Products

Abstract

There are many products in (co-)homology theory of spaces; we shall treat about eight here. All of them are combinations of the following ingredients: (i) Relations between ® and Hom which are familiar from (multi-)linear algebra; (ii) the mappings α: HC⊗HD→H(C⊗D) and α: H Hom(C, D)→ Hom(HC, HD) of VI, 9.11, 10.8; (III) the Eilenberg Zilber mappings VI, 12.1—plus, of course, the standard functorial properties of (co-)homology. The significance of products lies in the extra structure which they introduce in (co-)homology. The-product, for instance, turns H* (X ; R) into a graded ring (cohomology ring) and makes H* (X; R) a functor from gop to the category GRg of graded rings (R a ring with unit). This functor provides a much more accurate picture of Top than the mere cohomology group which is obtained by composing H* (-, R) with the forget-functor F : (GRg→GAG (F assigns to every ring its additive group).