & 0 \\ 0 & 0 \end{matrix}∥ ,

where f = qgp is the homomorphism from the bundle ξ to the vector bundle η which induces the homomorphism (6.11).

Now we prove the first statement. Let P be a finitely generated projective module over the algebra C(B). By definition the module P is the direct summand of the free module C(B)^{n} = Γ(n̄)

P ⊕ Q = Γ(n̄).

The module P is the image of a projection

φ : Γ(n̄)⟶Γ(n̄), φ^{2} = φ, P = Im φ.

The projection φ, a homomorphism of modules, is induced by a homomorphism g of the trivial vector bundle n̄. Hence g^{2} = g, that is, g is a fiberwise projection. Therefore, the image of g is a locally trivial vector bundle,

n̄ = ξ ⊕ η, ξ = Im g.

Thus P = Γ(ξ).

Theorem 56 says that the group K(B) is naturally isomorphic to the group K_{0}(C(B)) in the algebraic K–theory for the ring C(B) :

K(B)⟶K_{0}(C(B)).  (6.13)

We can generalize the situation considered in Theorem 56 to the case of a Cartesian product B = X × Y. Let ξ be a vector bundle over a base B. Fix a point y_{0} ∈ Y. Then the section space Γ(ξ, (X × y_{0})) is a projective module over the ring C(X). Thus we have a family of projective modules parametrized by points of Y. This leads us to the definition of a vector bundle over a base Y with fiber a projective module P = Γ(ξ, (X × y_{0})) and structure group the group of automorphisms of the module P.

More generally, let A be an arbitrary C^{∗}–algebra and P a projective A–module. Then a locally trivial bundle over the base Y with the structure group G = Aut_{A}(P) will be called a vector A–bundle. The corresponding Grothendieck group will be denoted by K_{A}(X). If X = pt is a one point space then the group K_{A}(pt) coincides with K_{0}(A) in algebraic K–theory. If B = X × Y then there is a natural isomorphism

K(X × 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 can also be generalized to the case of vector A–bundles. Namely, if ξ is a vector A–bundle over a base Y, then the section space Γ(ξ) is a projective module over the algebra C(Y, A) of continuous functions on the Y with values in A. Hence there is a natural isomorphism

K_{A}(Y)⟶K_{0}(C(Y, A)).

When B = X × Y,

C(X × Y) = C(Y, C(X))

and there is a commutative diagram

K(X × Y) ⟶ K_{0}(C(X × Y))
↓  ∥
K_{C(X)}(Y) ⟶ K_{0}(C(Y, C(X)))

The groups K_{A}(X) satisfy many properties similar to those of classical complex K–theory. The tensor product induces a natural homomorphism

K_{A}(X) ⊗ K(Y)⟶K_{A}(X × Y).

In particular, multiplication by the Bott element β ∈ K^{0}(S^{2}, s_{0}) defines an isomorphism

K_{A}(X, Y)⟶^{⊗β}K_{A}(X × D^{2}, (X × S^{1}) ∪ (Y × D^{2})),

and an isomorphism

K_{A}^{n}(X, Y)⟶^{⊗β}K_{A}^{n−2}(X, Y).

Consider the homomorphism

φ : K_{A}(pt) ⊗ K(X)⟶K_{A}(X),

induced by tensor multiplication. After tensoring with the rational number field Q, there is an isomorphism

φ^{−1} : K_{A}^{∗}(X) ⊗ Q = K^{∗}(X) ⊗ K_{A}^{∗}(pt) ⊗ Q.

Define

ch _{A} = (Id ⊗ ch )φ^{−1} : K_{A}^{∗}(X)⟶H^{∗} (X; K_{A}^{∗}(pt) ⊗ Q)  (6.15)

The homomorphism (6.15) is an analogue of the Chern character and fits into the following commutative diagram

K_{A}^{∗}(X) ⊗ K^{∗}(X)  ⟶^{⊗}  K_{A}^{∗}(X)
↓ch _{A} ⊗ ch  ↓ch _{A}
H^{∗} (X; K_{A}^{∗}(pt) ⊗ Q) ⊗ H^{∗}(X; Q) ⟶^{⊗} H^{∗} (X; K_{A}^{∗}(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 K_{C(H)}(X) it is sufficient to consider locally trivial bundles with free C(H)–module as fibers. In this case the structure group is GL(n, C(H)) — the group of invertible matrices with entries in C(H). The algebra End(C(H)^{n}) of n–dimensional 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

π : B(H)⟶C(H)

be the natural projection. Then the inverse image π^{−1}(GL(C(H)) coincides with the set F of Fredholm operators. The projection

π : 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

F = ⋃_{k} F_{k},

where F_{k} 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

π : 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

F̃ = (\begin{matrix}F & 0 \\ 0 & 1 \end{matrix}) ∈ 1 + Comp.

