Realizing dimension functions via homology

The following theorem is the main result of the paper: Theorem. Let G be an Abelian group and m > 0. Let G be a countable family of countable Abelian groups and let D : G → Z+ be a function. The following conditions are equivalent: 1. r any CW complex P and any a ϵ Hm(P;G) − {0} there is a compactum X and a map π : X → P such that dim X = m, dimH X ⩽ D(H) for each H ϵ G and a ϵ im(Ȟm(X ; G) → Ȟm(P;G)). k(K(H,D(H));G) = 0 for all k < m and all H ϵ G. application, we prove the existence of compacta realizing dimension functions, a result due to A.N. Dranishnikov.