1.1 Basic Definitions and Constructions

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Vector bundles are special sorts of fiber bundles with additional algebraic structure.

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

together with a real vector space structure on p−1.b. for each b 2 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_

_Rn taking

p−1.b. to fbg_Rn by a vector space isomorphism for each b 2 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 abbreviates

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

implicit. We could equally well take C in place of R 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

Rn 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_.!U_

_F taking p−1.b. to fbg_F for each

b 2 U_ .

Here are some examples of vector bundles:

(1) The product or trivial bundle E . B_Rn with p the projection onto the first

factor.

(2) If we let E be the quotient space of I_R under the identifications .0; t. _ .1;−t.,

then the projection I_R!I induces a map p : E!S1 which is a 1 dimensional vector

bundle, or line bundle. Since E is homeomorphic to a M¨obius band with its boundary

circle deleted, we call this bundle the M¨obius bundle.

(3) The tangent bundle of the unit sphere Sn in Rn.1 , a vector bundle p : E!Sn

where E . f.x;v. 2 Sn_Rn.1 j x ? v g and we think of v as a tangent vector to

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

2 Chapter 1 Vector Bundles

sends .x;v. to x. To construct local trivializations, choose any point b 2 Sn and

let Ub

_ Sn be the open hemisphere containing b and bounded by the hyperplane

through the origin orthogonal to b. Define hb :p−1.Ub.!Ub

_p−1.b. _ Ub

_Rn by

hb.x;v. . .x;_b.v.. where _b is orthogonal projection onto the tangent plane

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

p−1.x. onto p−1.b. for each x 2 Ub .

(4) The normal bundle to Sn in Rn.1 , a line bundle p : E!Sn with E consisting of

pairs .x;v. 2 Sn_Rn.1 such that v is perpendicular to the tangent plane to Sn at

x, i.e., v . tx for some t 2 R. The map p : E!Sn is again given by p.x;v. . x. As

in the previous example, local trivializations hb :p−1.Ub.!Ub

_R can be obtained

by orthogonal projection of the fibers p−1.x. onto p−1.b. for x 2 Ub .

(5) The canonical line bundle p : E!RPn . Thinking of RPn as the space of lines in

Rn.1 through the origin, E is the subspace of RPn_Rn.1 consisting of pairs .`;v.

with v 2 `, and p.`;v. . `. Again local trivializations can be defined by orthogonal

projection. We could also take n.1 and get the canonical line bundle E!RP1.

(6) The orthogonal complement E? . f.`;v. 2 RPn_Rn.1 j v ? ` g of the canonical

line bundle. The projection p : E?!RPn , 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 p1 : E1!B and p2 : E2!B over the same

base space B is a homeomorphism h: E1!E2 taking each fiber p−1

1 .b. to the corresponding

fiber p−1

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 E1 _ E2 to indicate that E1 and E2 are isomorphic.

For example, the normal bundle of Sn in Rn.1 is isomorphic to the product bundle

Sn_R by the map .x; tx.,.x; t.. The tangent bundle to S1 is also isomorphic

to the trivial bundle S1_R, via .ei_; itei_.,.ei_; t., for ei_ 2 S1 and t 2 R.

As a further example, the M¨obius bundle in (2) above is isomorphic to the canonical

line bundle over RP1 _ S1 . Namely, RP1 is swept out by a line rotating through

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

two ends f0g_R and f_g_R identified. The identification is .0;x. _ ._;−x. since

rotating a vector through an angle of _ produces its negative.

The zero section of a vector bundle p : E!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 ft.v. .

.1−t.v given by scalar multiplication of vectors v 2 E. Thus all vector bundles over

B have the same homotopy type.

One can sometimes distinguish nonisomorphic bundles by looking at the complement

of the zero section since any vector bundle isomorphism h: E1!E2 must take

Basic Definitions and Constructions Section 1.1 3

the zero section of E1 onto the zero section of E2 , hence the complements of the zero

sections in E1 and E2 must be homeomorphic. For example, the M¨obius bundle is not

isomorphic to the product bundle S1_R since the complement of the zero section

in the M¨obius 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 Hn in place of H0 .

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

bundles over B by Vectn.B., and its complex analogue by VectnC

.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 Vectn.B. and VectnC

.B. are indeed sets in

the strict sense when B is paracompact.

