Explicit Acyclic Models and (Co)Chain Operations

Abstract

We exploit a uniform recursive procedure to construct morphisms of chain complexes in a wide variety of situations, using free bases of domains and preferred null homotopies $h$ of targets. Preferred null homotopies satisfy $dh+hd=Id$ and $h^2=0$. In the presence of group actions we construct equivariant chain maps. The basic procedure extends to construct functorial maps $F_{\ast }(X_i)\to K_{\ast }(X_i)$, where $F_{\ast }$ and $K_{\ast }$ are chain complex functors of one or more simplicial set variables $X_i$, using contractible models for generators of the domain and null homotopies of $K_{\ast }$ applied to these models. Examples include classical Alexander-Whitney and Eilenberg-Zilber maps, chain maps related to homology and cohomology operations, operad structure maps for various chain complex operads, and chain maps that define morphisms between operads. Various uniqueness theorems characterize the chain maps produced by our procedures. We give new unified extended treatments of the operads known as the Barratt-Eccles operad and the surjection operads. In another paper we plan to use the results and methods of this paper to establish properties of the Steenrod algebra of mod p cohomology operations at the cochain level. The point is that an easily described explicit procedure clarifies aspects of chain maps that were important in the development of cohomology operations roughly seventy five years ago, as well as complicated chain maps related to operad structure maps and operad morphisms developed in a somewhat ad hoc manner roughly twenty five years ago.

v3 contains improved discussions of chain complexes, chain homotopies constructed by our methods, basic facts about operads, and operads in categories of chain complexes. More details are given in the proofs of the main theorems and additional discussion of applications to Steenrod operations has been included