A remark on the group-completion theorem

TL;DR

This paper provides an elementary proof that under certain conditions, the McDuff-Segal comparison map is acyclic, and explores implications for homotopy colimits of commutative monoids and their relation to Quillen plus-constructions.

Contribution

It offers a new elementary proof of the acyclicity of the McDuff-Segal comparison map under specific conditions and links homotopy colimits of commutative monoids to Quillen plus-constructions.

Findings

The comparison map is acyclic if left- and right-stabilisation commute on H_1(M).

If π_0M is central in the Pontryagin ring, the map always is acyclic.

Conditions are identified under which homotopy colimits relate to Quillen plus-constructions.

Abstract

Suppose that $M$ is a topological monoid satisfying $\pi_0M=\mathbb{N}$ to which the McDuff-Segal group-completion theorem applies. This implies that a certain map $f: \mathbb{M}_{\infty}\rightarrow \Omega BM$ defined on an infinite mapping telescope is a homology equivalence with integer coefficients. In this short note we give an elementary proof of the result that if left- and right-stabilisation commute on $H_1(M)$, then the "McDuff-Segal comparison map" $f$ is acyclic. For example, this always holds if $\pi_0M$ lies in the centre of the Pontryagin ring $H_{\ast}(M)$. As an application we describe conditions on a commutative $\mathbb{I}$-monoid $X$ under which $\text{hocolim}_{\mathbb{I}}X$ can be identified with a Quillen plus-construction.