Hence there is an inclusion

GL(n, C) ⊂ 1 + Comp,

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_{C(H)}^{−1}(X, x_{0})⟶K^{0}(X, x_{0}).

Proof.

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⟶F_{0}.

Let {U_{α}} be an atlas of sufficiently small charts on X. Then there are functions

φ_{α} : U_{α}⟶Comp(H)  (6.17)

such that for any α and x ∈ U_{α} the operator F (x) + φ_{α}(x) is invertible. Put

ψ_{αβ}(x) = (F (x) + φ_{α}(x)) (F (x) + φ_{β}(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_{C(H)}^{−1}(X, x_{0})⟶K^{0}(X, x_{0}).  (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_{α} : U_{α}⟶1 + Comp

such that

ψ_{αβ}(x) = u_{α}^{−1}(x)u_{β}(x).

Using (6.18),

u_{α}^{−1}(x) (F (x) + φ_{α}(x)) = u_{β}^{−1}(x) (F (x) + φ_{β}(x)) = Φ(x).  (6.20)

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 Φ(x, t), x ∈ X, t ∈ [0, 1] such that

Φ(x, 1) = Φ(x), Φ(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 SX.

Now let us prove that (6.19) is surjective. Consider a system of the transition functions ψ_{αβ} : U_{αβ}⟶1 + Comp. Using the contractibility of the group GL(H), we can find a system of functions h_{α} : U_{α}⟶1 + Comp such that

ψ_{αβ}(x) = h_{α}(x)h_{β}^{−1}(x).

This means that

h_{α}(x) = h_{β}(x) + Comp.

Hence there is a Fredholm operator valued function F (x) such that

F (x) − h_{α}(x) ∈ Comp.

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 locally trivial vector bundles [Ker F′] and [Coker F′]. The element

index F = [Ker F′] − [Coker F′] ∈ K(X)  (6.22)

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).

6.3 FAMILIES OF ELLIPTIC OPERATORS

Consider a continuous family of Fredholm operators

F_{x} : H⟶H, x ∈ X,

parametrized by a compact space X. Then index F_{x} is a locally constant function and hence gives no new invariant. But if dim ker F_{x} ≡ const, the kernels ker F_{x} 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

index F {def \atop =} [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 σ_{x}(D),

index σ(D) ∈ K(X)

can be expressed in terms of the symbol of the operator σ(D). Let

p : M⟶X

be a locally trivial bundle for which fiber is a manifold Y. Let T_{Y}^{∗} M be the vector bundle annihilating the tangent vectors to the fibers and

π : T_{Y}^{∗} M⟶M

be the correspondent projection.

Let ξ_{1}, ξ_{2} be two vector bundles over M. Consider the infinite dimensional vector bundles p_{!}(ξ_{1}) and p_{!}(ξ_{2}) over X with fibers Γ (ξ_{1}, p^{−1}(x)) and Γ (ξ_{2}, p^{−1}(x)), respectively, over the point x ∈ X. Notice that p^{−1}(x) is diffeomorphic to manifold Y. Let p_{!}(ξ_{i})_{s} denote the bundle associated to p_{!}(ξ_{i}) by substituting for the fiber Γ (ξ_{i}, p^{−1}(x)) its completion by s-Sobolev norm. Then a homomorphism of bundles

σ(D) : p_{!}(ξ_{1})⟶p_{!}(ξ_{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

σ : π^{∗}(ξ_{1})⟶π^{∗}(ξ_{2}).  (6.24)

and that the homomorphism (6.24) is an isomorphism away from the zero section of the bundle T_{Y}^{∗} (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

σ(D) : p_{!}(ξ_{1})_{s}⟶p_{!}(ξ_{2})_{s−m}

and, using (6.24), an element

[σ] ∈ K_{c} (T_{Y}^{∗} (M)) .

Theorem 59 For an elliptic family (6.23),

index σ(D) = p_{!}[σ] ∈ K(X) ⊗ Q.

Proof.

Let us prove this theorem for the case when M = X × Y. Then there is a second projection

q : M⟶Y.

