Explicit Acyclic Models and (Co)Chain Operations

TL;DR

This paper introduces a recursive, explicit method for constructing chain complex morphisms, unifying various classical and modern chain map constructions, with applications to operads and cohomology operations.

Contribution

It provides a uniform, explicit recursive procedure for constructing chain maps, clarifying and unifying many classical and operad-related chain map constructions.

Findings

Unified recursive procedure for chain maps

Explicit models for operad structures

Characterization theorems for chain maps

Abstract

We exploit a uniform recursive procedure using preferred contractions of targets $C_*$ to construct morphisms $B_* \to C_*$ between chain complexes in a wide variety of situations. 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. The procedure includes functorial chain maps $F_*(X_i) \to K_*(X_i)$, using models for generators of the domain and contractions of $K_*$ applied to these models. Various uniqueness theorems characterize the chain maps produced by our procedures. We give unified extended treatments of the operads known as the Barratt-Eccles operad and the surjection operads. In a subsequent 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. Classical treatments of Steenrod operations used less explicit acyclic model methods to deduce properties of operations at the cohomology level. Our paper is comprehensive and includes many examples and specific results of a somewhat technical nature. But the main point is that a rather easily described explicit procedure clarifies aspects of chain maps defined only up to homotopy that were important in the development of cohomology operations roughly seventy years ago, as well as complicated chain maps related to operad structure maps and operad morphisms developed in a rather ad hoc manner roughly twenty five years ago. Sections 0, 1, and 2 constitute an overview and summary of the paper.