3.3. Euler and Pontryagin Classes

Back

A characteristic class can be defined to be a function associating to each vector

bundle E!B of dimension n a class x.E. 2 Hk.B; G., for some fixed n and k, such

that the naturality property x.f_.E.. . f _.x.E.. is satisfied. In particular, for the

universal bundle En!Gn there is the class x . x.En. 2 Hk.Gn; G.. Conversely, any

element x 2 Hk.Gn; G. defines a characteristic class by the rule x.E. . f_.x. where

E _ f _.En. for f : B!Gn . Since f is unique up to homotopy, x.E. is well-defined,

and it is clear that the naturality property is satisfied. Thus characteristic classes

correspond bijectively with cohomology classes of Gn .

With Z2 coefficients all characteristic classes are simply polynomials in the Stiefel-

Whitney classes since we showed in Theorem 3.9 that H_.Gn; Z2. is the polynomial

ring Z2.w1; ___;wn.. Similarly for complex vector bundles all characteristic

classes with Z coefficients are polynomials in the Chern classes since H_.Gn.C.; Z. _

Z.c1; ___ ; cn.. Our goal in this section is to describe the more refined characteristic

classes for real vector bundles that arise when we take cohomology with integer coefficients

rather than Z2 coefficients.

The main tool we will use will be the Gysin exact sequence associated to an

n dimensional real vector bundle p : E!B. This is an easy consequence of the Thom

isomorphism Ø :Hi.B.!Hi.n.D.E.; S.E.. defined by Ø.b. . p_.b. ` c for a Thom

class c 2 Hn.D.E.; S.E.. having the property that its restriction to each fiber is a

generator of Hn.Dn; Sn−1.. The map Ø is an isomorphism whenever a Thom class

exists, as shown in Corollary 4D.9 of [AT]. In x3.2 we described an easy construction of

a Thom class which works for cohomology with Z2 coefficients or for complex vector

bundles with Z coefficients. We will eventually need the somewhat harder fact that

Thom classes with Z coefficients exist for all orientable real vector bundles. This is

shown in Theorem 4D.10 of [AT].

Once one has the Thom isomorphism, this gives the Gysin sequence as the lower

row of the following commutative diagram, whose upper row is the exact sequence

for the pair .D.E.; S.E..:

. . .¡¡!H (D(E),S(E)) ¡¡! H ( (E))¡! S(E) D(E),S(E) ¡¡!. . .

¡¡!

¡¡!

Ø

¡¡!

Ø

i i Hi( )¡!

Hi(S(E))

D Hi 1( ) j +

+ . . .¡¡¡!Hi n(B)¡¡¡¡¡¡¡!Hi(B )¡¡¡¡¡! ¡¡¡¡¡! ¡¡¡¡!. . . - e p Hi - n 1(B)

¤

¤

p¤

==

¼ ¼ ¼

Euler and Pontryagin Classes Section 3.3 85

The vertical map p_ is an isomorphism since p is a homotopy equivalence from

D.E. to B. The Euler class e 2 Hn.B. is defined to be .p_.−1j_.c., or in other words

the restriction of the Thom class to the zero section of E. The square containing

the map `e commutes since for b 2 Hi−n.B. we have j_Ø.b. . j_.p_.b. ` c. .

p_.b. ` j_.c., which equals p_.b ` e. . p_.b. ` p_.e. since p_.e. . j_.c.. The

Euler class can also be defined as the class corresponding to c ` c under the Thom

isomorphism, since Ø.e. . p_.e. ` c . j_.c. ` c . c ` c .

As a warm-up application of the Gysin sequence let us use it to give a different

proof of Theorem 3.9 computing H_.Gn; Z2. and H_.Gn.C.; Z.. Consider first the

real case. The proof will be by induction on n using the Gysin sequence for the universal

bundle En

--!-_ Gn . The sphere bundle S.En. is the space of pairs .v; `. where `

is an n dimensional linear subspace of R1 and v is a unit vector in `. There is a natural

map p : S.En.!Gn−1 sending .v; `. to the .n−1. dimensional linear subspace

v? _ ` orthogonal to v . It is an exercise to check that p is a fiber bundle. Its fiber is

S1, all the unit vectors in R1 orthogonal to a given .n − 1. dimensional subspace.