The crucial observation is that

Γ(ξ, M) = Γ (q_{!}(ξ), Y) = Γ (p_{!}(ξ), X) .

The family of pseudodifferential operators

σ(D) : p_{!}(ξ_{1}) = Γ(ξ_{1}, Y) × X⟶Γ(ξ_{2}, Y) × X  (6.25)

generates the homomorphism

Γ(σ(D)) : Γ (Γ(ξ_{1}, Y), X) = Γ(ξ_{1}, X × Y)⟶Γ(ξ_{2}, X × Y) = Γ (Γ(ξ_{1}, Y), X)

or

q_{!}(σ(D)) : Γ (Γ(ξ_{1}, X), Y) = Γ (q_{!}(ξ_{1}), Y) ⟶Γ(q_{!}(ξ_{2}), Y) = Γ (Γ(ξ_{1}, X), Y) . (6.26)

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 algebra C(X) are projective modules since they can be identified with spaces of sections of finite dimensional vector bundles:

Ker q_{!}(σ(D)) = Γ (Ker σ(D))
Coker q_{!}(σ(D)) = Γ (Coker σ(D))

Hence the index of the family of elliptic operators σ(D) as an element of the group K(X) is identical to the index of the elliptic operator q_{!}(σ(D)) as an element of the group K_{C(X)}(pt) = K(X).

On the other hand, the symbol σ of the family (6.25) generates a homomorphism of vector bundles

q_{!}(σ) : π^{∗}(q_{!}(ξ_{1}))⟶π^{∗}(q_{!}(ξ_{2}))

which is the symbol of a pseudodifferential operator (6.26) over C^{∗}–algebra C(X). Hence, in the elliptic case, the element [q_{!}(σ)] ∈ K_{C(X),c}(T^{∗}Y) is identified with [σ] by the isomorphism

K_{c}(T^{∗}Y × X) ≅ K_{C(X),c}(T^{∗}Y).

Thus to prove the theorem it is sufficient to generalize the Atiyah–Singer formula to the case of an elliptic operator over the C^{∗}–algebra C(X). This project 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 A–module 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, x, y ∈ M;

2. (x, x) ≥ 0, x ∈ M;

3. (λx, y) = λ(x, y), x, y ∈ M, λ ∈ A;

4. ∥x∥^{2} = ∥(x, x)∥, x ∈ M.

A model (infinite dimensional) Hilbert A–module l_{2}(A) can constructed as follows. By definition, a point x ∈ l_{2}(A) is a sequence

x = (x_{1}, x_{2}, …, x_{n}, …), x_{n} ∈ A

such that the series ∑_{k=1}^{∞} x_{k}x_{k}^{∗} converges in norm in the algebra A, that is, there is a limit

∥x∥^{2} = lim_{N⟶∞} ∑_{k=1}^{N} x_{k}x_{k}^{∗} ∈ A.

We define a ‘scalar’ product by

(x, y) =^{def} ∑_{k=1}^{∞} x_{k}y_{k}^{∗} ∈ A.  (6.27)

The following inequality

∥ ∑_{k} x_{k}y_{k}^{∗}∥^{2} ≤ ∥ ∑_{k} x_{k}x_{k}^{∗}∥ · ∥ ∑_{k} y_{k}y_{k}^{∗}∥.

implies that the series on the right hand side of (6.27) converges. The space l_{2}(A) has a decreasing sequence of subspaces

[l_{2}(A)]_{n} = {x : x_{1} = x_{2} = … = x_{n} = 0}.

A bounded A–linear operator

K : l_{2}(A)⟶l_{2}(A)

is said to be A–compact if

lim_{n⟶∞} ∥K|_{[l_{2}(A)]_{n}}∥ = 0.

A bounded A–linear operator

F : l_{2}(A)⟶l_{2}(A)

is said to be A–Fredholm if there exists a bounded A–linear operator

G : l_{2}(A)⟶l_{2}(A)

such that both F G − 1 and GF − 1 are A–compact.

This terminology is justified by the following properties:

1. Any finitely generated projective A–module admits the structure of a Hilbert module.

2. Let ξ be a A–bundle over a compact manifold X with fiber a finitely generated projective A–module. Then the Sobolev completion H^{s}(ξ) of the section space is isomorphic to a direct summand of l_{2}(A).