For example, Vect1.S1. contains exactly two elements, the M¨obius 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 . 11, or equivalently,

s.b. 2 p−1.b. for all b 2 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 Sn . Here a section is just a tangent vector

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

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

tangent bundle of Sn 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!B is isomorphic to the trivial bundle iff

it has n sections s1; ___ ; sn such that s1.b.; ___ ; sn.b. are linearly independent in

each fiber p−1.b.. For if one has such sections si , the map h: B_Rn!E given by

h.b; t1; ___ ; tn. .

P

i tisi.b. is a linear isomorphism in each fiber, and is continuous,

as can be verified by composing with a local trivialization p−1.U.!U_Rn . Hence h

is an isomorphism by the following useful technical result:

Lemma 1.1. A continuous map h: E1!E2 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.

Proof: 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

U _ B over which E1 and E2 are trivial. Composing with local trivializations reduces

to the case of an isomorphism h:U_Rn!U_Rn of the form h.x;v. . .x;gx.v...

4 Chapter 1 Vector Bundles

Here gx is an element of the group GLn.R. of invertible linear transformations of

Rn which depends continuously on x. This means that if gx is regarded as an n_n

matrix, its n2 entries depend continuously on x. The inverse matrix g−1

x also depends

continuously on x since its entries can be expressed algebraically in terms of the

entries of gx , namely, g−1

x is 1=.det gx. times the classical adjoint matrix of gx .

Therefore h−1.x;v. . .x;g−1

x .v.. is continuous. tu

As an example, the tangent bundle to S1 is trivial because it has the section

.x1;x2.,.−x2;x1. for .x1;x2. 2 S1 . In terms of complex numbers, if we set

z . x1 . ix2 then this section is z,iz since iz . −x2 .ix1 .

There is an analogous construction using quaternions instead of complex numbers.

Quaternions have the form z . x1.ix2.jx3.kx4 , and form a division algebra

H via the multiplication rules i2 . j2 . k2 . −1, ij . k, jk . i, ki . j , ji . −k,

kj . −i, and ik . −j. If we identify H with R4 via the coordinates .x1;x2;x3;x4.,

then the unit sphere is S3 and we can define three sections of its tangent bundle by

the formulas

z,iz or .x1;x2;x3;x4.,.−x2;x1;−x4;x3.

z,jz or .x1;x2;x3;x4.,.−x3;x4;x1;−x2.

z,kz or .x1;x2;x3;x4.,.−x4;−x3;x2;x1.

It is easy to check that the three vectors in the last column are orthogonal to each other

and to .x1;x2;x3;x4., so we have three linearly independent nonvanishing tangent

vector fields on S3 , and hence the tangent bundle to S3 is trivial.

The underlying reason why this works is that quaternion multiplication satisfies

jzwj . jzjjwj , where j_j is the usual norm of vectors in R4 . Thus multiplication by a

quaternion in the unit sphere S3 is an isometry of H. The quaternions 1; i; j; k form

the standard orthonormal basis for R4 , so when we multiply them by an arbitrary unit

quaternion z 2 S3 we get a new orthonormal basis z; iz; jz; kz.

The same constructions work for the Cayley octonions, a division algebra structure

on R8 . Thinking of R8 as H_H, multiplication of octonions is defined by

.z1; z2..w1;w2. . .z1w1−w2z2; z2w1.w2z1. and satisfies the key property jzwj .

jzjjwj . This leads to the construction of seven orthogonal tangent vector fields on

the unit sphere S7 , so the tangent bundle to S7 is also trivial. As we shall show in

x2.3, the only spheres with trivial tangent bundle are S1 , S3 , and S7 .

One final general remark before continuing with our next topic: Another way of

characterizing the trivial bundle E _ B_Rn is to say that there is a continuous projection

map E!Rn which is a linear isomorphism on each fiber, since such a projection

together with the bundle projection E!B gives an isomorphism E _ B_Rn .

Basic Definitions and Constructions Section 1.1 5

Direct Sums

As a preliminary to defining a direct sum operation on vector bundles, we make

two simple observations:

(a) Given a vector bundle p : E!B and a subspace A _ B, then p :p−1.A.!A is

clearly a vector bundle. We call this the restriction of E over A .