Since S1 is contractible, p induces an isomorphism on all homotopy groups, hence

also on all cohomology groups. Using this isomorphism p_ the Gysin sequence, with

Z2 coefficients, has the form

___!- Hi.Gn. --`-------!-e Hi.n.Gn. --!-_ Hi.n.Gn−1. -!Hi.1.Gn. -!___

where e 2 Hn.Gn; Z2. is the Z2 Euler class.

We show first that _.wj.En.. . wj.En−1.. By definition the map _ is the composition

H_.Gn.!H_.S.En.. _---- H_.Gn−1. induced from Gn−1

p---- S.En. _ ----!Gn .

The pullback __.En. consists of triples .v;w; `. where ` 2 Gn and v;w 2 ` with

v a unit vector. This pullback splits naturally as a sum L_p_.En−1. where L is the

subbundle of triples .v; tv; `., t 2 R, and p_.En−1. consists of the triples .v;w; `.

with w 2 v? . The line bundle L is trivial, having the section .v; v; `.. Thus the cohomology

homomorphism __ takes wj.En.. to wj.L_p_.En−1.. . wj.p_.En−1.. .

p_.wj.En−1.., so _.wj.En.. . wj.En−1..

By induction on n, H_.Gn−1. is the polynomial ring on the classes wj.En−1.,

j < n. The induction can start with the case n . 1, where G1 . RP1 and H_.RP1. _

Z2.w1. since w1.E1. is a generator of H1.RP1; Z2.. Or we could start with the trivial

case n . 0. Since _.wj.En.. . wj.En−1., the maps _ are surjective and the Gysin

sequence splits into short exact sequences

0!- Hi.Gn. --`-------!-e Hi.n.Gn. --!-_ Hi.n.Gn−1. -!0

The image of `e :H0.Gn.!Hn.Gn. is a Z2 generated by e. By exactness, this Z2

is the kernel of _:Hn.Gn.!Hn.Gn−1.. The class wn.En. lies in this kernel since

wn.En−1. . 0. Moreover, wn.En. . 0, since if wn.En. . 0 then wn is zero for all

n dimensional vector bundles, but the bundle E!RP1 which is the direct sum of n

86 Chapter 3 Characteristic Classes

copies of the canonical line bundle has total Stiefel-Whitney class w.E. . .1 . _.n ,

where _ generates H1.RP1., hence wn.E. . _n . 0. Thus e and wn.En. generate

the same Z2, so e .wn.En..

Now we argue that each element _ 2 Hk.Gn. can be expressed as a unique polynomial

in the classes wi

. wi.En., by induction on k. First, _._. is a unique polynomial

f in the wi.En−1.’s by the basic induction on n. Then _−f.w1; ___;wn−1. is in

Ker_ . Im.`wn., hence has the form _`wn for _ 2 Hk−n.Gn. which is unique since

`wn is injective. By induction on k, _ is a unique polynomial g in the wi ’s. Thus

we have _ expressed uniquely as a polynomial f.w1; ___;wn−1..wng.w1; ___;wn..

Since every polynomial in w1; ___;wn has a unique expression in this form, the theorem

follows in the real case.

Virtually the same argument works in the complex case. We noted earlier that

complex vector bundles always have a Gysin sequence with Z coefficients. The only

elaboration needed to extend the preceding proof to the complex case is at the step

where we showed the Z2 Euler class is wn . The argument from the real case shows

that cn is a multiple me for some m 2 Z, e being now the Z Euler class. Then

for the bundle E!CP1 which is the direct sum of n copies of the canonical line

bundle, classified by f :CP1!Gn.C1., we have _n . cn.E. . f _.cn. . mf_.e. in

H2n.CP1; Z. _ Z, with _n a generator, hence m . _1 and e . _cn. The rest of the

proof goes through without change.

