3.1. Stiefel-Whitney and Chern Classes

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Stiefel-Whitney classes are defined for real vector bundles, Chern classes for complex

vector bundles. The two cases are quite similar, but for concreteness we shall emphasize

the real case, with occasional comments on the minor modifications needed

to treat the complex case.

A technical point before we begin: We shall assume without further mention that

all base spaces of vector bundles are paracompact, so that the fundamental results

of Chapter 1 apply. For the study of characteristic classes this is not an essential

64 Chapter 3 Characteristic Classes

restriction since one can always pass to pullbacks over a CW approximation to a given

base space, and CW complexes are paracompact.

Axioms and Construction

Here is the main result giving axioms for Stiefel-Whitney classes:

Theorem 3.1. There is a unique sequence of functions w1;w2; ___ assigning to each

real vector bundle E!B a class wi.E. 2 Hi.B; Z2., depending only on the isomorphism

type of E, such that

(a) wi.f _.E.. . f _.wi.E.. for a pullback f _.E..

(b) w.E1

_E2. . w.E1.`w.E2. for w . 1 .w1 .w2 . ___ 2 H_.B; Z2..

(c) wi.E. . 0 if i > dim E.

(d) For the canonical line bundle E!RP1, w1.E. is a generator of H1.RP1; Z2..

The sum w.E. . 1.w1.E..w2.E..___ is the total Stiefel-Whitney class. Note that

(c) implies that the sum 1.w1.E..w2.E..___ has only finitely many nonzero terms,

so this sum does indeed lie in H_.B; Z2., the direct sum of the groups Hi.B; Z2.. From

the formal identity

.1 .w1 .w2 . ___..1 .w0

1 .w0

2 . ___. . 1..w1 .w0

1. . .w2 .w1w0

1 .w0

2. . ___

it follows that the formula w.E1

_E2. . w.E1. ` w.E2. is just a compact way of

writing the relations wn.E1

_E2. .

P

i.j.nwi.E1. ` wj.E2., where w0 . 1. This

relation is sometimes called the Whitney sum formula.

For complex vector bundles there are analogous Chern classes:

Theorem 3.2. There is a unique sequence of functions c1; c2; ___ assigning to each

complex vector bundle E!B a class ci.E. 2 H2i.B; Z., depending only on the isomorphism

type of E, such that

(a) ci.f _.E.. . f _.ci.E.. for a pullback f _.E..

(b) c.E1

_E2. . c.E1. ` c.E2. for c . 1 . c1 . c2 . ___ 2 H_.B; Z..

(c) ci.E. . 0 if i > dim E.

(d) For the canonical line bundle E!CP1, c1.E. is a generator of H2.CP1; Z. specified

in advance.

As in the real case, the formula in (b) for the total Chern classes can be rewritten

in the form cn.E1

_E2. .

P

i.j.n ci.E1. ` cj.E2., where c0 . 1.

Proof of 3.1 and 3.2: Associated to a vector bundle _ : E!B with fiber Rn is the

projective bundle P._. : P.E.!B, where P.E. is the space of all lines through the

origin in all the fibers of E, and P._. is the natural projection sending each line in

_−1.b. to b 2 B. We topologize P.E. as a quotient of the complement of the zero

section of E, the quotient obtained by factoring out scalar multiplication in each fiber.

Stiefel-Whitney and Chern Classes Section 3.1 65

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 .

We would like to apply the Leray-Hirsch theorem for cohomology with Z2 coefficients

to this bundle P.E.!B. To do this we need classes xi

2 Hi.P.E.; Z2.

restricting to generators of Hi.RPn−1; Z2. in each fiber RPn−1 for i . 0; ___;n − 1.

Recall from the proof of Theorem 1.8 that there is a map g : E!R1 that is a linear

injection on each fiber. Projectivizing the map g by deleting zero vectors and then

factoring out scalar multiplication produces a map P.g. : P.E.!RP1. Let _ be a generator

of H1.RP1; Z2. and let x . P.g._._. 2 H1.P.E.; Z2.. Then the powers xi for

i . 0; ___;n − 1 are the desired classes xi since a linear injection Rn!R1 induces

an embedding RPn−1>RP1 for which _ pulls back to a generator of H1.RPn−1; Z2.,

hence _i pulls back to a generator of Hi.RPn−1; Z2.. Note that any two linear injections

Rn!R1 are homotopic through linear injections, so the induced embeddings

RPn−1>RP1 of different fibers of P.E. are all homotopic. We showed in the proof

of Theorem 1.8 that any two choices of g are homotopic through maps that are linear

injections on fibers, so the classes xi are independent of the choice of g.