3. A pseudodifferential A–operator σ(D) of order m that maps from the section space Γ(ξ_{1}) to the section space Γ(ξ_{2}) can be extended to a bounded operator of the Sobolev spaces

σ(D) : H_{s}(ξ_{1})⟶H^{s−m}(ξ_{2}).  (6.28)

4. The natural inclusion

H^{s+1}(ξ) ↪ H^{s}(ξ)

is an A–compact operator.

5. If the operator (6.28) is elliptic then it is A–Fredholm.

6. If F is an A–Fredholm operator then it defines a homotopy invariant

index F ∈ K_{0}(A).

In fact, for such an operator F,

F : H_{1} = l_{2}(A)⟶H_{2} = l_{2}(A),

there exist decompositions

H_{1} = V_{10} ⊕ V_{11}, H_{2} = V_{20} ⊕ V_{21},

such that the corresponding matrix of the operator F has the form

F = ∥ \begin{matrix}F_{0} & 0 \\ 0 & F_{1} \end{matrix} ∥

where F_{1} is an isomorphism, the modules V_{10} and V_{20} are finitely generated projective modules and

index F = [V_{10}] − [V_{20}] ∈ K_{0}(A).

7. If σ(A) is an elliptic pseudodifferential A–operator on a compact manifold

X then

index σ(D) = p_{∗}([σ]) ∈ K_{0} ⊗ Q,

where

p : X⟶pt

and

p_{∗} : K_{A,c}(T^{∗}X)⟶K_{A}(pt) = K_{0}(A)

is the direct image in K_{A}–theory.

The proofs of these properties are very similar to those for classical pseudodifferential operators.

6.4 FREDHOLM REPRESENTATIONS AND ASYMPTOTIC REPRESENTATIONS OF DISCRETE GROUPS

Let

ρ : G⟶A

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(ρ) ∈ K_{A}(BG) giving a homomorphism

b : R(G; A)⟶K_{A}(BG),  (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(BG).

6.4.1 Fredholm representations

There is a so called relative version of representation theory associated with a homomorphism

φ : A_{1}⟶A_{2}.

We use the term relative representation for a triple ρ = (ρ_{1}, F, ρ_{2}) where

ρ_{1}, ρ_{2} : G⟶A_{1}

are two representations and F is an operator interlacing the representations φρ_{1} and φρ_{2}. In other words, F is an invertible element in A_{2} such that

F^{−1}φρ_{1}(g)F = φρ_{2}(g), g ∈ G.  (6.30)

The representations ρ_{1} and ρ_{2} define vector A_{1}–bundles over BG, ξ_{1} = b(ρ_{1}) and ξ_{2} = b(ρ_{2}). The associated A_{2}–bundles ξ_{1} ⊗_{φ} A_{2} and ξ_{2} ⊗_{φ} A_{2} are isomorphic:

b(F) : ξ_{1} ⊗_{φ} A_{1}⟶ξ_{2} ⊗_{φ} A_{2}.

The triple (ξ_{1}, b(F), ξ_{2}) defines an element of the relative K–theory

b(ρ) = (ξ_{1}, b(F), ξ_{2}) ∈ K_{φ}(BG).  (6.31)

On the other hand, if ρ_{1} = ρ_{2} then the condition (6.30) means that the triple (ρ_{1}, F, ρ_{2}) defines a classical representation ρ̄ of the group G×Z into the algebra A_{2}:

ρ̄(g) = φρ(g) ∈ A_{2}, g ∈ G;
ρ̄(a) = F ∈ A_{2}, where a ∈ Z is the generator.

Hence the representation ρ̄ defines an element

b(ρ̄) ∈ K_{A_{2}} (B(G × Z)) = K_{A_{2}} (BG × S^{1}) .  (6.32)

The two elements (6.31) and (6.32) are connected in the classical exact sequence for the homomorphism φ : A_{1}⟶A_{2}:

K_{A_{2}} (BG × S^{1}) ⟶ K_{A_{2}} (S(BG))  b(ρ)
∪ ↓= ∩
b(ρ̄)  K_{A_{2}}^{1}(BG) ⟶ K_{φ}(BG) ⟶ K_{A_{1}}(BG)

For an illustration let

φ : B(H)⟶B(H)/Comp(H) = C(H)  (6.33)

be the natural projection to the Calkin algebra. As the bundles ξ_{1} and ξ_{2} are trivial the isomorphism b(F) generates a continuous family of Fredholm operators, that is, an element of the group K(BG):

K_{φ}(BG) ∼ K(BG).