We can also compute H_.eG

n; Z2. where eG

n is the oriented Grassmannian. To

state the result, let _ : eG

n!Gn be the covering projection, so e En

. __.En., and let

fwi

. wi. e En. . __.wi. 2 Hi.eG

n; Z2., where wi

. wi.En..

Proposition 3.25. __ :H_.Gn; Z2.!H_.eG

n; Z2. is surjective with kernel the ideal

generated by w1 , hence H_.eG

n; Z2. _ Z2.fw2; ___ ;fwn..

This is just the answer one would hope for. Since eG

n is simply-connected, fw1 has

to be zero, so the isomorphism H_.eGn; Z2. _ Z2.fw2; ___ ;fwn. is the simplest thing

that could happen.

Proof: The 2 sheeted covering _ : eG

n!Gn can be regarded as the unit sphere bundle

of a 1 dimensional vector bundle, so we have a Gysin sequence beginning

0 -!H0.Gn; Z2. -!H0.eG

n; Z2.!- H0.Gn; Z2. -`--------!-x H1.Gn; Z2.

where x 2 H1.Gn; Z2. is the Z2 Euler class. Since eG

n is connected, H0.eG

n; Z2. _

Z2 and so the map `x is injective, hence x . w1 , the only nonzero element of

H1.Gn; Z2.. Since H_.Gn; Z2. _ Z2.w1; ___;wn., the map `w1 is injective in all

dimensions, so the Gysin sequence breaks up into short exact sequences

0 -!Hi.Gn; Z2. -`----------w-----!- 1 Hi.Gn; Z2. __ -----------!Hi.eG

n; Z2.!- 0

Euler and Pontryagin Classes Section 3.3 87

from which the conclusion is immediate. tu

The goal for the rest of this section is to determine H_.Gn; Z. and H_.eG

n; Z.,

or in other words, to find all characteristic classes for real vector bundles with Z

coefficients, rather than the Z2 coefficients used for Stiefel-Whitney classes. It turns

out that H_.Gn; Z., modulo elements of order 2 which are just certain polynomials

in Stiefel-Whitney classes, is a polynomial ring Z.p1;p2; ___. on certain classes pi of

dimension 4i, called Pontryagin classes. There is a similar statement for H_.eG

n; Z.,

but with one of the Pontryagin classes replaced by an Euler class when n is even.

The Euler Class

Recall that the Euler class e.E. 2 Hn.B; Z. of an orientable n dimensional vector

bundle E!B is the restriction of a Thom class c 2 Hn.D.E.; S.E.; Z. to the zero

section, that is, the image of c under the composition

Hn.D.E.; S.E.; Z.!Hn.D.E.; Z.!Hn.B; Z.

where the first map is the usual passage from relative to absolute cohomology and the

second map is induced by the inclusion B>D.E. as the zero section. By its definition,

e.E. depends on the choice of c . However, the assertion (*) in the construction of a

Thom class in Theorem 4D.10 of [AT] implies that c is determined by its restriction to

each fiber, and the restriction of c to each fiber is in turn determined by an orientation

of the bundle, so in fact e.E. depends only on the choice of an orientation of E.

Choosing the opposite orientation changes the sign of c . There are exactly two choices

of orientation for each path-component of B.

Here are the basic properties of Euler classes e.E. 2 Hn.B; Z. associated to oriented

n dimensional vector bundles E!B:

Proposition 3.26.

(a) An orientation of a vector bundle E!B induces an orientation of a pullback

bundle f _.E. such that e.f_.E.. . f _.e.E...

(b) Orientations of vector bundles E1!B and E2!B determine an orientation of

the sum E1

_E2 such that e.E1

_E2. . e.E1. ` e.E2..

(c) For an orientable n dimensional real vector bundle E, the coefficient homomorphism

Hn.B; Z.!Hn.B; Z2. carries e.E. to wn.E.. For an n dimensional complex

vector bundle E there is the relation e.E. . cn.E. 2 H2n.B; Z., for a suitable

choice of orientation of E.

(d) e.E. . −e.E. if the fibers of E have odd dimension.

