`Group-completion', local coefficient systems and perfection (Q2851202)

Let \(M\) be a homotopy commutative monoid, where one imposes suitable conditions on the monoid of path components of \(M\). The group completion theorem, as stated by \textit{D. McDuff} and \textit{G. Segal} in [Invent. Math. 31, 279--284 (1976; Zbl 0306.55020)], describes the relationship between the homology of \(M\) and that of \(\Omega BM\). At the level of topological spaces it states that a certain telescope \(M_\infty\) obtained by iterating multiplication on \(M\) by representatives of path-components is homology equivalent to \(\Omega BM\).NEWLINENEWLINEIn many applications it turns out that one can even identify the loop space \(\Omega BM\) with the Quillen plus-construction \(M_\infty^+\). That is, the homology equivalence is in fact an \textit{acyclic map}. The aim of this short and nice article is precisely this: to prove that the McDuff-Segal comparison map is not only a homology equivalence with integral coefficients, but a homology equivalence with all systems of local coefficients, hence acyclic. As a consequence, the commutator subgroup of \(\pi_1(M_\infty)\) must be perfect.