The Leray-Hirsch theorem then says that H_.P.E.; Z2. is a free H_.B; Z2. module

with basis 1;x; ___;xn−1 . Consequently, xn can be expressed uniquely as a linear

combination of these basis elements with coefficients in H_.B; Z2.. Thus there is a

unique relation of the form

xn .w1.E.xn−1 . ___.wn.E. _ 1 . 0

for certain classes wi.E. 2 Hi.B; Z2.. Here wi.E.xi means P._._.wi.E.. ` xi, by

the definition of the H_.B; Z2. module structure on H_.P.E.; Z2.. For completeness

we define wi.E. . 0 for i > n and w0.E. . 1.

To prove property (a), consider a pullback f _.E. . E0 , fitting

into the diagram at the right. If g : E!R1 is a linear injection on

¡¡!

¡¡!

0¡¡¡¡¡!

0

E E

0¡¡¡¡¡! B B f

f

»

¼ ¼

fibers then so is g e f , and it follows that P. e f._ takes the canonical

class x . x.E. for P.E. to the canonical class x.E0. for P.E0.. Then

P. e f._

_X

i

P._._􀀀

wi.E.

_

` x.E.n−i

_

.

X

i

P. e f._P._._􀀀

wi.E.

_

` P. e f._􀀀

x.E.n−i_

.

X

i

P._0._f_􀀀

wi.E.

_

` x.E0.n−i

so the relation x.E.n . w1.E.x.E.n−1 . ___ .wn.E. _ 1 . 0 defining wi.E. pulls

back to the relation x.E0.n . f_.w1.E..x.E0.n−1 . ___.f_.wn.E.. _ 1 . 0 defining

wi.E0.. By the uniqueness of this relation, wi.E0. . f _.wi.E...

Proceeding to property (b), the inclusions of E1 and E2 into E1

_E2 give inclusions

of P.E1. and P.E2. into P.E1

_E2. with P.E1. \ P.E2. . ;. Let U1 .

P.E1

_E2. − P.E1. and U2 . P.E1

_E2. − P.E2.. These are open sets in P.E1

_E2.

that deformation retract onto P.E2. and P.E1., respectively. A map g : E1

_E2!R1

66 Chapter 3 Characteristic Classes

which is a linear injection on fibers restricts to such a map on E1 and E2 , so the

canonical class x 2 H1.P.E1

_E2.; Z2. for E1

_E2 restricts to the canonical classes

for E1 and E2. If E1 and E2 have dimensions m and n, consider the classes !1 .

P

j wj.E1.xm−j and !2 .

P

j wj.E2.xn−j in H_.P.E1

_E2.; Z2., with cup product

!1!2 .

P

j

_P

r.s.j wr .E1.ws.E2.

_

xm.n−j . By the definition of the classes wj.E1.,

the class !1 restricts to zero in Hm.P.E1.; Z2., hence !1 pulls back to a class in

the relative group Hm.P.E1

_E2.; P.E1.; Z2. _ Hm.P.E1

_E2.;U2; Z2., and similarly

for !2 . The following commutative diagram, with Z2 coefficients understood, then

shows that !1!2 . 0:

¡!

¡!

Hn P E U ¡¡¡¡¡!

1 E2 1 ( ( © ), ) Hm n P E U

1 E2 1 U2 0 Hm P E U ( ( © ), )

1 E2 2 ( ( © ), ) £ + [ =

Hn P E ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡!

1 E2

( ( © )) Hm n P E

1 E2 Hm P E ( ( © ))

1 E2 ( ( © )) £ +

Thus !1!2 .

P

j

_P

r.s.j wr .E1.ws.E2.

_

xm.n−j . 0 is the defining relation for the

Stiefel-Whitney classes of E1

_E2 , and so wj.E1

_E2. .

P

r.s.j wr .E1.ws.E2..

Property (c) holds by definition. For (d), recall that the canonical line bundle is

E . f.`;v. 2 RP1_R1 j v 2 ` g. The map P._. in this case is the identity. The

map g : E!R1 which is a linear injection on fibers can be taken to be g.`;v. . v .