(e) e.E. . 0 if E has a nowhere-zero section.

Proof: (a) For an n dimensional vector bundle E, let E0 _ E be the complement of

the zero section. A Thom class for E can be viewed as an element of Hn.E; E0; Z.

88 Chapter 3 Characteristic Classes

which restricts to a generator of Hn.Rn;Rn −f0g;Z. in each fiber Rn . For a pullback

f _.E., we have a map e f : f _.E.!E which is a linear isomorphism in each fiber, so

e f _.c.E.. restricts to a generator of Hn.Rn;Rn − f0g;Z. in each fiber Rn of f _.E..

Thus e f _.c.E.. . c.f_.E... Passing from relative to absolute cohomology classes

and then restricting to zero sections, we get e.f_.E.. . f _.e.E...

(b) There is a natural projection p1 : E1

_E2!E1 which is linear in each fiber, and

likewise we have p2 : E1

_E2!E2. If E1 is m dimensional we can view a Thom class

c.E1. as lying in Hm.E1; E0

1. where E0

1 is the complement of the zero section in E1 .

Similarly we have a Thom class c.E2. 2 Hn.E2; E0

2. if E2 has dimension n. Then

the product p_

1 .c.E1.. ` p_

2 .c.E2.. is a Thom class for E1

_E2 since in each fiber

Rm_Rn . Rm.n we have the cup product

Hm.Rm.n;Rm.n − Rn._Hn.Rm.n;Rm.n − Rm.z!Hm.n.Rm.n;Rm.n − f0g.

which takes generator cross generator to generator by the calculations in Example 3.11

of [AT]. Passing from relative to absolute cohomology and restricting to the zero section,

we get the relation e.E1

_E2. . e.E1. ` e.E2..

(c) We showed this for the universal bundle in the calculation of the cohomology of

Grassmannians a couple pages back, so by the naturality property in (a) it holds for

all bundles.

(d) When we defined the Euler class we observed that it could also be described as the

element of Hn.B; Z. corresponding to c ` c 2 H2n.D.E.; S.E.; Z. under the Thom

isomorphism. If n is odd, the basic commutativity relation for cup products gives

c ` c . −c`c, so e.E. . −e.E..

(e) A nowhere-zero section of E gives rise to a section s : B!S.E. by normalizing

vectors to have unit length. Then in the exact sequence

Hn.D.E.; S.E.; Z. j_ ----!Hn.D.E.; Z. i_ ----!Hn.S.E.; Z.

the map i_ is injective since the composition D.E.!- B --!-s S.E. --!-i D.E. is homotopic

to the identity. Since i_ is injective, the map j_ is zero by exactness, and hence

e.E. . 0 from the definition of the Euler class. tu

Consider the tangent bundle TSn to Sn . This bundle is orientable since its base

Sn is simply-connected if n > 1, while if n . 1, TS1 is just the product S1_R.

When n is odd, e.TSn. . 0 either by part (d) of the proposition since H_.Sn; Z. has

no elements of order two, or by part (e) since there is a nonzero tangent vector field

to Sn when n is odd, namely s.x1; ___;xn.1. . .−x2;x1; ___ ;−xn.1;xn.. However,

when n is even e.TSn. is nonzero:

Proposition 3.27. For even n, e.TSn. is twice a generator of Hn.Sn; Z..

Proof: Let E0 _ E . TSn be the complement of the zero section. Under the Thom

isomorphism the Euler class e.TSn. corresponds to the square of a Thom class

Euler and Pontryagin Classes Section 3.3 89

c 2 Hn.E; E0., so it suffices to show that c2 is twice a generator of H2n.E; E0.. Let

A _ Sn_Sn consist of the antipodal pairs .x;−x.. Define

a homeomorphism f : Sn_Sn − A!E sending a pair

.x;y. 2 Sn_Sn − A to the vector from x to the point of

intersection of the line through −x and y with the tangent

plane at x. The diagonal D . f.x;x.g corresponds under

x

y

- x

f to the zero section of E. Excision then gives the first of

the following isomorphisms:

