UCalgary Peripatetic Seminar: Blake Whiting

The Eilenberg-Zilber (EZ) theorem is a powerful tool in homological algebra and algebraic topology, being a key ingredient in the Kunneth theorem. It also serves as the basis for defining the cup product, which in turn establishes cohomology as a graded ring. Since its initial proof in 1953, the main generalization beyond simplicial R-modules has been to bisimplicial objects in abelian categories.

The method of proof, acyclic models, was introduced together with the first EZ theorem in 1953. Acyclic models is actually a theorem of Eilenberg and Maclane’s, and was originally used to prove the equivalence of singular homology based, respectively, on simplices and on cubes. In his 2002 monograph “Acyclic models”, Barr collects various versions of the acyclic models theorem, then shows its applications in Cartan-Eilenberg cohomology and homology on manifolds (among other things). 

In this talk I will introduce all of the concepts required to understand EZ theorems, as well as acyclic models. We will examine recent developments in these concepts to motivate a conjecture for an EZ theorem with weaker hypotheses. Finally, we will also examine how acyclic models is used to prove the most recent EZ theorem, revealing potential obstacles to generalizing it further.