Constructing equivariant homotopy equivalences via extension of scalars (Q721497)

The paper is dedicated to prove the following result: Let \(R\) be a differential non-negatively graded algebra over \(\mathbb{Z}\) whose underlying chain complex is a \(\mathbb{Z}\)-complex with \(1\) as generator. Let \(M\) be a free locally effective \((R,d)\)-module and let \(N\) be a locally effective and locally finite \(\mathbb{Z}\)-complex. Given a locally effective strong homotopy equivalence of \(\mathbb{Z}\)-complexes \(M \Leftrightarrow N\), there is an algorithm which constructs a free locally effective and locally finite \((R, d)\)-module \(N'\) and a locally effective \(R\)-linear strong homotopy equivalence \(M \Leftrightarrow N'\). The most relevant statement in the paper is the particular case \(R = \mathbb{Z}G\) in the above main result, which describes a constructive algorithmic process to obtain the strong homotopy equivalence.