Cohomology classes as homotopy classes: CW-complexes

Abstract

We consider filtered spaces and especially CW-complexes. Using the cofibre constructions, we discuss the Whitehead mapping theorem. This characterization of homotopy equivalence was already used in the study of the uniqueness properties of classifying spaces. This completes a question left open in the previous chapter. The usual definition of cohomology arises in terms of the dual of homology either directly for coefficients in a field or in terms of chains and cochains as linear forms on chain groups. These chains can arise very geometrically as cell chains or more generally as simplicial chains. The cochains can be algebraic linear functionals or as in the case of manifolds they can be differential forms which become linear functionals on chains of simplexes upon integration over the simplexes and summing over the chain. There is another perspective on the cohomology of X discovered by Eilenberg and MacLane in terms of homotopy classes of maps of X into an Eilenberg - MacLane space K(G,n). This is a topological space which has the homotopy type of a CW-complex and is characterized by its homotopy groups A formula is presented. There is a canonical cohomology class ιn ∈ Hn(K(G,n),G) corresponding to the identity morphism in Hom(G,G), where Hom(G,G) = Hom(Hn(G,n),G) = Hn (K(G,n),G) is part of the universal coefficient theorem. In this chapter, we use the notion of cofibre maps and the cofibre sequence for a map. This is needed to establish cohomology properties of the K-theory functor and in the discussion of the above result that cohomology is defined also by maps into a classifying space which in this case it is a K(π,n). Spanier (1963) is a reference for this chapter. © Springer-Verlag Berlin Heidelberg 2008.