Resolutions for Metrizable Compacta in Extension Theory

This is a summary of research which appears in a preprint of the same title. 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 dimX ≤ K (in the sense of extension theory), n is the first element of N such that G = πn(K) 6= 0, and it is also true that πn+1(K) = 0, then there exists a metrizable compactum Z and a surjective map π : Z → X such that: (a) π is G-acyclic, (b) dimZ ≤ n+ 1, and (c) dimZ ≤ K. If additionally, πn+2(K) = 0, then we may improve (a) to the statement, (aa) π is K-acyclic. To say that a map π is K-acyclic means that each map of each fiber π−1(x) to K is nullhomotopic. In case πn+1(K) 6= 0, we obtain a resolution theorem with a weaker outcome. Nevertheless, it implies the G-resolution theorem for arbitrary abelian groups G in cohomological dimension dimGX ≤ n when n ≥ 2. The Edwards-Walsh resolution theorem, the first resolution theorem for cohomological dimension, was proved in [Wa] (see also [Ed]). It states that ifX is a metrizable compactum and dimZ X ≤ n (n ≥ 0), then there exists a metrizable compactum Z with dimZ ≤ n and a surjective cell-like map π : Z → X. This result, in conjunction with Dranishnikov’s work ([Dr1]) showing that in the class of metrizable 1991 Mathematics Subject Classification. 55P55, 54F45.