So P.g. is also the identity, hence x.E. is a generator of H1.RP1; Z2.. The defining

relation x.E. .w1.E. _ 1 . 0 then says that w1.E. is a generator of H1.RP1; Z2..

The proof of uniqueness of the classes wi will use a general property of vector

bundles called the splitting principle:

Proposition 3.3. For each vector bundle _ : E!B there is a space F.E. and a map

p : F.E.!B such that the pullback p_.E.!F.E. splits as a direct sum of line bundles,

and p_ :H_.B; Z2.!H_.F.E.; Z2. is injective.

Proof: Consider the pullback P._._.E. of E via the map P._. : P.E.!B. This pullback

contains a natural one-dimensional subbundle L . f.`;v. 2 P.E._E j v 2 ` g.

An inner product on E pulls back to an inner product on the pullback bundle, so we

have a splitting of the pullback as a sum L_L? with the orthogonal bundle L? having

dimension one less than E. As we have seen, the Leray-Hirsch theorem applies

to P.E.!B, so H_.P.E.; Z2. is the free H_.B; Z2. module with basis 1;x; ___;xn−1

and in particular the induced map H_.B; Z2.!H_.P.E.; Z2. is injective since one of

the basis elements is 1.

This construction can be repeated with L?!P.E. in place of E!B. After finitely

many repetitions we obtain the desired result. tu

Looking at this construction a little more closely, L? consists of pairs .`;v. 2

P.E._E with v ? `. At the next stage we form P.L?., whose points are pairs .`; `0.

where ` and `0 are orthogonal lines in E. Continuing in this way, we see that the

Stiefel-Whitney and Chern Classes Section 3.1 67

final base space F.E. is the space of all orthogonal splittings `1 _ ____`n of fibers

of E as sums of lines, and the vector bundle over F.E. consists of all n tuples of

vectors in these lines. Alternatively, F.E. can be described as the space of all chains

V1 _ ___ _ Vn of linear subspaces of fibers of E with dimVi

. i. Such chains are

called flags, and F.E.!B is the flag bundle associated to E. Note that the description

of points of F.E. as flags does not depend on a choice of inner product in E.

Now we can finish the proof of Theorem 3.1. Property (d) determines w1.E.

for the canonical line bundle E!RP1. Property (c) then determines all the wi ’s for

this bundle. Since the canonical line bundle is the universal line bundle, property (a)

therefore determines the classes wi for all line bundles. Property (b) extends this

to sums of line bundles, and finally the splitting principal implies that the wi ’s are

determined for all bundles.

For complex vector bundles we can use the same proof, but with Z coefficients

since H_.CP1; Z. _ Z._., with _ now two-dimensional. The defining relation for the

ci.E.’s is modified to be

xn − c1.E.xn−1 . ___..−1.ncn.E. _ 1 . 0

with alternating signs. This is equivalent to changing the sign of _, so it does not

affect the proofs of properties (a)–(c), but it has the advantage that the canonical line

bundle E!CP1 has c1.E. . _ rather than −_, since the defining relation in this

case is x.E. − c1.E. _ 1 . 0 and x.E. . _. tu

Note that in property (d) for Stiefel-Whitney classes we could just as well use the

canonical line bundle over RP1 instead of RP1 since the inclusion RP1>RP1 induces

an isomorphism H1.RP1; Z2. _ H1.RP1; Z2.. The analogous remark for Chern classes

is valid as well.

Example 3.4. Property (a), the naturality of Stiefel-Whitney classes, implies that a

product bundle E . B_Rn has wi.E. . 0 for i > 0 since a product is the pullback

of a bundle over a point, which must have wi

. 0 for i > 0 since a point has trivial

cohomology in positive dimensions.

Example 3.5: Stability. Property (b) implies that taking the direct sum of a bundle

with a product bundle does not change its Stiefel-Whitney classes. In this sense Stiefel-

Whitney classes are stable. For example, the tangent bundle TSn to Sn is stably

trivial since its direct sum with the normal bundle to Sn in Rn.1 , which is a trivial

line bundle, produces a trivial bundle. Hence the Stiefel-Whitney classes wi.TSn. are

zero for i > 0.