Thus we have a homomorphism

b : R(G; φ)⟶K(BG).  (6.34)

Relative representations for the projection (6.33) are called Fredholm representations of the group G. Since the group K(BG) 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 :

ρ_{1}F = F ρ_{2}; g ∈ G.

Then the representations ρ_{1} and ρ_{2} induce two finite dimensional representations in Ker F and Coker F. It is clear that

b(Ker F) − b(Coker F) = b(ρ_{1}, F, ρ_{2}) ∈ K(BG).

For example, consider the discrete group G = Z^{n}. We showed earlier that for any finite dimensional representation ρ the associated vector bundle b(ρ) is trivial, that is, the homomorphism

b : R(Z^{n})⟶K(BZ^{n})  (6.35)

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 BG can be represented by a smooth complete Riemannian manifold with nonpositive section curvature. Then the homomorphism

b : R(G, φ)⟶K(BG)

is an epimorphism.

When G = Z^{n} we have a special case of Theorem 60 since BZ^{n} is the torus with trivial sectional curvature. (see for example Mishchenko [?], Solov’yov [?], Kasparov [?])

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

ρ : G⟶U(N)

is a map that is not actually a representation but is an approximation defined by the conditions:

1. ρ(g^{−1}) = (ρ(g))^{−1}, g ∈ G,

2. ∥ρ(g_{1}g_{2}) − ρ(g_{1})ρ(g_{2})∥ ≤ ε for g_{1}, g_{2} ∈ K ⊂ G.

If K is a finite subset including a set of generators of the group G and the number ε is sufficiently small we can say that ρ is an almost representation of the group G. Then in the following section 3.4 we construct candidates for the transition functions on BG,

ϕ_{αβ} = ρ ∘ ψ_{αβ},  (6.36)

where ψ_{αβ} are the transition functions of the universal covering of BG. 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

∥ρ∥_{K} = sup_{g_{1},g_{2}∈K} ∥ρ(g_{1})ρ(g_{2}) − ρ(g_{1}g_{2})∥.

Consider an increasing sequence of integers

N_{k} : N_{k} < N_{k+1}, N_{k}⟶∞

and a sequence of maps

ρ = {ρ_{k} : G⟶U(N_{k})}.  (6.37)

The norms of matrices for different N_{k} can be compared by using the natural inclusions

U(N_{k})⟶U(N_{k+1}) : U(N_{k}) ∋ X ↦ X ⊕ 1_{N_{k+1}−N_{k}} ∈ U(N_{k+1}).

Then the sequence (6.37) is called an asymptotic representation if the following conditions hold:

1. ρ_{k}(g^{−1}) = (ρ_{k}(g))^{−1} , g ∈ G,

2. lim_{k⟶∞} ∥ρ_{k}∥_{K} = 0,

3. lim_{k⟶∞} ∥ρ_{k+1}(g) − ρ_{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 Fiberwise homotopy equivalence of vector bundles

Let S(ξ) be the bundle formed by all unit vectors in the total space of a vector bundle ξ over a base X. Two vector bundles ξ and η are said to be fiberwise homotopy equivalent if there are two fiberwise maps

f : S(ξ)⟶S(η); g : S(η)⟶S(ξ),

such that the two compositions

gf : S(ξ)⟶S(ξ); fg : S(η)⟶S(η)

are homotopic to the identity maps.

This definition is justified by the following theorem (Atiyah, [4]):

Theorem 61 If

f : X⟶Y

is a homotopy equivalence of closed compact manifolds, ν(X) and ν(Y) are normal bundles ( of the same dimension) of X and Y, respectively, then the bundles ν(X) and f^{∗}(ν(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 projection K(X)⟶J(X) can be solved by using the Adams cohomology operations Ψ^{k}. This description is reduced to the so called the Adams conjecture. Namely, let f(k) ∈ Z be an arbitrary positive function , where W_{f}(X) ⊂ K(X) is the subgroup generated by elements of the form

k^{f(k)} (Ψ^{k} − 1) y, y ∈ K(X).

If

W (X) = ⋂_{f} W_{f}(X).

Theorem 62 (Quillen [?], Sullivan [?])

J(X) = K(X)/W (X).

In particular, the group J(X) is finite.

6.5.2 Immersions of manifolds

Let

f : X⟶Y

be an immersion of a compact manifold. Then, clearly, the normal bundle ν(X⟶Y) has dimension

dim ν(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.

