Equivariant K-homology and restriction to finite cyclic subgroups

For a discrete group G, we prove that a G-map between proper G-CW-complexes induces an isomorphism in G-equivariant K-homology if it induces an isomorphism in C-equivariant K-homology for every finite cyclic subgroup C of G. As an application, we show that the source of the Baum-Connes assembly map, namely K-not asymptotic to(G)(E(G, F in)), is isomorphic to K-not asymptotic to(G) (E(G, FC)), where E(G, FC) denotes the classifying space for the family of finite cyclic subgroups of G. Letting VC be the family of virtually cyclic subgroups of G, we also establish that K-not asymptotic to(G) (E(G, F in)) congruent to K-not asymptotic to(G) (E(G, VC)) and related results.