A note on the equivariant Dold–Thom theorem

Abstract

In this note we prove a version of the classical Dold–Thom theorem for the RO( G)-graded equivariant homology functors H ∗ G(−; M ̄ ) , where G is a finite group, M is a discrete Z[G] -module, and M̱ is the Mackey functor associated to M. In the case where M= Z with the trivial G-action, our result says that, for a G-CW-complex X, and for a finite dimensional G-representation V, there is a natural isomorphism [S V, Z 0(X)] G≅H V G(X; Z ), where Z 0(X) denotes the free abelian group on X.