(b) Given vector bundles p1 : E1!B1 and p2 : E2!B2 , then p1_p2 : E1_E2!B1_B2

is also a vector bundle, with fibers the products p−1

1 .b1._p−1

2 .b2.. For if we have

local trivializations h_ :p−1

1 .U_.!U_

_Rn and h_ :p−1

2 .U_.!U_

_Rm for E1 and

E2 , then h_

_h_ is a local trivialization for E1_E2 .

Now suppose we are given two vector bundles p1 : E1!B and p2 : E2!B over

the same base space B. The restriction of the product E1_E2 over the diagonal B .

f.b; b. 2 B_Bg is then a vector bundle, called the direct sum E1

_E2!B. Thus

E1

_E2 . f.v1; v2. 2 E1_E2 j p1.v1. . p2.v2. g

The fiber of E1

_E2 over a point b 2 B is the product, or direct sum, of the vector

spaces p−1

1 .b. and p−1

2 .b..

The direct sum of two trivial bundles is again a trivial bundle, clearly, but the

direct sum of nontrivial bundles can also be trivial. For example, the direct sum of

the tangent and normal bundles to Sn in Rn.1 is the trivial bundle Sn_Rn.1 since

elements of the direct sum are triples .x; v; tx. 2 Sn_Rn.1_Rn.1 with x ? v , and

the map .x; v; tx.,.x;v .tx. gives an isomorphism of the direct sum bundle with

Sn_Rn.1 . So the tangent bundle to Sn is stably trivial: it becomes trivial after taking

the direct sum with a trivial bundle.

As another example, the direct sum E_E? of the canonical line bundle E!RPn

with its orthogonal complement, defined in example (6) above, is isomorphic to the

trivial bundle RPn_Rn.1 via the map .`; v;w.,.`;v . w. for v 2 ` and w ? `.

Specializing to the case n . 1, both E and E? are isomorphic to the M¨obius bundle

over RP1 . S1 , so the direct sum of the M¨obius bundle with itself is the trivial bundle.

This is just saying that if one takes a slab I_R2 and glues the two faces f0g_R2 and

f1g_R2 to each other via a 180 degree rotation of R2 , the resulting vector bundle

over S1 is the same as if the gluing were by the identity map. In effect, one can

gradually decrease the angle of rotation of the gluing map from 180 degrees to 0

without changing the vector bundle.

Pullback Bundles

Next we describe a procedure for using a map f :A!B to transform vector

bundles over B into vector bundles over A. Given a vector bundle p : E!B, let

6 Chapter 1 Vector Bundles

f _.E. . f.a;v. 2 A_E j f .a. . p.v. g. This subspace of A_E fits into the commutative

diagram at the right where _.a;v. . a and e f .a;v. . v . It is

¡!

¡!

f E ¡¡¡¡¡!E

A¡¡¡¡¡!B f

f

¼ p

¤

»

not hard to see that _ : f ( ) _.E.!A is also a vector bundle with fibers

of the same dimension as in E. For example, we could say that

f _.E. is the restriction of the vector bundle 11_p :A_E!A_B

over the graph of f , f.a; f .a.. 2 A_Bg, which we identify with A via the projection

.a; f .a..,a. The vector bundle f _.E. is called the pullback or induced bundle.

As a trivial example, if f is the inclusion of a subspace A _ B, then f _.E. is

isomorphic to the restriction p−1.A. via the map .a;v.,v , since the condition

f .a. . p.v. just says that v 2 p−1.a.. So restriction over subspaces is a special

case of pullback.

An interesting example which is small enough to be visualized completely is the

pullback of the M¨obius bundle E!S1 by the two-to-one covering map f : S1!S1 ,

f.z. . z2 . In this case the pullback f _.E. is a two-sheeted covering space of E

which can be thought of as a coat of paint applied to ‘both sides’ of the M¨obius bundle.

Since E has one half-twist, f _.E. has two half-twists, hence is the trivial bundle. More

generally, if En is the pullback of the M¨obius bundle by the map z,zn , then En is

the trivial bundle for n even and the M¨obius bundle for n odd.

Some elementary properties of pullbacks, whose proofs are one-minute exercises

in definition-chasing, are:

(i) .fg._.E. _ g_.f _.E...

(ii) If E1 _ E2 then f _.E1. _ f _.E2..

