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Resolutions for metrizable compacta in extension theory

We prove a K-resolution theorem for simply connected CW-complexes K in extension theory in the class of metrizable compacta X. This means that if K is a connected CW-complex, G is an abelian group, n ∈ N≥2, G = π n (K), π k (K) = 0 for 0 ≤ k 2. Thus, in case K is an Eilenberg-Mac Lane complex of type K(G, n), then (c) becomes dime Z < n. If in addition π n+1 (K) = 0, then (a) can be replaced by the stronger statement, (aa) π is K-acyclic. To say that a map π is K-acyclic means that for each x ∈ X, every map of the fiber π -1 (x) to K is nullhomotopic.