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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Cohomological dimension and metrizable spaces


Author: Jerzy Dydak
Journal: Trans. Amer. Math. Soc. 337 (1993), 219-234
MSC: Primary 55M10; Secondary 54F45, 55M15, 55U20
MathSciNet review: 1153013
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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,

$\displaystyle {\dim _G}X \leq \max \{ {\dim _{{G_s}}}X\vert s \in S\} $

.

Theorem B. Let $ X$ be a metrizable space and let $ G$ be an abelian group. Let $ l = \{ p\vert 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\vert 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\vert 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].


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1153013-X
PII: S 0002-9947(1993)1153013-X
Keywords: Cohomological dimension, absolute extensors, Eilenberg-Mac Lane complexes, metrizable spaces, $ p$-adic integers
Article copyright: © Copyright 1993 American Mathematical Society