H_.E; E0. _ H_.Sn_Sn; Sn_Sn −D. _ H_.Sn_Sn;A. _ H_.Sn_Sn;D.;

The second isomorphism holds since Sn_Sn − D deformation retracts onto A by

sliding a point y . _x along the great circle through x and y to −x, and the third

comes from the homeomorphism .x;y.,.x;−y. of Sn_Sn interchanging D and

A. From the long exact sequence of the pair .Sn_Sn;D. we extract a short exact

sequence

0!Hn.Sn_Sn;D.!Hn.Sn_Sn.!Hn.D.!0

The middle group Hn.Sn_Sn. has generators _, _ which are pullbacks of a generator

of Hn.Sn. under the two projections Sn_Sn!Sn . Both _ and _ restrict to

the same generator of Hn.D. since the two projections Sn_Sn!Sn restrict to the

same homeomorphism D _ Sn, so _−_ generates Hn.Sn_Sn;D., the kernel of the

restriction map Hn.Sn_Sn.!Hn.D.. Thus _ − _ corresponds to the Thom class

and ._−_.2 . −__−__, which equals −2__ if n is even. This is twice a generator

of H2n.Sn_Sn;D. _ H2n.Sn_Sn.. tu

It is a fairly elementary theorem in differential topology that the Euler class of

the unit tangent bundle of a closed, connected, orientable smooth manifold Mn is

j_.M.j times a generator of Hn.M., where _.M. is the Euler characteristic of M;

see for example [Milnor-Stasheff]. This agrees with what we have just seen in the case

M . Sn , and is the reason for the name ‘Euler class.’

One might ask which elements of Hn.Sn. can occur as Euler classes of vector

bundles E!Sn in the nontrivial case that n is even. If we form the pullback of the

tangent bundle TSn by a map Sn!Sn of degree d, we realize 2d times a generator,

by part (a) of the preceding proposition, so all even multiples of a generator of Hn.Sn.

are realizable. To investigate odd multiples, consider the Thom space T.E.. This

has integral cohomology consisting of Z’s in dimensions 0, n, and 2n by the Thom

isomorphism, which also says that the Thom class c is a generator of Hn.T .E... We

know that the Euler class corresponds under the Thom isomorphism to c`c, so e.E.

is k times a generator of Hn.Sn. iff c `c is k times a generator of H2n.T .E... This

is precisely the context of the Hopf invariant, and the solution of the Hopf invariant

one problem in Chapter 2 shows that c ` c can be an odd multiple of a generator

90 Chapter 3 Characteristic Classes

only if n . 2, 4, or 8. In these three cases there is a bundle E!Sn for which c ` c

is a generator of H2n.T .E.., namely the vector bundle whose unit sphere bundle is

the complex, quaternionic, or octonionic Hopf bundle, and whose Thom space, the

mapping cone of the sphere bundle, is the associated projective plane CP2 , HP2, or

OP2. Since we can realize a generator of Hn.Sn. as an Euler class in these three cases,

we can realize any multiple of a generator by taking pullbacks as before.

Pontryagin Classes

The easiest definition of the Pontryagin classes pi.E. 2 H4i.B; Z. associated to

a real vector bundle E!B is in terms of Chern classes. For a real vector bundle

E!B, its complexification is the complex vector bundle EC!B obtained from the

real vector bundle E_E by defining scalar multiplication by the complex number i

in each fiber Rn_Rn via the familiar rule i.x;y. . .−y;x.. Thus each fiber Rn

of E becomes a fiber Cn of EC . The Pontryagin class pi.E. is then defined to be

.−1.ic2i.EC. 2 H4i.B; Z.. The sign .−1.i is introduced in order to avoid a sign in the

formula in (b) of the next proposition. The reason for restricting attention to the even

Chern classes c2i.EC. is that the odd classes c2i.1.EC. turn out to be expressible in

terms of Stiefel-Whitney classes, and hence give nothing new. The exercises at the

end of the section give an explicit formula.

