An equivariant generalisation of McDuff-Segal's group-completion theorem

TL;DR

This paper extends McDuff-Segal's group-completion theorem to the G-equivariant setting for finite groups, addressing new localization challenges with a condition on homotopy groups of E-infinity-rings in G-spectra.

Contribution

It introduces a G-equivariant generalization of the group-completion theorem, resolving localization issues with a homotopy group condition and applying it to G-spherical group rings.

Findings

Established a G-equivariant group-completion theorem.

Identified a homotopy group condition for localization.

Applied results to G-spherical group rings.

Abstract

In this short note, we prove a G-equivariant generalisation of McDuff-Segal's group-completion theorem for finite groups G. A new complication regarding genuine equivariant localisations arises and we resolve this by isolating a simple condition on the homotopy groups of E-infinity-rings in G-spectra. We check that this condition is satisfied when our inputs are a suitable variant of E-infinity-monoids in G-spaces via the existence of multiplicative norm structures, thus giving a localisation formula for their associated G-spherical group rings.