Products

Abstract

We have now described homology theories in general and three important particular examples. Let us not loose sight of one of the reasons for studying homology functors: one wants to investigate the existence or nonexistence of maps f: X → Y by looking at the corresponding algebraic morphisms f_*:k_*(X) → k_*(Y). As we have said before, the richer the algebraic structure on k_*(X), the more useful k_* will be for these investigations. In this chapter we introduce products, so that under appropriate assumptions k*(X) will be a ring for all X. In Chapters 17, 18 and 19 we introduce another very useful algebraic structure: we make E_*(X) into a comodule over a certain Hopf algebra (and E*(X) into a module over the dual Hopf algebra).