Constructing homotopy equivalences of chain complexes of free ZG-modules

TL;DR

This paper introduces a general algorithmic method to construct G-equivariant chain homotopy equivalences from non-equivariant ones, enabling computation of equivariant (co)homology for certain spaces and advancing equivariant homotopy theory.

Contribution

It presents a novel algorithmic approach for constructing G-equivariant chain homotopy equivalences from non-equivariant ones, applicable to modules over graded differential graded algebras.

Findings

Algorithm for computing equivariant (co)homology of Eilenberg-MacLane spaces

Method for constructing G-equivariant chain homotopy equivalences

Framework applicable to modules over graded differential graded algebras

Abstract

We describe a general method for algorithmic construction of G-equivariant chain homotopy equivalences from non-equivariant ones. As a consequence, we obtain an algorithm for computing equivariant (co)homology of Eilenberg-MacLane spaces K(pi,n), where pi is a finitely generated ZG-module. This result will be used in a forthcoming paper to construct equivariant Postnikov towers of simply connected spaces with free actions of a finite group G and further to compute stable equivariant homotopy classes of maps between such spaces. The methods of this paper work for modules over any non-negatively graded differential graded algebra, whose underlying graded abelian group is free with 1 as one of the generators.