(iii) f _.E1

_E2. _ f _.E1._f _.E2..

Now we come to our first important result:

Theorem 1.2. Given a vector bundle p : E!B and homotopic maps f0; f1 :A!B,

then the induced bundles f _

0 .E. and f _

1 .E. are isomorphic if A is paracompact.

All the spaces one ordinarily encounters in algebraic and geometric topology are

paracompact, for example compact Hausdorff spaces and CW complexes; see the Appendix

to this chapter for more information about this.

Proof: Let F :A_I!B be a homotopy from f0 to f1 . The restrictions of F_.E. over

A_f0g and A_f1g are then f _

0 .E. and f _

1 .E.. So the theorem will follow from: tu

Proposition 1.3. The restrictions of a vector bundle E!X_I over X_f0g and

X_f1g are isomorphic if X is paracompact.

Proof: We need two preliminary facts:

(1) A vector bundle p : E!X_.a; b. is trivial if its restrictions over X_.a; c. and

X_.c; b. are both trivial for some c 2 .a; b.. To see this, let these restrictions

be E1 . p−1.X_.a; c.. and E2 . p−1.X_.c; b.., and let h1 : E1!X_.a; c._Rn

Basic Definitions and Constructions Section 1.1 7

and h2 : E2!X_.c; b._Rn be isomorphisms. These isomorphisms may not agree on

p−1.X_fcg., but they can be made to agree by replacing h2 by its composition with

the isomorphism X_.c; b._Rn!X_.c; b._Rn which on each slice X_fxg_Rn is

given by h1h−1

2 :X_fcg_Rn!X_fcg_Rn . Once h1 and h2 agree on E1 \ E2 , they

define a trivialization of E.

(2) For a vector bundle p : E!X_I , there exists an open cover fU_

g of X so that each

restriction p−1.U_

_I.!U_

_I is trivial. This is because for each x 2 X we can find

open neighborhoods Ux;1; ___;Ux;k in X and a partition 0 . t0 < t1 < ___ < tk

. 1 of

.0;1. such that the bundle is trivial over Ux;i

_.ti−1; ti., using compactness of .0; 1..

Then by (1) the bundle is trivial over U_

_I where U_

. Ux;1 \ ___ \ Ux;k .

Now we prove the proposition. By (2), we can choose an open cover fU_

g of X so

that E is trivial over each U_

_I . Lemma 1.19 in the Appendix to this chapter asserts

that there is a countable cover fVk

g

k_1 of X and a partition of unity f'k

g with 'k

supported in Vk , such that each Vk is a disjoint union of open sets each contained in

some U_ . This means that E is trivial over each Vk

_I .

For k _ 0, let k

. '1 . ___ .'k, with 0 . 0. Let Xk be the graph of k ,

so Xk

. f.x; k.x.. 2 X_I g, and let pk : Ek!Xk be the restriction of the bundle

E over Xk . Choosing a trivialization of E over Vk

_I , the natural projection

homeomorphism Xk!Xk−1 lifts to an isomorphism hk : Ek!Ek−1 which is the identity

outside p−1

k .Vk.. The infinite composition h . h1h2 ___ is then a well-defined

isomorphism from the restriction of E over X_f0g to the restriction over X_f1g

since near each point x 2 X only finitely many 'i ’s are nonzero, which implies that

for large enough k, hk

. 11 over a neighborhood of x. tu

Corollary 1.4. A homotopy equivalence f :A!B of paracompact spaces induces a

bijection f _ : Vectn.B.!Vectn.A.. In particular, every vector bundle over a contractible

paracompact base is trivial.

Proof: If g is a homotopy inverse of f then we have f _g_ . 11_ . 11 and g_f _ .

11_ . 11. tu

Theorem 1.2 holds for fiber bundles as well as vector bundles, with the same

proof.

Inner Products

An inner product on a vector bundle p : E!B is a map h ; i : E_E!R which

restricts in each fiber to an inner product, i.e., a positive definite symmetric bilinear

form.

Proposition 1.5. An inner product exists for a vector bundle p : E!B if B is paracompact.

8 Chapter 1 Vector Bundles

Proof: An inner product for p : E!B can be constructed by first using local trivializations

h_ :p−1.U_.!U_

_Rn , to pull back the standard inner product in Rn to an

inner product h_; _i

_ on p−1.U_., then setting hv;wi .