For example, for n = 2^{r}, the real projective space RP^{n} cannot be immersed into RP^{2n−2} since then the normal bundle ν = ν(RP^{n}⟶RP^{2n−2}) would have a nontrivial Stieffel–Whitney class w_{n−1}(ν). Hence, the dimension cannot equal n − 2.

More generally, let ν(X) be a normal bundle for which the dimension is determined by the dimension of Euclidean space R^{N} in which the manifold X is embedded. Put

k = min dim ν_{0}(X)

where the minimum is taken over all possible decompositions

ν = ν_{0}(X) ⊕ n̄,

where n̄ is a trivial bundle. The number k is called the geometric dimension of the normal bundle ν(X). It is clear that the manifold X cannot be embedded in the Euclidean space R^{dim 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) =^{def} 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

α : K^{∗}(X × Y)⟶K^{∗}(Y)  (6.38)

which is a homomorphism of modules over the ring K^{∗}(Y).

Clearly, all such transformations form a Z_{2}–graded group, which we denote by

K_{∗}(X) = K_{0}(X) ⊕ K_{1}(X).  (6.39)

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:

⟨α, y⟩ = α(y).

Theorem 63 Let X be an almost complex manifold. Then there is a homomorphism (Poincare duality)

D : K_{c}^{∗}(X)⟶K_{∗}(X)

satisfying the condition

⟨D(x), y⟩ = ⟨xy, D(1)⟩

Atiyah ([5]) noticed that each elliptic operator σ(D) defines an element of the homology K–theory dual to the element [α] ∈ K_{c}(T^{∗}X). The construction is as follows. A continuous family of elliptic operators

σ(D) ⊗ η : Γ(ξ_{1} ⊗ η_{y})⟶Γ(ξ_{2} ⊗ η_{y}), y ∈ Y

is constructed for each vector bundle η over X × Y. Then the index of this family induces a homomorphism

index : K(X × Y)⟶K(Y)

which is by definition an element of the homology group K_{∗}(X).

The notion of an elliptic operator can be extended as follows. Consider a Fredholm operator F : H⟶H. Assume that the algebra C(X) acts on the Hilbert space H in such a way such that F commutes with this action, modulo compact operators. Let ρ denote the action of the algebra. Clearly, the pair (F, ρ) gives a well defined family of Fredholm operators

F_{y} = F ⊗ η : H ⊗_{C(X)} Γ(η_{y})⟶H ⊗_{C(X)} Γ(η_{y}), y ∈ Y,

and hence defines a homomorphism

index : K(X × Y)⟶K(Y),

that is, an element

[F, ρ] ∈ K_{∗}(X).

Moreover, if f : X_{1}⟶X_{2} is a continuous map then the pair (F, ρ ∘ f^{∗}) defines an element of K_{∗}(X_{2}) for which

f_{∗} ([F, ρ]) = [F, ρ ∘ f^{∗}].

When Y_{2} = pt,

K_{∗}(X_{2}) = Z

and

[F, ρ ∘ f^{∗}] = index F.

This last formula in some sense explains why, in the Atiyah–Singer formula, the direct image appears.

6.5.4 Historical notes

The study of bundles was probably started by Poincaré with the study of nontrivial 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 Stieffel–Whitney 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 discovery by Leray of spectral sequences which were applied to the calculation of the homology groups of bundles. Another was at the discovery by Bott [13] of the periodicity of the homotopy groups of the unitary and orthogonal 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 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 .

\overline{INDEX}

Atiyah–Singer formula, 213 Atlas of charts, 3 Atlas, 46 Base of the bundle, 2 Bilinear form on the bundle, 30 Bott homomorphism, 148 Bundle, 2

KR–bundle, 163 KRS–bundle, 167 atlas of charts, 3 base, 2 bundle complex, 142 charts, 3 coordinate homeomorphism, 3 cross–section, 15 equivalent bundles, 11 fiber, 2 fiberspace, 2 inverse image, 18 isomorphism compatible with

the structure group, 11 locally trivial, 2 principal, 77 projection, 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 Stieffel–Whitney 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 Cross–section, 15 Difference construction, 144 Differential of the mapping, 50 Differential operator, 198

full symbol, 198

order, 198

principal symbol, 198

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 Hirzebruch–Pontryagin 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, 48–49 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 G–bundle principal, 77

Principal symbol, 198

Projection, 2

Projective coordinates, 66

Projective space, 65 projective 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 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 Stieffel–Whitney 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, 48–49 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 G–bundle, 77 Vector A–bundle, 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

REFERENCES

[1] Adams J.F., Vector fields on spheres. — Ann.Math., Vol.75, (1962) p. 603–632.

