Universal acyclic resolutions for finitely generated coefficient groups

We prove that for every compactum X and every integer n ≥ 2 there are a compactum Z of dim ≤ n and a surjective UV n−1-map r : Z −→ X having the property that: for every finitely generated abelian group G and every integer k ≥ 2 such that dimG X ≤ k ≤ n we have dimG Z ≤ k and r is G-acyclic, or equivalently: for every simply connected CW-complex K with finitely generated homotopy groups such that e− dimX ≤ K we have e− dimZ ≤ K and r is K-acyclic. (A space is K-acyclic if every map from the space to K is nullhomotopic. A map is K-acyclic if every fiber is K-acyclic.)