P

_'_p.v.hv;wi

_._. where

f'_

g is a partition of unity with the support of '_ contained in U_._. . tu

In the case of complex vector bundles one can construct Hermitian inner products

in the same way.

Having an inner product on a vector bundle E, lengths of vectors are defined,

and so we can speak of the associated unit sphere bundle S.E.!B, a fiber bundle

with fibers the spheres consisting of all vectors of length 1 in fibers of E. Similarly

there is a disk bundle D.E.!B with fibers the disks of vectors of length less than

or equal to 1. It is possible to describe S.E. without reference to an inner product,

as the quotient of the complement of the zero section in E obtained by identifying

each nonzero vector with all positive scalar multiples of itself. It follows that D.E.

can also be defined without invoking a metric, namely as the mapping cylinder of the

projection S.E.!B.

The canonical line bundle E!RPn has as its unit sphere bundle S.E. the space

of unit vectors in lines through the origin in Rn.1 . Since each unit vector uniquely

determines the line containing it, S.E. is the same as the space of unit vectors in

Rn.1 , i.e., Sn . It follows that canonical line bundle is nontrivial if n > 0 since for the

trivial bundle RPn_R the unit sphere bundle is RPn_S0 , which is not homeomorphic

to Sn .

Similarly, in the complex case the canonical line bundle E!CPn has S.E. equal

to the unit sphere S2n.1 in Cn.1 . Again if n > 0 this is not homeomorphic to the unit

sphere bundle of the trivial bundle, which is CPn_S1 , so the canonical line bundle is

nontrivial.

Subbundles

A vector subbundle of a vector bundle p : E!B has the natural definition: a subspace

E0 _ E intersecting each fiber of E in a vector subspace, such that the restriction

p : E0!B is a vector bundle.

Proposition 1.6. If E!B is a vector bundle over a paracompact base B and E0 _ E

is a vector subbundle, then there is a vector subbundle E?

0 _ E such that E0

_E?

0 _ E.

Proof: With respect to a chosen inner product on E, let E?

0 be the subspace of E

which in each fiber consists of all vectors orthogonal to vectors in E0 . We claim

that the natural projection E?

0!B is a vector bundle. If this is so, then E0

_E?

0 is

isomorphic to E via the map .v;w.,v .w, using Lemma 1.1.

To see that E?

0 satisfies the local triviality condition for a vector bundle, note

first that we may assume E is the product B_Rn since the question is local in B.

Basic Definitions and Constructions Section 1.1 9

Since E0 is a vector bundle, of dimension m say, it has m independent local sections

b,.b; si.b.. near each point b0 2 B. We may enlarge this set of m independent

local sections of E0 to a set of n independent local sections b,.b; si.b.. of E by

choosing sm.1; ___ ; sn first in the fiber p−1.b0., then taking the same vectors for all

nearby fibers, since if s1; ___ ; sm; sm.1; ___ ; sn are independent at b0 , they will remain

independent for nearby b by continuity of the determinant function. Apply the Gram-

Schmidt orthogonalization process to s1; ___ ; sm; sm.1; ___ ; sn in each fiber, using the

given inner product, to obtain new sections s0

i . The explicit formulas for the Gram-

Schmidt process show the s0

i ’s are continuous. The sections s0

i allow us to define

a local trivialization h:p−1.U.!U_Rn with h.b; s0

i.b.. equal to the ith standard

basis vector of Rn . This h carries E0 to U_Rm and E?

0 to U_Rn−m, so hjjE?

0 is a

local trivialization of E?

0 . tu

Tensor Products

In addition to direct sum, a number of other algebraic constructions with vector

spaces can be extended to vector bundles. One which is particularly important

for K–theory is tensor product. For vector bundles p1 : E1!B and p2 : E2!B, let

E1

E2 , as a set, be the disjoint union of the vector spaces p−1

1 .x.p−1

2 .x. for

x 2 B. The topology on this set is defined in the following way. Choose isomorphisms

hi :p−1

i .U.!U_Rni for each open set U _ B over which E1 and E2 are trivial. Then

a topology TU on the set p−1

1 .U.p−1

2 .U. is defined by letting the fiberwise tensor

product map h1

h2 :p−1

1 .U.p−1

2 .U.!U_.Rn1 Rn2. be a homeomorphism. The

topology TU is independent of the choice of the hi ’s since any other choices are obtained

by composing with isomorphisms of U_Rni of the form .x;v.,.x;gi.x..v..

