An Equivariant Generalisation of McDuff–Segal’s Group–Completion Theorem

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 $\mathbb{E}_{\infty }$–rings in $G$–spectra. We check that this condition is satisfied when our inputs are a suitable variant of $\mathbb{E}_{\infty }$–monoids in $G$–spaces via the existence of multiplicative norm structures, thus giving a localisation formula for their associated $G$–spherical group rings.