2.4. Further Calculations

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In this section we give computations of the K–theory of some other interesting

spaces.

The Thom Isomorphism

The relative form of the Leray-Hirsch theorem for disk bundles is a useful technical

result known as the Thom isomorphism:

Proposition 2.27. Let p : E!B be a fiber bundle with fibers Dn and with base B a

finite cell complex, and let E0!B be the sphere subbundle with fibers the boundary

spheres of the fibers of E. If there is a class c 2 K_.E; E0. which restricts to a

generator of K_.Dn; Sn. _ Z in each fiber, then the map Ø :K_.B.!K_.E; E0.,

Ø.b. . p_.b. _ c , is an isomorphism.

The class c is called a Thom class for the bundle. As we will show below, the unit

disk bundle in every complex vector bundle has a Thom class.

Proof: Let b E!B be the bundle with fiber Sn obtained as a quotient of E by collapsing

each fiber of the subbundle E0 to a point. The union of these points is a copy of B

in b E forming a section of b E. The long exact sequence for the pair . b E; B. then splits,

giving an isomorphism K_. b E. _ K_. b E; B._K_.B.. Under this isomorphism the class

c 2 K_.E; E0. . K_. b E; B. corresponds to a class bc 2 K_. b E., which, together with the

62 Chapter 2 Complex K–Theory

element 1 2 K_. b E., allows us to define the left-hand Ø in the following commutative

diagram, where _ is a point.

¡¡!

¤ ¡¡¡! ­

Ø

¡¡!

Ø Ø

K (B ) , ¤ K (S ) ¤ K (B )­ ¤ ¤ K (

¤ ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡! K (E ) ¤ K (B)

n Sn ) ¤ K (B )­ ¤ ¤ © K ( )

©

©

b ¤ K (Eb,B)

¼

¼

The Leray-Hirsch theorem implies that the left-hand Ø is an isomorphism, hence both

Ø’s on the right-hand side of the diagram are isomorphisms as well. tu

Example 2.28. For a complex vector bundle E!X with X compact Hausdorff we will

now show how to find a Thom class U 2 eK

.D.E.; S.E.., where D.E. and S.E. are the

unit disk and sphere bundles in E. We can also regard U as an element of e K.T.E..

where the Thom space T.E. is the quotient D.E.=S.E.. Since X is compact, T.E. can

also be described as the one-point compactification of E. We may view T.E. as the

quotient P.E_1.=P.E. since in each fiber Cn of E we obtain P.Cn_C. . CPn from

P.Cn. . CPn−1 by attaching the 2n cell Cn_f1g, so the quotient P.Cn_C.=P.Cn.

is S2n , which is the part of T.E. coming from this fiber Cn . From Example 2.24 we

know that K_.P.E_1.. is the free K_.X. module with basis 1; L; ___ ; Ln, where L is

the canonical line bundle over P.E_1.. Restricting to P.E. _ P.E_1.;K_.P.E.. is

the free K_.X. module with basis the restrictions of 1; L; ___ ; Ln−1 to P.E.. So we

have a short exact sequence

0 -!eK

_.T .E.. -!K_.P.E_1.. --!-_ K_.P.E.. -!0

and Ker _ must be generated as a K_.X. module by some polynomial of the form

Ln . an−1Ln−1 . ___ . a01 with coefficients ai

2 K_.X., namely the polynomial

P

i.−1.i_i.E.Ln−i in Proposition 2.25, regarded now as an element of K.P.E_1...

The class U 2 e K.T.E.. mapping to

P

i.−1.i_i.E.Ln−i is the desired Thom class

since when we restrict over a point of X the preceding considerations still apply, so

the kernel of K.CPn.!K.CPn−1. is generated by the restriction of

P

i.−1.i_i.E.Ln−i

to a fiber.

[More applications will be added later: the Gysin Sequence, the K¨unneth formula,

and calculations of the K–theory of various spaces including Grassmann manifolds,

flag manifolds, the group U.n., real projective space, and lens spaces.]

Characteristic classes are cohomology classes in H_.B; R. associated to vector

bundles E!B by some general rule which applies to all base spaces B. The four

classical types of characteristic classes are:

1. Stiefel-Whitney classes wi.E. 2 Hi.B; Z2. for a real vector bundle E.

2. Chern classes ci.E. 2 H2i.B; Z. for a complex vector bundle E.

3. Pontryagin classes pi.E. 2 H4i.B; Z. for a real vector bundle E.

4. The Euler class e.E. 2 Hn.B; Z. when E is an oriented n dimensional real vector

bundle.

The Stiefel-Whitney and Chern classes are formally quite similar. Pontryagin classes

can be regarded as a refinement of Stiefel-Whitney classes when one takes Z rather

than Z2 coefficients, and the Euler class is a further refinement in the orientable case.

Stiefel-Whitney and Chern classes lend themselves well to axiomatization since

in most applications it is the formal properties encoded in the axioms which one uses

rather than any particular construction of these classes. The construction we give,

using the Leray-Hirsch theorem (proved in x4.D of [AT]), has the virtues of simplicity

and elegance, though perhaps at the expense of geometric intuition into what properties

of vector bundles these characteristic classes are measuring. There is another

definition via obstruction theory which does provide some geometric insights, and

this will be described in the Appendix to this chapter.

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