[2] Adams J.F., On the groups K(X), I. — Topology, Vol.2, (1963) p. 181–195. II. — Topology, Vol.3, (1965) p. 137–171. III. — Topology, Vol.3, (1965) p. 193–222.

[3] Anderson D.W., The real K– theory of classifying spaces . — Proc. Nat. Acad. Sci. U.S.A., Vol.51, (1964) p. 634–636.

[4] Atiyah M.F., Thom complexes. — Proc. London Math. Soc., Vol.11, (1961) No.3, p. 291– 310.

[5] Atiyah M.F., Global theory of elliptic operators. — Proc. Intern. Conf. on Functional Analysis and Related Topics (Tokyo 1969). Tokyo, Univ. Tokyo Press, (1970) p. 21–30.

[6] Atiyah M.F., K–theory. — Benjamin, New York, (1967)

[7] Atiyah M.F., Vector bundles and the Kunneth formula. — Topology, Vol.1, (1963) p. 245–248.

[8] Atiyah M.F., Bott R., On the periodicity theorem for complex vector bundles. — Acta Math., Vol.112, (1964) p. 229–247.

[9] Atiyah M.F., Singer I.M., The index of elliptic operators on compact manifolds. — Bull. Amer. Math. Soc., Vol.69, (1963) p. 422–433.

[10] Atiyah M.F., Singer I.M., The index of elliptic operators, I.

— Ann. of Math., Vol.87, (1968) No.2, p. 484–530. , II.

— p. 531–545. , III. — p. 546–604.

[11] Atiyah M.F., Hirzebruch F., Vector bundles and homogeneous spaces in differential geometry. — Amer. Math. Soc. Proc. Symp. Pure Math., Vol.3, (1961) p. 7–38.

[12] Borel A., Hirzebruch F., Characteristic classes and homogeneous spaces, I.. — Amer. J. Math., Vol.80, (1958) p. 458–538. , II.. — Vol.81, (1959) p. 315–382. , III.. — Vol.82, (1960) p. 491–504.

[13] Bott R., The stable homotopy of the classical groups. —

252 Vector bundles and their applications

Ann. Math., Vol.70, (1959) p. 313–337.

[14] Bott R., Lectures on K(X).

— Cambridge, Mass. Harvard Univ., (1962)

[15] Chern S.S., On the multiplication in the characteristic ring of a sphere bundle. — Ann. Math., Vol.49, (1948) p. 362–372.

[16] Connes A., Gromov M., Moscovici H., Conjecture de Novicov et fibres presque plats. — C.R.Acad.Sci. Paris, Vol.310, (1990) p. 273–277.

[17] Connes A., Moscovici H., Cyclic Cohomology, The Novikov Conjecture and Hyperbolic Groups.

— Topology, Vol.29, (1990) No.3, p. 345–388.

[18] Godement R., Topologie algébrique et theorie des faisceaux. — Hermann & Cie, Paris, (1958)

[19] Hirzebruch F., A Riemann– Roch theorem for differential manifolds. — Seminar Bourbaki, (1959) p. 177.

[20] Hirzebruch F., Topological Methods in Algebraic Geometry. — Berlin, Springer– Verlag, (1966)

[21] Hirsch M.W., Immersions of manifolds. — Trans. Amer. Math. Soc., Vol.93, (1959) p. 242–276.

[22] Husemoller D., Fiber Bundles. — McGraw–Hill, (1966)

[23] Karoubi M., K–Theory, An Introduction. — Springer–Verlag, Berlin, Heidelberg, New York, (1978)

[24] Kasparov G., Equivariant KK–theory and the Novikov conjecture . — Invent. Math., Vol.91, (1988) p. 147–201.

[25] Kuiper N.H., The homotopy type of the unitary group of Hibert space. — Topology, Vol.3, (1965) p. 19–30.

[26] Milnor J., Construction of universal bundles,I. — Ann. Math., Vol.63, (1965) p. 272–284. ,II. — p. 430–436.

[27] Milnor J., Stasheff J.D., Characteristic Classes. — Annals of Math. Studies No. 76, Princeton , (1974)

[28] Mishchenko A.S., Infinite dimensional representations of discrete groups and homotopy invariants of non simply connected manifolds. — Soviet math. survey, Vol.28, (1973) No.2, p. 239–240.

