Cohomological dimension and acyclic resolutions

Let G be an Abelian group admitting a homomorphism α :Z→G such that the induced homomorphisms α⊗id :Z⊗G→G⊗G and α∗ :Hom(G,G)→Hom(Z,G) are isomorphisms. We show that for every simplicial complex L there exists an Edwards–Walsh resolution ω :EWG(L,n)→|L|. As applications of it we give several resolution theorems. In particular, we have m. Let G be an arbitrary Abelian group. For every compactum X with c-dimGX⩽n there exists a G-acyclic map f :Z→X from a compactum Z with dimZ⩽n+2 and c-dimGZ⩽n+1. thods determine other results as well. If the group G is cyclic, then one can obtain Z with dimZ⩽n. In certain other cases, depending on G, we may resolve X in such a manner that dimZ⩽n+1 and c-dimGZ⩽n.