Hatcher A. - Vector Bundles and K-Theory.pdf - Topologia - besia86

Version 1.3, July 2001

Allen Hatcher

2001 by Allen Hatcher

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Table of Contents

Chapter 1. Vector Bundles

1.1. Basic Deﬁnitions and Constructions ............ 1

Sections 3. Direct Sums 5. Pullback Bundles 5. Inner Products 7.

Subbundles 8. Tensor Products 9. Associated Bundles 11.

1.2. Classifying Vector Bundles ................. 12

The Universal Bundle 12. Vector Bundles over Spheres 16.

Orientable Vector Bundles 21. A Cell Structure on Grassmann Manifolds 22.

Appendix: Paracompactness 24.

Chapter 2. Complex K-Theory

2.1. The Functor K(X) ....................... 28

Ring Structure 31. Cohomological Properties 32.

2.2. Bott Periodicity ........................ 38

Clutching Functions 38. Linear Clutching Functions 43.

Conclusion of the Proof 45.

2.3. Adams’ Hopf Invariant One Theorem ........... 48

Adams Operations 51. The Splitting Principle 55.

2.4. Further Calculations ..................... 61

The Thom Isomorphism 61.

Chapter 3. Characteristic Classes

3.1. Stiefel-Whitney and Chern Classes ............ 63

Axioms and Construction 64. Cohomology of Grassmannians 69.

Applications of w 1 and c 1 72.

3.2. The Chern Character ..................... 73

The J–Homomorphism 76.

3.3. Euler and Pontryagin Classes ................ 83

The Euler Class 87. Pontryagin Classes 90.

1. Basic Deﬁnitions and Constructions

Vector bundles are special sorts of ﬁber bundles with additional algebraic struc-

ture. Here is the basic deﬁnition. An n dimensional vector bundle is a map p : E

B , such that the

following local triviality condition is satisﬁed: There is a cover of B by open sets

U for each of which there exists a homeomorphism h : p − 1 U

U R

n taking

U . Such an h is

called a local trivialization of the vector bundle. The space B is called the base space ,

E is the total space , and the vector spaces p − 1 b are the ﬁbers . Often one abbrevi-

ates terminology by just calling the vector bundle E , letting the rest of the data be

implicit. We could equally well take

g R

n by a vector space isomorphism for each b

in place of

as the scalar ﬁeld here, obtaining

the notion of a complex vector bundle .

If we modify the deﬁnition by dropping all references to vector spaces and replace

n by an arbitrary space F , then we have the deﬁnition of a ﬁber bundle: a map

p : E

B such that there is a cover of B by open sets U for each of which there

exists a homeomorphism h : p − 1 U

F taking p − 1 b to

F for each

U .

Here are some examples of vector bundles:

(1) The product or trivial bundle E

n with p the projection onto the ﬁrst

factor.

(2) If we let E be the quotient space of I

under the identiﬁcations 0 ;t

1 ;

−

t ,

S 1 which is a 1 dimensional vector

bundle, or line bundle . Since E is homeomorphic to a Mobius band with its boundary

circle deleted, we call this bundle the Mobius bundle .

(3) The tangent bundle of the unit sphere S n in

R !

I induces a map p : E

1 , a vector bundle p : E

S n

and we think of v as a tangent vector to

S n by translating it so that its tail is at the head of x ,on S n . The map p : E

x;v

S n

n 1

S n

together with a real vector space structure on p − 1 b for each b

p − 1 b to

then the projection I

where E

Chapter 1

Vector Bundles

sends x;v to x . To construct local trivializations, choose any point b

S n and

S n be the open hemisphere containing b and bounded by the hyperplane

through the origin orthogonal to b . Deﬁne h b : p − 1 U b

U b

p − 1 b

U b R

n by

x; b v where b is orthogonal projection onto the tangent plane

p − 1 b . Then h b is a local trivialization since b restricts to an isomorphism of

p − 1 x onto p − 1 b for each x

U b .

(4) The normal bundle to S n in

1 , a line bundle p : E

S n with E consisting of

pairs x;v

S n

such that v is perpendicular to the tangent plane to S n at

x , i.e., v

tx for some t

2 R

. The map p : E

S n is again given by px;v

x .As

in the previous example, local trivializations h b : p − 1 U b

U b R

can be obtained

by orthogonal projection of the ﬁbers p − 1 x onto p − 1 b for x

U b .

(5) The canonical line bundle p : E

! R P n . Thinking of R P n as the space of lines in

through the origin, E is the subspace of

P n

consisting of pairs `;v

` . Again local trivializations can be deﬁned by orthogonal

projection. We could also take n

` , and p`;v

and get the canonical line bundle E

! R P 1 .