From the identity

.1 .w1 .w2 . ___..1 .w0

1 .w0

2 . ___. . 1..w1 .w0

1. . .w2 .w1w0

1 .w0

2. . ___

68 Chapter 3 Characteristic Classes

we see that w.E1. and w.E1

_E2. determine w.E2. since the equations

w1 .w0

1 . a1

w2 .w1w0

1 .w0

2 . a2

___

P

iwn−iw0

i

. an

can be solved successively for the w0

i ’s in terms of the wi ’s and ai ’s. In particular, if

E1

_E2 is the trivial bundle, then we have the case that ai

. 0 for i > 0 and so w.E1.

determines w.E2. uniquely by explicit formulas that can be worked out. For example,

w0

1 . −w1 and w0

2 . −w1w0

1−w2 .w2

1 −w2. Of course for Z2 coefficients the signs

do not matter, but the same reasoning applies to Chern classes, with Z coefficients.

Example 3.6. Let us illustrate this principle by showing that there is no bundle

E!RP1 whose sum with the canonical line bundle E1.R1. is trivial. For we have

w.E1.R1.. . 1 .! where ! is a generator of H1.RP1; Z2., and hence w.E. must

be .1.!.−1 . 1.!.!2.___ since we are using Z2 coefficients. Thus wi.E. . !i ,

which is nonzero in H_.RP1; Z2. for all i. However, this contradicts the fact that

wi.E. . 0 for i > dim E.

This shows the necessity of the compactness assumption in Proposition 1.9. To

further delineate the question, note that Proposition 1.9 says that the restriction

E1.Rn.1. of the canonical line bundle to the subspace RPn _ RP1 does have an

‘inverse’ bundle. In fact, the bundle E?

1 .Rn.1. consisting of pairs .`;v. where `

is a line through the origin in Rn.1 and v is a vector orthogonal to ` is such an

inverse. But for any bundle E!RPn whose sum with E1.Rn.1. is trivial we must

have w.E. . 1 . ! . ___ .!n, and since wn.E. . !n . 0, E must be at least

n dimensional. So we see there is no chance of choosing such bundles E for varying

n so that they fit together to form a single bundle over RP1.

Example 3.7. Let us describe an n dimensional vector bundle E!B with wi.E.

nonzero for each i _ n. This will be the n fold Cartesian product .E1.n!.G1.n of

the canonical line bundle over G1 . RP1 with itself. This vector bundle is the direct

sum __

1 .E1._ ___ ___

n .E1. where _i : .G1.n!G1 is projection onto the ith factor, so

w..E1.n. .

Q

i.1 . _i. 2 Z2._1; ___;_n. _ H_..RP1.n; Z2.. Hence wi..E1.n. is the

ith elementary symmetric polynomial _i in the _j ’s, the sum of all the products of i

different _j ’s. For example, if n . 3 then _1 . _1._2._3 , _2 . _1_2._1_3._2_3 ,

and _3 . _1_2_3 . Since each _i with i _ n is nonzero in Z2._1; ___;_n., we have

an n dimensional bundle whose first n Stiefel-Whitney classes are all nonzero.

The same reasoning applies in the complex case to show that the n fold Cartesian

product of the canonical line bundle over CP1 has its first n Chern classes nonzero.

In this example we see that the wi ’s and ci ’s can be identified with elementary

symmetric functions, and in fact this can be done in general using the splitting principle.

Given an n dimensional vector bundle E!B we know that the pullback to F.E.

Stiefel-Whitney and Chern Classes Section 3.1 69

splits as a sum L1

_ ___ _Ln!F.E.. Letting _i

. w1.Li., we see that w.E. pulls

back to w.L1

_ ___ _Ln. . .1 . _1. ___ .1 . _n. . 1 . _1 . ___._n, so wi.E. pulls

back to _i . Thus we have embedded H_.B; Z2. in a larger ring H_.F.E.; Z2. such that

wi.E. becomes the ith elementary symmetric polynomial in the elements _1; ___;_n

of H_.F.E.; Z2..

Besides the evident formal similarity between Stiefel-Whitney and Chern classes

there is also a direct relation:

Proposition 3.8. Regarding an n dimensional complex vector bundle E!B as a

2n dimensional real vector bundle, then w2i.1.E. . 0 and w2i.E. is the image of

ci.E. under the coefficient homomorphism H2i.B; Z.!H2i.B; Z2..

For example, since the canonical complex line bundle over CP1 has c1 a generator

of H2.CP1; Z., the same is true for its restriction over S2 . CP1 , so by the proposition

this 2 dimensional real vector bundle E!S2 has w2.E. . 0.

