A functor converting equivariant homology to homotopy

Abstract

In this paper, an equivariant version of the classical Dold–Thom theorem is proved. Let G be a finite group, X a G-space, and k a covariant coefficient system on G. Then a topological abelian group 𝒢X ⊗Gℱ k is constructed by the coend construction. For a G-CW complex X, it is proved that there is a natural isomorphism π i ( 𝒢 X ⊗ G ℱ k ) ≅ H i G ( X k ), where the right-hand side is the Bredon equivariant homology of X with coefficients in k. At the end, several examples of this result are presented.