Characteristic Classes

Abstract

In Chapter 11 we saw that isomorphism classes of vector bundles with structure group G(n) over a CW-complex X were classified by homotopy classes of maps f:X→ BG(n)of X into the classifying space BG(n) for G(n)-bundles. If ξ, η are two G(n)-bundles with classifying maps f_ξ, f_η:X→BG(n), then ξ ≃ ηif and only if f_ξ ≃f _η . Suppose we wanted to prove ξ, η were not isomorphic. We might try to show that f_ξ, and f_η were not homotopic. There are two disadvantages to this approach: i) given a vector bundle ξ, as vector bundle it is usually very difficult to describe its classifying map f_ξ; ii) the problem of showing directly that two given functions are not homotopic is at least as difficult in general as showing two vector bundles are not isomorphic. However, we do have one standard trick for showing two functions are not homotopic: for any cohomology theory k* if f*_ξ ≠ f*_η:k*(BG(n)) → k*(X), then f_ξ ≄f_η and hence ξ ≄ η. Therefore we look for an appropriate k* and some x ∈ k*(BG(n))such that f*_ξ(x) ≠ f*_η(x) ∈ k*(X).