[29] Mishchenko A.S., On Fredholm representations of discrete groups. — Funk. an. and its applications, Vol.9, (1975) No.2, p. 229–256.

REFERENCES 253

[30] Mishchenko A.S., Fomenko A.T., Index of elliptic operators over C^{∗}–algebras .

— Math. USSR – Izvestia, Vol.15, (1980) p. 87–112.

[31] Mishchenko A.S., Solov’yov Ju.P., Representations of Banach algebras and Hirzebruch type formulas. — Math.USSR– sbornik, Vol.39, (1981) p. 189–205.

[32] Novikov S.P., Pontrjagin classes, the fundamental group, and some problems of stable algebra. — In “Essay on Topology and Related Topics. Mémoires dédiés à Georges de Rham”, Springer, (1970) p. 147–155.

[33] Novikov S.P., Algebraic construction and properties of Hermitian analogues of K– theory over rings with involution from the point of view of Hamilton formalism,I. . — Izvestia Acad. Nauk SSSR, sert. mat., Vol.34, (1970) p. 253–288. ,II. . — p. 475–500.

[34] Palais R., Seminar on the Atiyah–Singer index theorem .

— Ann. of Math. Stud. 57, Princeton Univ. Press, Princeton, N.J., (1965)

[35] Paschke W., Inner product modules over B^{∗}–algebras.

— Trans. Amer. Math. Soc., Vol.182, (1973) p. 443–468.

[36] Pontryagin L.S., Characteristic cycles of differentiable manifolds. — Doklady Acad.Sci. USSR, Vol.19, (1938) p. 361–363.

[37] Pontryagin L.S., Classification of some skew products.

— Doklady Acad. Sci. USSR, Vol.43, (1944) p. 91–94.

[38] Puppe D., Homotopiemengen und ihre induzierten Abbildungen. — Math. Z., Vol.69, (1958) p. 299–344.

[39] Quillen D., Cohomology of groups. — Actes Congres internat. Math., Vol.2, (1970) p. 47–51.

[40] Quillen D., The Adams conjecture. — Topology, Vol.10, (1971) p. 67–80.

[41] Rieffel, M.A. Morita equivalence for C^{∗}–algebras and W^{∗}– algebras. — J.Pure Appl. Algebra, Vol.5, (1974) p. 51– 96.

[42] Rochlin V.A., Three dimensional manifold is a boundary of four dimensional one.

— Dokl. Akad. Nauk SSSR, Vol.81, (1951) p. 355–357.

[43] Rochlin V.A., New results in the theory of four dimensional manifolds. — Doklady Akad. Nauk. USSR, Vol.84, (1952) p. 221–224.

[44] Serre J.-P., Homologie singulière des espaces fibrès. —

254 Vector bundles and their applications

Ann. Math., Vol.54, (1951) p. 425–505.

[45] Shih Weishu, Fiber cobordism and the index of elliptic differential operators. — Bull. Amer. Math. Soc., Vol.72, (1966) No.6, p. 984–991.

[46] Solov’yov Ju.P., Discrete groups, Bruhat–Tits complexes and homotopy invariance of higher signatures. — Uspechi Math. Nauk, Vol.31, No.1, (1976) p. 261–262.

[47] Spanier E.H., Algebraic Topology. — N.Y., San Francisko, St. Luis, Toronto, London, Sydney, Mc Graw–Hill, (1966)

[48] Steenrod N.F., The topology of fiber bundles. — Princeton Univ. Press, Princeton, New Jersey, (1951)

[49] Steenrod N.F., Classification of sphere bundles. — Ann. Math., Vol.45, (1944) p. 294–311.

[50] Stiefel E., Richtungsfelder und Fernparallelismus in Mannigfaltigkeiten. — Comm. Math. Helv., Vol.8, (1936) p. 3–51.

[51] Sullivan D., Geometric Topology. — MIT, (1970)

[52] Whitney H., Sphere spaces. — Proc. Nat. Acad. Sci. U.S.A., Vol.21, (1935) p. 462–468.

[53] Whitney H., On the theory of sphere bundles. — Proc. Nat. Acad. sci. U.S.A., Vol.26, (1940) No.26, p. 148–153.

[54] Wood R., Banach algebras and Bott periodicity. — Topology, Vol.4, (1966) p. 371–389.