A unified approach of characterizations and resolutions for cohomological dimension modulo \(p\)
type: Item
instance of: scholarly article
title: A unified approach of characterizations and resolutions for cohomological dimension modulo \(p\)
zbMATH Open document ID: 0851.55001
author: Akira Koyama
author: Katsuya Yokoi
published in: Tsukuba Journal of Mathematics
publication date: 1996-11-24T00:00:00Z
review text: The authors prove the following theorem: Let \(X\) be either compact Hausdorff or metrizable, with \(a\)-\(\dim_G X \leq n\). Then there exists a proper map \(f : Z \to X\) (\(Z\) either compact Hausdorff or metrizable resp.) \(\dim Z \leq n\) such that \(H^n (f^{-1} (x); G) = 0\) for all \(x\in X\). The new concept is that of approximable dimension with respect to an abelian group \(G\), \(a\)-\(\dim_G X\), constituting a new version of cohomology dimension of a space \(X\). Ordinary cohomology dimension and ordinary dimension of compact Hausdorff or metrizable spaces are related by the inequality \[ c \text{-dim}_G X \leq a \text{-dim}_G X \leq \dim X. \] Moreover for \(G = \mathbb{Z}\) or \(G = \mathbb{Z}_p\), \(p\) being a prime, and space \(X\) as above one has \[ a \text{-dim}_G X = c \text{-dim}_G X. \] As a result the main theorem of this paper comprises results of Edwards and J. J. Walsh, R. L. Rubin, A. N. Dranishnikov and S. Mardešić which were accomplished for different kinds of compact Hausdorff or metrizable spaces and the coefficient groups \(\mathbb{Z}\) resp. \(\mathbb{Z}_p\), \(p\) being a prime.
reviewed by: Friedrich Wilhelm Bauer
Mathematics Subject Classification ID: 55M10
zbMATH DE Number: 729213
zbMATH Keywords: approximable dimension
zbMATH Keywords: cohomology dimension
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496162605
OpenAlex ID: W2342188267
DOI: 10.21099/TKBJM/1496162605
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description: scientific article; zbMATH DE number 729213