Topological Abelian Groups and Equivariant Homology

Abstract

We prove an equivariant version of the Dold-Thom theorem by giving an explicit isomorphism between Bredon-Illman homology H G * (X; L) and equivariant homotopical homology π * (F G (X, L)), where G is a finite group and L is a G-module. We use the homotopical definition to obtain several properties of this theory and we do some calculations.

The transfer for ramified covering G-maps

“…This generalizes to the equivariant case the definition in [11]. For any G-module L, using the topology on F G (X, L) described in [4], we prove that the transfer constructed in the previous section is continuous for any p (3.6). In Section 4, using the continuity of the transfer in the case of coefficients in a G-module L, we prove the continuity of the transfer with coefficients in a homological Mackey functor M , provided that E and X are strong ρ-spaces (4.7).…”

“…As already remarked, the group F G (Y, L) has a natural topology that was studied in [4]. This topology is defined as follows.…”

“…Remark 1. 4 We can always assume that an n-fold G-function with multiplicity is pointed by adding isolated points * to A and to C which remain fixed under the G-action and by defining µ( * ) = n. Therefore we shall always consider pointed n-fold G-functions with multiplicity without saying it explicitly.…”

“…By [4], ψ G L is a homeomorphism, and since X is a strong ρ-space, ρ G X• is an identification. Since by definition, π G X is also an identification, the result follows.…”

“…To construct the transfer, we shall use the homotopical definition of BredonIllman homology H G * (−; L) given in [4], when L is a G-module, and of H G * (X; M ) given in [5] when M is a homological Mackey functor. Namely, to each pointed G-space X , a G-module L and a homological Mackey functor M for G, we associate topological abelian groups F G (X, L) and F G (X, M ) such that π q (F G (X, L)) ∼ = H G (X; L) and π q (F G (X, M )) ∼ = H G (X; M ).…”