Proof: The bundle E has two projectivizations RP.E. and CP.E., consisting of all the

real and all the complex lines in fibers of E, respectively. There is a natural projection

p :RP.E.!CP.E. sending each real line to the complex line containing it, since a real

line is all the real scalar multiples of any nonzero vector in it and a complex line is all

the complex scalar multiples. This projection p fits into a commutative diagram

¡!

¡!

¡!

RP2 ¡¡¡¡¡¡¡¡¡!RP(E )¡¡¡¡¡¡¡¡g¡¡!

p

RP( ) n - 1 RP1

CP ¡¡¡¡¡¡¡¡¡! CP(E )¡¡¡¡C¡P¡¡(¡g¡¡)! n - 1 CP1

where the left column is the restriction of p to a fiber of E and the maps RP.g.

and CP.g. are obtained by projectivizing, over R and C, a map g : E!C1 which

is a C linear injection on fibers. It is easy to see that all three vertical maps in

this diagram are fiber bundles with fiber RP1 , the real lines in a complex line. The

Leray-Hirsch theorem applies to the bundle RP1!CP1, with Z2 coefficients, so if

_ is the standard generator of H2.CP1; Z., the Z2 reduction _ 2 H2.CP1; Z2. pulls

back to a generator of H2.RP1; Z2., namely the square _2 of the generator _ 2

H1.RP1; Z2.. Hence the Z2 reduction xC.E. . CP.g._._. 2 H2.CP.E.; Z2. of the

basic class xC.E. . CP.g._._. pulls back to the square of the basic class xR.E. .

RP.g._._. 2 H1.RP.E.; Z2.. Consequently the Z2 reduction of the defining relation

for the Chern classes of E, which is xC.E.n . c1.E.xC.E.n−1 . ___.cn.E. _ 1 . 0,

pulls back to the relation xR.E.2n.c1.E.xR.E.2n−2.___.cn.E. _1 . 0, which is the

defining relation for the Stiefel-Whitney classes of E. This means that w2i.1.E. . 0

and w2i.E. . ci.E.. tu

70 Chapter 3 Characteristic Classes

Cohomology of Grassmannians

From Example 3.7 and naturality it follows that the universal bundle En!Gn

must also have all its Stiefel-Whitney classes w1.En.; ___;wn.En. nonzero. In fact a

much stronger statement is true. Let f : .RP1.n!Gn be the classifying map for the

n fold Cartesion product .E1.n of the canonical line bundle E1 , and for notational

simplicity let wi

. wi.En.. Then the composition

Z2.w1; ___;wn. -!H_.Gn; Z2. f _ ----!H_􀀀

.RP1.n; Z2

_

_ Z2._1; ___;_n.

sends wi to _i , the ith elementric symmetric polynomial. It is a classical algebraic result

that the polynomials _i are algebraically independent in Z2._1; ___;_n.. Proofs

of this can be found in [van der Waerden, x26] or [Lang, p. 134] for example. Thus

the composition Z2.w1; ___;wn.!Z2._1; ___;_n. is injective, hence also the map

Z2.w1; ___;wn.!H_.Gn; Z2.. In other words, the classes wi.En. generate a polynomial

subalgebra Z2.w1; ___;wn. _ H_.Gn; Z2.. This subalgebra is in fact equal to

H_.Gn; Z2., and the corresponding statement for Chern classes holds as well:

Theorem 3.9. H_.Gn; Z2. is the polynomial ring Z2.w1; ___;wn. on the Stiefel-

Whitney classes wi

. wi.En. of the universal bundle En!Gn . Similarly, in the

complex case H_.Gn.C1.; Z. _ Z.c1; ___ ; cn. where ci

. ci.En.C1.. for the universal

bundle En.C1.!Gn.C1..

The proof we give here for this basic result will be a fairly quick application of the

CW structure on Gn constructed at the end of x1.2. A different proof will be given

in x3.3 where we also compute the cohomology of Gn with Z coefficients, which is

somewhat more subtle.

Proof: Consider a map f : .RP1.n!Gn which pulls En back to the bundle .E1.n

considered above. We have noted that the image of f _ contains the symmetric polynomials

in Z2._1; ___;_n. _ H_..RP1.n; Z2.. The opposite inclusion holds as well,

since if _ : .RP1.n!.RP1.n is an arbitrary permutation of the factors, then _ pulls

.E1.n back to itself, so f_ ' f , which means that f _ . __f _ , so the image of f _ is

invariant under __ :H_..RP1.n; Z2.!H_..RP1.n; Z2., but the latter map is just the

same permutation of the variables _i .

