A remark on the group-completion theorem

2017/09/07 by Simon Gritschacher, Gritschacher, Simon
Mathematics · #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1709.02036

openalex publication_date 2017/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose that M is a topological monoid satisfying π₀M=ℕ to which the McDuff-Segal group-completion theorem applies. This implies that a certain map f: \mathbbM_∞→ ΩBM 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 H₁(M), then the "McDuff-Segal comparison map" f is acyclic. For example, this always holds if π₀M lies in the centre of the Pontryagin ring H_∗(M). As an application we describe conditions on a commutative \mathbbI-monoid X under which hocolim_\mathbbIX can be identified with a Quillen plus-construction.

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