Here is how Pontryagin classes are related to Stiefel-Whitney and Euler classes:

Proposition 3.28.

(a) For a real vector bundle E!B, pi.E. maps to w2i.E.2 under the coefficient

homomorphism H4i.B; Z.!H4i.B; Z2..

(b) For an orientable real 2n dimensional vector bundle with Euler class e.E. 2

H2n.B; Z., pn.E. . e.E.2 .

Note that statement (b) is independent of the choice of orientation of E since the

Euler class is squared.

Proof: (a) By Proposition 3.4, c2i.EC. reduces mod 2 to w4i.E_E., which equals

w2i.E.2 since w.E_E. . w.E.2 and squaring is an additive homomorphism mod 2.

(b) First we need to determine the relationship between the two orientations of EC _

E_E, one coming from the canonical orientation of the complex bundle EC , the

other coming from the orientation of E_E determined by an orientation of E. If

v1;___ ; v2n is a basis for a fiber of E agreeing with the given orientation, then EC

is oriented by the ordered basis v1; iv1; ___ ; v2n; iv2n, while E_E is oriented by

v1; ___ ; v2n; iv1; ___ ; iv2n. To make these two orderings agree requires .2n − 1. .

.2n − 2. . ___.1 . 2n.2n − 1.=2 . n.2n − 1. transpositions, so the two orientations

differ by a sign .−1.n.2n−1. . .−1.n . Thus we have pn.E. . .−1.nc2n.EC. .

.−1.ne.EC. . e.E_E. . e.E.2 . tu

Euler and Pontryagin Classes Section 3.3 91

Pontryagin classes can be used to describe the cohomology of Gn and eG

n with Z

coefficients. First let us remark that since Gn has a CW structure with finitely many

cells in each dimension, so does eG

n , hence the homology and cohomology groups of

Gn and eG

n are finitely generated. For the universal bundles En!Gn and e En!eG

n

let pi

. pi.En., e pi

. pi. e En., and e . e. e En., the Euler class being defined via the

canonical orientation of e En .

Theorem 3.29.

(a) All torsion in H_.Gn; Z. consists of elements of order 2, and H_.Gn; Z.=torsion

is the polynomial ring Z.p1; ___;pk. for n . 2k or 2k . 1.

(b) All torsion in H_.eG

n; Z. consists of elements of order 2, and H_.eG

n; Z.=torsion

is Z. e p1; ___ ; e pk. for n . 2k.1 and Z. e p1; ___ ; e pk−1; e. for n . 2k, with e2 . e pk

in the latter case.

The torsion subgroup of H_.Gn; Z. therefore maps injectively to H_.Gn; Z2.,

with image the image of the Bockstein _:H_.Gn; Z2.!H_.Gn; Z2., which we shall

compute in the course of proving the theorem; for the definition and basic properties

of Bockstein homomorphisms see x3.E of [AT]. The same remarks apply to H_.eG

n; Z..

The theorem implies that Stiefel-Whitney and Pontryagin classes determine all characteristic

classes for unoriented real vector bundles, while for oriented bundles the

only additional class needed is the Euler class.

Proof: We shall work on (b) first since for orientable bundles there is a Gysin sequence

with Z coefficients. As a first step we compute H_.eG

n; R. where R . Z.1=2. _ Q, the

rational numbers with denominator a power of 2. Since we are dealing with finitely

generated integer homology groups, changing from Z coefficients to R coefficients

eliminates any 2 torsion in the homology, that is, elements of order a power of 2, and

Z summands of homology become R summands. The assertion to be proved is that

H_.eG

n; R. is R. e p1; ___ ; e pk. for n . 2k . 1 and R. e p1; ___ ; e pk−1; e. for n . 2k. This

implies that H_.eG

n; Z. has no odd-order torsion and that H_.eG

n; Z.=torsion is as

stated in the theorem. Then it will remain only to show that all 2 torsion in H_.eG

n; Z.

consists of elements of order 2.