To finish the proof in the real case it remains to see that f _ is injective. It suffices

to find a CW structure on Gn in which the r cells are in one-to-one correspondence

with monomials wr1

1 ___wrn

n of dimension r . r1 .2r2 .___.nrn , since the number

of r cells in a CW complex X is an upper bound on the dimension of Hr.X; Z2. as a

Z2 vector space, and a surjective linear map between finite-dimensional vector spaces

is injective if the dimension of the domain is not greater than the dimension of the

range.

Monomials wr1

1 ___wrn

n of dimension r correspond to n tuples .r1; ___ ; rn. with

r . r1 . 2r2 . ___.nrn . Such n tuples in turn correspond to partitions of r into at

Stiefel-Whitney and Chern Classes Section 3.1 71

most n integers, via the correspondence

.r1; ___ ; rn. ! rn

_ rn

.rn−1 _ ___ _ rn

. rn−1 . ___.r1:

Such a partition becomes the sequence _1−1 _ _2−2 _ ___ _ _n

−n, corresponding

to the strictly increasing sequence 0 < _1 < _2 < ___ < _n. For example, when n . 3

we have:

.r1; r2; r3. ._1 − 1;_2 −2;_3 −3. ._1;_2;_3. dimension

1 0 0 0 0 0 0 1 2 3 0

w1 1 0 0 0 0 1 1 2 4 1

w2 0 1 0 0 1 1 1 3 4 2

w2

1 2 0 0 0 0 2 1 2 5 2

w3 0 0 1 1 1 1 2 3 4 3

w1w2 1 1 0 0 1 2 1 3 5 3

w3

1 3 0 0 0 0 3 1 2 6 3

The cell structure on Gn constructed in x1.2 has one cell of dimension ._1 − 1. .

._2 − 2. . ___ . ._n

− n. for each increasing sequence 0 < _1 < _2 < ___ < _n. So

we are done in the real case.

The complex case is entirely similar, keeping in mind that ci has dimension 2i

rather than i. The CW structure on Gn.C1. described in x1.2 also has these extra factors

of 2 in the dimensions of its cells. In particular, the cells are all even-dimensional,

so the cellular boundary maps for Gn.C1. are all trivial and the cohomology with Z

coefficients consists of a Z summand for each cell. Injectivity of f _ then follows

from the algebraic fact that a surjective homomorphism between free abelian groups

of finite rank is injective if the rank of the domain is not greater than the rank of the

range. tu

One might guess that the monomial wr1

1 ___wrn

n corresponding to a given cell of

Gn in the way described above was the cohomology class dual to this cell, represented

by the cellular cochain assigning the value 1 to the cell and 0 to all the other cells.

This is true for the classes wi themselves, but unfortunately it is not true in general.

For example the monomial wi1

corresponds to the cell whose associated partition is

the trivial partition i . i, but the cohomology class dual to this cell is w0

i where

1.w0

1.w0

2 .___ is the multiplicative inverse of 1.w1.w2 .___. If one replaces the

basis of monomials by the more geometric basis of cohomology classes dual to cells,

the formulas for multiplying these dual classes become rather complicated. In the

parallel situation of Chern classes this question has very classical roots in algebraic

geometry, and the rules for multiplying cohomology classes dual to cells are part of

the so-called Schubert calculus. Accessible expositions of this subject from a modern

viewpoint can be found in [Fulton] and [Hiller].

72 Chapter 3 Characteristic Classes

Applications of w1 and c1

We saw in x1.1 that the set Vect1.X. of isomorphism classes of line bundles

over X forms a group with respect to tensor product. We know also that Vect1.X. .

.X;G1.R1.., and G1.R1. is just RP1, an Eilenberg-MacLane space K.Z2; 1.. It is a

basic fact in algebraic topology that .X;K.G; n.. _ Hn.X; G. when X has the homotopy

type of a CW complex; see Theorem 4.56 of [AT], for example. Thus one might

ask whether the groups Vect1.X. and H1.X; Z2. are isomorphic. For complex line

bundles we have G1.C1. . CP1, and this is a K.Z; 2., so the corresponding question

is whether Vect1

C.X. is isomorphic to H2.X; Z..

