Stable equivariant abelianization, its properties, and applications

TL;DR

This paper introduces a new stable equivariant abelianization for based G-spaces with Mackey functor coefficients, establishing its properties and applications, including an equivariant Dold-Thom theorem and a model for Eilenberg-Mac Lane spectra.

Contribution

It constructs the stable equivariant abelianization, proves it forms an infinite loop space for G-CW complexes, and provides models for Eilenberg-Mac Lane spectra in the equivariant setting.

Findings

X~⊗M is an infinite loop space for G-CW complexes

Provides a version of the RO(G)-graded equivariant Dold-Thom theorem

Constructs a model for the Eilenberg-Mac Lane spectrum HM

Abstract

Let $G$ be a finite group. For a based $G$-space $X$ and a Mackey functor $M$, a topological Mackey functor $X\widetilde\otimes M$ is constructed, which will be called the stable equivariant abelianization of $X$ with coefficients in $M$. When $X$ is a based $G$-CW complex, $X\widetilde\otimes M$ is shown to be an infinite loop space in the sense of $\mathcal{G}$-spaces. This gives a version of the $RO(G)$-graded equivariant Dold-Thom theorem. Applying a variant of Elmendorf's construction, we get a model for the Eilenberg-Mac Lane spectrum $HM$. The proof uses a structure theorem for Mackey functors and our previous results.