(6) The orthogonal complement E ? f

`;v

2 R

P n

of the canonical

! R P n , p`;v` , is a vector bundle with ﬁbers

the orthogonal subspaces ` ? , of dimension n . Local trivializations can be obtained

once more by orthogonal projection.

An isomorphism between vector bundles p 1 : E 1

B and p 2 : E 2

B over the same

E 2 taking each ﬁber p − 1 b to the cor-

responding ﬁber p − 2 b by a linear isomorphism. Thus an isomorphism preserves

all the structure of a vector bundle, so isomorphic bundles are often regarded as the

same. We use the notation E 1

E 2 to indicate that E 1 and E 2 are isomorphic.

For example, the normal bundle of S n in

n 1

is isomorphic to the product bun-

dle S n

by the map x;tx

x;t . The tangent bundle to S 1

is also isomorphic

As a further example, the Mobius bundle in (2) above is isomorphic to the canon-

ical line bundle over

, via e i ;ite i

e i ;t , for e i

S 1

and t

2 R

P 1 is swept out by a line rotating through

an angle of , so the vectors in these lines sweep out a rectangle 0 ;

P 1

S 1 . Namely,

with the

two ends

g R

and

g R

identiﬁed. The identiﬁcation is 0 ;x

;

−

x since

rotating a vector through an angle of produces its negative.

B is the union of the zero vectors in

all the ﬁbers. This is a subspace of E which projects homeomorphically onto B by

p . Moreover, E deformation retracts onto its zero section via the homotopy f t v

The zero section of a vector bundle p : E

−

tv given by scalar multiplication of vectors v

E . Thus all vector bundles over

B have the same homotopy type.

One can sometimes distinguish nonisomorphic bundles by looking at the comple-

ment of the zero section since any vector bundle isomorphism h : E 1

E 2 must take

let U b

h b x;v

with v

line bundle. The projection p : E ?

base space B is a homeomorphism h : E 1

to the trivial bundle S 1

Basic Deﬁnitions and Constructions

Section 1.1 3

since the complement of the zero section

in the Mobius bundle is connected while for the product bundle the complement of

the zero section is not connected. This method for distinguishing vector bundles can

also be used with more reﬁned topological invariants such as H n in place of H 0 .

We shall denote the set of isomorphism classes of n dimensional real vector

bundles over B by Vect n B , and its complex analogue by Vect n

B . For those who

worry about set theory, we are using the term ‘set’ here in a naive sense. It follows

from Theorem 1.8 later in the chapter that Vect n B and Vect n

B are indeed sets in

the strict sense when B is paracompact.

For example, Vect 1 S 1 contains exactly two elements, the Mobius bundle and the

product bundle. This will be a rather trivial application of later theory, but it might

be an interesting exercise to prove it now directly from the deﬁnitions.

Sections

A section of a bundle p : E

B is a map s : B

E such that ps

1 , or equivalently,

B . We have already mentioned the zero section, which

is the section whose values are all zero. At the other extreme would be a section

whose values are all nonzero. Not all vector bundles have such a nonvanishing section.

Consider for example the tangent bundle to S n . Here a section is just a tangent vector

ﬁeld to S n . One of the standard ﬁrst applications of homology theory is the theorem

that S n has a nonvanishing vector ﬁeld iff n is odd. From this it follows that the

tangent bundle of S n is not isomorphic to the trivial bundle if n is even and nonzero,

since the trivial bundle obviously has a nonvanishing section, and an isomorphism

between vector bundles takes nonvanishing sections to nonvanishing sections.

In fact, an n dimensional bundle p : E

p − 1 b for all b

B is isomorphic to the trivial bundle iff

;s n b are linearly independent in

each ﬁber p − 1 b . For if one has such sections s i , the map h : B

;s n such that s 1 b;

E given by

P i t i s i b is a linear isomorphism in each ﬁber, and is continuous,

as can be veriﬁed by composing with a local trivialization p − 1 U

hb;t 1 ;

;t n

n . Hence h

is an isomorphism by the following useful technical result:

E 2 between vector bundles over the same

base space B is an isomorphism if it takes each ﬁber p − 1

1 b to the corresponding

ﬁber p − 1

2 b by a linear isomorphism.

P roof : The hypothesis implies that h is one-to-one and onto. What must be checked

is that h − 1

is continuous. This is a local question, so we may restrict to an open set

B over which E 1 and E 2 are trivial. Composing with local trivializations reduces

to the case of an isomorphism h : U

n of the form hx;v

x;g x v .

the zero section of E 1 onto the zero section of E 2 , hence the complements of the zero

sections in E 1 and E 2 must be homeomorphic. For example, the Mobius bundle is not

isomorphic to the product bundle S 1

it has n sections s 1 ;

L emma 1.1. A continuous map h : E 1

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