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Resolutions for metrizable compacta in extension theory

Authors:	Leonard R Rubin Philip J Schapiro

Institution:	Department of Mathematics, University of Oklahoma, 601 Elm Ave., Rm. 423, Norman, Oklahoma 73019 ; Department of Mathematics, Langston University, Langston, Oklahoma 73050

Abstract:	We prove a -resolution theorem for simply connected CW- complexes in extension theory in the class of metrizable compacta . This means that if is a connected CW-complex, is an abelian group, $n\in \mathbb N _{\geq 2}$ , $G=\pi _{n}(K)$ , $\pi _{k}(K)=0$ for $0\leq k<n$ , and $\operatorname{extdim} X\leq K$ (in the sense of extension theory, that is, is an absolute extensor for ), then there exists a metrizable compactum and a surjective map $\pi :Z\rightarrow X$ such that: (a) $\pi$ is -acyclic, (b) $\dim Z\leq n+1$ , and (c) $\operatorname{extdim} Z\leq K$ . This implies the -resolution theorem for arbitrary abelian groups for cohomological dimension $\dim _{G} X\leq n$ when $n\in \mathbb N_{\geq 2}$ . Thus, in case is an Eilenberg-MacLane complex of type , then (c) becomes $\dim _{G} Z\leq n$ . If in addition $\pi _{n+1}(K)=0$ , then (a) can be replaced by the stronger statement, (aa) $\pi$ is -acyclic. To say that a map $\pi$ is -acyclic means that for each $x\in X$ , every map of the fiber $\pi ^{-1}(x)$ to is nullhomotopic.

Keywords:	Bockstein basis Bockstein inequalities \v{C}ech cohomology cell-like map cohomological dimension CW-complex dimension Edwards-Walsh resolution Eilenberg-Mac\ Lane complex $G$-acyclic resolution inverse sequence $K$-acyclic resolution Moore space shape of a point simplicial complex

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