Proposition 3.10. The function w1 : Vect1.X.!H1.X; Z2. is a homomorphism, and

is an isomorphism if X has the homotopy type of a CW complex. The same is also

true for c1 : Vect1

C.X.!H2.X; Z..

Proof: The argument is the same in both the real and complex cases, so for definiteness

let us describe the complex case. To show that c1 : Vect1

C.X.!H2.X. is

a homomorphism, we first prove that c1.L1

L2. . c1.L1. . c1.L2. for the bundle

L1

L2!G1_G1 where L1 and L2 are the pullbacks of the canonical line bundle

L!G1 . CP1 under the projections p1;p2 :G1_G1!G1 onto the two factors. Since

c1.L. is the generator _ of H2.CP1., we know that H_.G1_G1. _ Z._1;_2. where

_i

. p_

i ._. . c1.Li.. The inclusion G1_G1 _ G1_G1 induces an isomorphism on H2 ,

so to compute c1.L1

L2. it suffices to restrict to G1_G1 . Over the first G1 the bundle

L2 is the trivial line bundle, so the restriction of L1

L2 over this G1 is L1

1 _ L1 .

Similarly, L1

L2 restricts to L2 over the second G1. So c1.L1

L2. restricted to

G1_G1 is _1._2 restricted to G1_G1 . Hence c1.L1

L2. . _1._2 . c1.L1..c1.L2..

The general case of the formula c1.E1

E2. . c1.E1..c1.E2. for line bundles E1

and E2 now follows by naturality: We have E1 _ f _

1 .L. and E2 _ f _

2 .L. for maps

f1; f2 :X!G1 . For the map F . .f1; f2. :X!G1_G1 we have F_.Li. . f _

i .L. _ Ei ,

so

c1.E1

E2. . c1.F_.L1.F_.L2.. . c1.F_.L1

L2.. . F_.c1.L1

L2..

. F_.c1.L1. . c1.L2.. . F_.c1.L1.. . F_.c1.L2..

. c1.F_.L1.. . c1.F_.L2.. . c1.E1. . c1.E2.:

As noted above, if X is a CW complex, there is a bijection .X;CP1. _ H2.X; Z.,

and the more precise statement is that this bijection is given by the map .f .,f _.u.

for some class u 2 H2.CP1; Z.. The class u must be a generator, otherwise the map

would not always be surjective. Which of the two generators we choose for u is

not important, so we may take it to be the class _. The map .f .,f _._. factors

as the composition .X;CP1.!Vect1

C.X.!H2.X; Z., .f .,f _.L.,c1.f _.L.. .

f _.c1.L.. . f _._.. The first map in this composition is a bijection, so since the

composition is a bijection, the second map c1 must be a bijection also. tu

Stiefel-Whitney and Chern Classes Section 3.2 73

The first Stiefel-Whitney class w1 is closely related to orientability:

Proposition 3.11. A vector bundle E!X is orientable iff w1.E. . 0, assuming that

X is homotopy equivalent to a CW complex.

Thus w1 can be viewed as the obstruction to orientability of vector bundles. An

interpretation of the other classes wi as obstructions will be given in the Appendix

to this chapter.

Proof: Without loss we may assume X is a CW complex. By restricting to pathcomponents

we may further assume X is connected. There are natural isomorphisms

._. H1.X; Z2. _ -----------!Hom.H1.X.; Z2. _ -----------!Hom._1.X.; Z2.

from the universal coefficient theorem and the fact that H1.X. is the abelianization of

_1.X.. When X . Gn we have _1.Gn. _ Z2 , and w1.En. 2 H1.Gn; Z2. corresponds

via ._. to this isomorphism _1.Gn. _ Z2 since w1.En. is the unique nontrivial element

of H1.Gn; Z2.. By naturality of ._. it follows that for any map f :X!Gn ,

f _.w1.En.. corresponds under ._. to the homomorphism f_ :_1.X.!_1.Gn. _

Z2 . Thus if we choose f so that f _.En. is a given vector bundle E, we have w1.E.

corresponding under ._. to the induced map f_ :_1.X.!_1.Gn. _ Z2 . Hence

w1.E. . 0 iff this f_ is trivial, which is exactly the condition for lifting f to the

universal cover eG

n , i.e., orientability of E. tu

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