Cohomology Theories and Brown Representability

Abstract

In Chapter 7 we presented cohomology theory, and in Chapter 9 we introduced K-theory. Both theories have some properties in common. In this chapter we unify these properties and define the generalized cohomology theories. Prom this point of view we shall be able to obtain several results that follow from the formal properties rather than from the specific characteristics of the theory in question. Further, we shall prove a theorem that shows that our approach to both theories is quite general. Namely, we prove the Brown representability theorem, which shows that in an adequate category of spaces every generalized cohomology theory is represented by some classifying spaces, such as the Eilenberg-Mac Lane spaces in the case of cohomology and the spaces BU × ℤ and BU in the case of K-theory. Thus cohomology can always be expressed in homotopical terms. Finally we see that the representability of the cohomology theories implies the existence of certain objects, called spectra, which topologically, or better, homotopically, encode all the information concerning their associated cohomology and homology theories.