Representation Theorems

Abstract

In Chapter 8 we saw how to associate a homology theory and a co-homology theory (both satisfying the wedge axiom) to a spectrum E. In this chapter we shall prove a converse result: given a cohomology theory k^* satisfying the wedge axiom on PW’ we shall construct a spectrum E and a natural equivalence of cohomology theories T: E* →p E* on PW’. In fact, we shall do somewhat more than that; for any cofunctor F*: PW’ → PJ satisfying the wedge axiom and a suitable exactness axiom we shall find a CW-complex (Y, y₀) and a natural equivalence T: [-; T, y₀] → F, *. We shall also prove such a theorem for cofunctors F, * defined only on the category PW’_F of finite CW-complexes provided F* takes values in G. F is called a classifying spacefor F *.