As in the calculation of H_.Gn; Z2. via the Gysin sequence, consider the sphere

bundle Sn−1 -!S. eEn. _ ----! eG

n , where S. eEn. is the space of pairs .v; `. where ` is

an oriented n dimensional linear subspace of R1 and v is a unit vector in `. The

orthogonal complement v? _ ` of v is then naturally oriented, so we get a projection

p : S. eEn.!eG

n−1 . The Gysin sequence with coefficients in R has the form

___ -!Hi.eG

n. --`-------!-e Hi.n.eG

n. _ ----!Hi.n.eG

n−1.!- Hi.1.eG

n.!- ___

where _ takes e pi. e En. to e pi. e En−1..

92 Chapter 3 Characteristic Classes

If n . 2k, then by induction H_.eG

n−1. _ R. e p1; ___ ; e pk−1., so _ is surjective and

the sequence splits into short exact sequences. The proof in this case then follows

the H_.Gn; Z2. model.

If n . 2k . 1, then e is zero in Hn.eG

n; R. since with Z coefficients it has order

2. The Gysin sequence now splits into short exact sequences

0 -!Hi.n.eG

n. --!-_ Hi.n.eG

n−1.!- Hi.1.eG

n.!- 0

Thus _ injects H_.eG

n. as a subring of H_.eG

n−1. _ R. e p1; ___ ; e pk−1; e., where e now

means e. e En−1.. The subring Im_ contains R. e p1; ___ ; e pk. and is torsionfree, so we

can show it equals R. e p1; ___ ; e pk. by comparing ranks of these R modules in each

dimension. Let rj be the rank of R. e p1; ___ ; e pk. in dimension j and r 0

j the rank

of Hj.eG

n.. Since R. e p1; ___ ; e pk−1; e. is a free module over R. e p1; ___ ; e pk. with basis

f1; eg, the rank of H_.eG

n−1. _ R. e p1; ___ ; e pk−1; e. in dimension j is rj

. rj−2k , the

class e . e. e En−1. having dimension 2k. On the other hand, the exact sequence above

says this rank also equals r 0

j

. r 0

j−2k . Since r 0m

_ rm for each m, we get r 0

j

. rj , and

so H_.eG

n. . R. e p1; ___ ; e pk., completing the induction step. The induction can start

with the case n . 1, with eG

1 _ S1.

Before studying the remaining 2 torsion question let us extend what we have just

done to H_.Gn; Z., to show that for R . Z.1=2., H_.Gn; R. is R.p1; ___;pk., where

n . 2k or 2k . 1. For the 2 sheeted covering _ : eG

n!Gn consider the transfer homomorphism

__ :H_.eG

n; R.!H_.Gn; R. defined in x3.G of [AT]. The main feature

of __ is that the composition ____ :H_.Gn; R.!H_.eG

n; R.!H_.Gn; R. is multiplication

by 2, the number of sheets in the covering space. This is an isomorphism

for R . Z.1=2., so __ is injective. The image of __ contains R. e p1; ___ ; e pk. since

__.pi. . e pi . So when n is odd, __ is an isomorphism and we are done. When n

is even, observe that the image of __ is invariant under the map __ induced by the

deck transformation _ : eG

n!eG

n interchanging sheets of the covering, since __ . _

implies ____ . __ . The map _ reverses orientation in each fiber of e En!eG

n, so __

takes e to −e. The subring of H_.eGn; R. _ R. e p1; ___ ; e pk−1; e. invariant under __ is

then exactly R. e p1; ___ ; e p.n=2.., finishing the proof that H_.Gn; R. . R.p1; ___;pk..

To show that all 2 torsion in H_.Gn; Z. and H_.eG

n; Z. has order 2 we use the

Bockstein homomorphism _ associated to the short exact sequence of coefficient

groups 0!Z2!Z4!Z2!0. The goal is to show that Ker _= Im_ consists exactly

of the mod 2 reductions of nontorsion classes in H_.Gn; Z. and H_.eG

n; Z., that is,

polynomials in the classes w2

2i in the case of Gn and eG

2k.1 , and for eG

2k , polynomials

in the w2

2i ’s for i < k together with w2k , the mod 2 reduction of the Euler class. By

general properties of Bockstein homomorphisms proved in x3.E of [AT] this will finish

the proof.

