Cohomological dimension and metrizable spaces

Abstract:

The purpose of this paper is to address several problems posed by V. I. Kuzminov [Ku] regarding cohomological dimension of noncompact spaces. In particular, we prove the following results: Theorem A. Suppose $X$ is metrizable and $G$ is the direct limit of the direct system $\{ {G_s},{h_{s\prime ,s}},S\}$ of abelian groups. Then, \[ {\dim _G}X \leq \max \{ {\dim _{{G_s}}}X|s \in S\} \] . Theorem B. Let $X$ be a metrizable space and let $G$ be an abelian group. Let $l = \{ p|p \cdot (G/\operatorname {Tor}G) \ne G/\operatorname {Tor}G\}$. (a) If $G = \operatorname {Tor}G$, then ${\dim _G}X = \max \{ {\dim _H}X|H \in \sigma (G)\}$, (b) ${\dim _G}X = \max \{ {\dim _{\operatorname {Tor}G}}X,{\dim _{G/\operatorname {Tor}G}}X\}$, (c) ${\dim _G}X \geq {\dim _\mathbb {Q}}X$ if $G \ne \operatorname {Tor}G$, (d) ${\dim _G}X \geq {\dim _{{{\hat {\mathbb {Z}}}_l}}}X$, where ${\hat {\mathbb {Z}}_l}$ is the group of $l$-adic integers, (e) $\max ({\dim _G}X,{\dim _\mathbb {Q}}X + 1) \geq \max \{ {\dim _H}X|H \in \sigma (G)\}$, (f) ${\dim _G}X \leq {\dim _{{\mathbb {Z}_l}}}X \leq {\dim _G}X + 1$ if $G \ne 0$ is torsion-free. Theorem B generalizes a well-known result of M. F. Bockstein [B].