for continuous maps gi :U!GLni.R., hence h1

h2 changes by composing with

analogous isomorphisms of U_.Rn1 Rn2. whose second coordinates g1

g2 are

continuous maps U!GLn1n2.R., since the entries of the matrices g1.x.g2.x. are

the products of the entries of g1.x. and g2.x.. When we replace U by an open subset

V , the topology on p−1

1 .V.p−1

2 .V. induced by TU is the same as the topology

TV since local trivializations over U restrict to local trivializations over V . Hence we

get a well-defined topology on E1

E2 making it a vector bundle over B.

There is another way to look at this construction that takes as its point of departure

a general method for constructing vector bundles we have not mentioned previously.

If we are given a vector bundle p : E!B and an open cover fU_

g of B with local

trivializations h_ :p−1.U_.!U_

_Rn , then we can reconstruct E as the quotient

space of the disjoint union

`

_.U_

_Rn. obtained by identifying .x;v. 2 U_

_Rn

with h_h−1

_ .x;v. 2 U_

_Rn whenever x 2 U_

\ U_ . The functions h_h−1

_ can

be viewed as maps g__ :U_

\ U_!GLn.R.. These satisfy the ‘cocycle condition’

g_g__

. g_ on U_

\ U_

\ U . Any collection of ‘gluing functions’ g__ satisfying

this condition can be used to construct a vector bundle E!B.

10 Chapter 1 Vector Bundles

In the case of tensor products, suppose we have two vector bundles E1!B and

E2!B. We can choose an open cover fU_

g with both E1 and E2 trivial over each U_ ,

and so obtain gluing functions gi

__ :U_

\ U_!GLni.R. for each Ei . Then the gluing

functions for the bundle E1

E2 are the tensor product functions g1

__ g2

__ assigning

to each x 2 U_

\ U_ the tensor product of the two matrices g1

__.x. and g2

__.x..

It is routine to verify that the tensor product operation for vector bundles over a

fixed base space is commutative, associative, and has an identity element, the trivial

line bundle. It is also distributive with respect to direct sum.

If we restrict attention to line bundles, then Vect1.B. is an abelian group with

respect to the tensor product operation. The inverse of a line bundle E!B is obtained

by replacing its gluing matrices g__.x. 2 GL1.R. with their inverses. The cocycle

condition is preserved since 1_1 matrices commute. If we give E an inner product,

we may rescale local trivializations h_ to be isometries, taking vectors in fibers of E

to vectors in R1 of the same length. Then all the values of the gluing functions g__

are _1, being isometries of R. The gluing functions for EE are the squares of these

g__ ’s, hence are identically 1, so EE is the trivial line bundle. Thus each element of

Vect1.B. is its own inverse. As we shall see in x3.1, the group Vect1.B. is isomorphic

to H1.B; Z2. when B is homotopy equivalent to a CW complex.

These tensor product constructions work equally well for complex vector bundles.

Tensor product again makes Vect1

C.B. into an abelian group, but after rescaling the

gluing functions g__ for a complex line bundle E, the values are complex numbers

of norm 1, not necessarily _1, so we cannot expect EE to be trivial. In x3.1 we

will show that the group Vect1

C.B. is isomorphic to H2.B; Z. when B is homotopy

equivalent to a CW complex.

We may as well mention here another general construction for complex vector

bundles E!B, the notion of the conjugate bundle E!B. As a topological space, E

is the same as E, but the vector space structure in the fibers is modified by redefining

scalar multiplication by the rule _.v. . _v where the right side of this equation

means scalar multiplication in E and the left side means scalar multiplication in E.

This implies that local trivializations for E are obtained from local trivializations for

E by composing with the coordinatewise conjugation map Cn!Cn in each fiber. The

effect on the gluing maps g__ is to replace them by their complex conjugates as

well. Specializing to line bundles, we then have EE isomorphic to the trivial line

bundle since its gluing maps have values zz . 1 for z a unit complex number. Thus

conjugate bundles provide inverses in Vect1

C.B..