Lemma 3.30. _w2i.1 . w1w2i.1 and _w2i

. w2i.1 .w1w2i .

Exercises 93

Proof: By naturality it suffices to prove this for the universal bundle En!Gn with

wi

. wi.En.. As observed in x3.1, we can view wk as the kth elementary symmetric

polynomial _k in the polynomial algebra Z2._1; ___;_n. _ H_..RP1.n; Z2.. Thus

to compute _wk we can compute __k . Using the derivation property _.x ` y. .

_x ` y . x ` _y and the fact that __i

. _2i

, we see that __k is the sum of all

products _i1 ____2

ij

____ik for i1 < ___ < ik and j . 1; ___ ; k. On the other hand,

multiplying _1_k out, one obtains __k

. .k . 1._k.1 . tu

Now for the calculation of Ker _= Im_. First consider the case of G2k.1 . The ring

Z2.w1; ___;w2k.1. is also the polynomial ring Z2.w1;w2; _w2; ___;w2k; _w2k. since

the substitution w1,w1;w2i,w2i;w2i.1,w2i.1 .w1w2i

. _w2i for i > 0 is

invertible, being its own inverse in fact. Thus Z2.w1; ___;w2k.1. splits as the tensor

product of the polynomial rings Z2.w1. and Z2.w2i; _w2i., each of which is invariant

under _. Moreover, viewing Z2.w1; ___;w2k.1. as a chain complex with boundary

map _, this tensor product is a tensor product of chain complexes. According to

the algebraic K¨unneth theorem, the homology of Z2.w1; ___;w2k.1. with respect to

the boundary map _ is therefore the tensor product of the homologies of the chain

complexes Z2.w1. and Z2.w2i; _w2i..

For Z2.w1. we have _.w`

1 . . `w`.1

1 , so Ker_ is generated by the even powers

of w1 , all of which are also in Im_, and hence the _ homology of Z2.w1. is trivial in

positive dimensions; we might remark that this had to be true since the calculation is

the same as for RP1. For Z2.w2i; _w2i. we have _.w`

2i._w2i.m. . `w`−1

2i ._w2i.m.1 ,

so Ker_ is generated by the monomials w`

2i._w2i.m with ` even, and such monomials

with m>0 are in Im_. Hence Ker _= Im_ . Z2.w2

2i..

For n . 2k, Z2.w1; ___;w2k. is the tensor product of the Z2.w2i; _w2i.’s for

i < k and Z2.w1;w2k., with _.w2k. . w1w2k . We then have the formula _.w`

1wm

2k. .

`w`.1

1 wm

2k

.mw`.1

1 wm

2k

. .` .m.w`.1

1 wm

2k . For w`

1wm

2k to be in Ker_ we must have

` . m even, and to be in Im_ we must have in addition ` > 0. So Ker _= Im_ .

Z2.w2

2k..

Thus the homology of Z2.w1; ___;wn. with respect to _ is the polynomial ring in

the classes w2

2i , the mod 2 reductions of the Pontryagin classes. By general properties

of Bocksteins this finishes the proof of part (a) of the theorem.

The case of eG

n is simpler since w1 . 0, hence _w2i

. w2i.1 and _w2i.1 . 0.

Then we can break Z2.w2; ___;wn. up as the tensor product of the chain complexes

Z2.w2i;w2i.1., plus Z2.w2k. when n . 2k. The calculations are quite similar to those

we have just done, so further details will be left as an exercise. tu

Exercises

1. Show that every class in H2k.CP1. can be realized as the Euler class of some vector

bundle over CP1 that is a sum of complex line bundles.

94 Chapter 3 Characteristic Classes

2. Show that c2i.1.EC. . _.w2i.E.w2i.1.E...

3. For an oriented .2k . 1. dimensional vector bundle E show that e.E. . _w2k.E..

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B. L. van der Waerden, Modern Algebra, Ungar, 1949.

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