Equivariant Dold-Thom topological groups

TL;DR

This paper introduces new equivariant Dold-Thom topological groups that model Bredon-Illman G-equivariant homology, proving their uniqueness and applying them to verify axioms and compute low-dimensional homology.

Contribution

It constructs and analyzes new equivariant topological groups whose homotopy matches G-equivariant homology, establishing their uniqueness and applications.

Findings

New equivariant Dold-Thom groups constructed

Proved these groups are unique up to homotopy

Applied groups to verify the infinite-wedge axiom and compute 0th homology

Abstract

Let $M$ be a covariant coefficient system for a finite group $G$. In this paper we analyze several topological abelian groups, some of them new, whose homotopy groups are isomorphic to the Bredon-Illman $G$-equivariant homology theory with coefficients in $M$. We call these groups equivariant Dold-Thom topological groups and we show that they are unique up to homotopy. We use one of the new groups to prove that the Bredon-Illman homology satisfies the infinite-wedge axiom and to make some calculations of the 0th equivariant homology.