Besides tensor product of vector bundles, another construction useful in K–theory

is the exterior power _k.E. of a vector bundle E. Recall from linear algebra that

the exterior power _k.V. of a vector space V is the quotient of the k fold tensor

product V  ___ V by the subspace generated by vectors of the form v1

 ___ vk

−

sgn._.v_.1.  ___ v_.k. where _ is a permutation of the subscripts and sgn._. .

Basic Definitions and Constructions Section 1.1 11

_1 is its sign, .1 for an even permutation and −1 for an odd permutation. If V has

dimension n then _k.V. has dimension

_

nk

_

. Now to define _k.E. for a vector bundle

p : E!B the procedure follows closely what we did for tensor product. We first form

the disjoint union of the exterior powers _k.p−1.x.. of all the fibers p−1.x., then we

define a topology on this set via local trivializations. The key fact about tensor product

which we needed before was that the tensor product ' of linear transformations

' and depends continuously on ' and . For exterior powers the analogous fact

is that a linear map ':Rn!Rn induces a linear map _k.'. : _k.Rn.!_k.Rn. which

depends continuously on '. This holds since _k.'. is a quotient map of the k fold

tensor product of ' with itself.

Associated Bundles

There are a number of geometric operations on vector spaces which can also

be performed on vector bundles. As an example we have already seen, consider the

operation of taking the unit sphere or unit disk in a vector space with an inner product.

Given a vector bundle E!B with an inner product, we can then perform the operation

in each fiber, producing the sphere bundle S.E.!B and the disk bundle D.E.!B.

Here are some more examples:

(1) Associated to a vector bundle E!B is the projective bundle P.E.!B, where P.E.

is the space of all lines through the origin in all the fibers of E. We topologize P.E.

as the quotient of the sphere bundle S.E. obtained by factor out scalar multiplication

in each fiber. Over a neighborhood U in B where E is a product U_Rn , this quotient

is U_RPn−1, so P.E. is a fiber bundle over B with fiber RPn−1 , with respect to the

projection P.E.!B which sends each line in the fiber of E over a point b 2 B to

b. We could just as well start with an n dimensional vector bundle over C, and then

P.E. would have fibers CPn−1 .

(2) For an n dimensional vector bundle E!B, the associated flag bundle F.E.!B

has total space F.E. the subspace of the n fold product of P.E. with itself consisting

of n tuples of orthogonal lines in fibers of E. The fiber of F.E. is thus the flag

manifold F.Rn. consisting of n tuples of orthogonal lines through the origin in Rn .

Local triviality follows as in the preceding example. More generally, for any k _ n one

could take k tuples of orthogonal lines in fibers of E and get a bundle Fk.E.!B.

(3) As a refinement of the last example, one could form the Stiefel bundle Vk.E.!B,

where points of Vk.E. are k tuples of orthogonal unit vectors in fibers of E, so Vk.E.

is a subspace of the product of k copies of S.E.. The fiber of Vk.E. is the Stiefel

manifold Vk.Rn. of orthonormal k frames in Rn .

(4) Generalizing P.E., there is the Grassmann bundle Gk.E.!B of k dimensional

linear subspaces of fibers of E. This is the quotient space of Vk.E. obtained by

identifying two k frames if they span the same subspace of a fiber. The fiber of

Gk.E. is the Grassmann manifold Gk.Rn. of k planes through the origin in Rn .

12 Chapter 1 Vector Bundles

Some of these associated fiber bundles have natural vector bundles lying over

them. For example, there is a canonical line bundle L!P.E. where L . f.`;v. 2

P.E._E j v 2 ` g. Similarly, over the flag bundle F.E. there are n line bundles Li

consisting of all vectors in the ith line of an n tuple of orthogonal lines in fibers of E.

The direct sum L1_____Ln is then equal to the pullback of E

over F.E. since a point in the pullback consists of an n tuple

¡!

¡!

¡¡¡¡¡!E

F (E )¡¡¡¡¡!B

L1© ©Ln

. . .

of lines `1 ? ___ ? `n in a fiber of E together with a vector v

in this fiber, and v can be expressed uniquely as a sum v . v1.___.vn with vi

2 `i .

Thus we see an interesting fact: For every vector bundle there is a pullback which splits

as a direct sum of line bundles. This observation plays a role in the so-called ‘splitting

principle,’ as we shall see in Corollary 2.23 and Proposition 3.3.

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