Resolutions for metrizable compacta in extension theory

Abstract:

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, $n\in \mathbb N _{\geq 2}$, $G=\pi _{n}(K)$, $\pi _{k}(K)=0$ for $0\leq k<n$, and $\operatorname {extdim} X\leq K$ (in the sense of extension theory, that is, $K$ is an absolute extensor for $X$), then there exists a metrizable compactum $Z$ and a surjective map $\pi :Z\rightarrow X$ such that: (a) $\pi$ is $G$-acyclic, (b) $\dim Z\leq n+1$, and (c) $\operatorname {extdim} Z\leq K$. This implies the $G$-resolution theorem for arbitrary abelian groups $G$ for cohomological dimension $\dim _{G} X\leq n$ when $n\in \mathbb N_{\geq 2}$. Thus, in case $K$ is an Eilenberg-Mac Lane complex of type $K(G,n)$, 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 $K$-acyclic. To say that a map $\pi$ is $K$-acyclic means that for each $x\in X$, every map of the fiber $\pi ^{-1}(x)$ to $K$ is nullhomotopic.