On (co)homology locally connected spaces
type: Item
instance of: scholarly article
title: On (co)homology locally connected spaces
zbMATH Open document ID: 1039.55006
DOI: 10.1016/S0166-8641(01)00087-6
author: Katsuya Eda
author: Umed H. Karimov
author: Dušan Repovš
published in: Topology and its Applications
publication date: 2002-08-22T00:00:00Z
review text: A topological space \(X\) is homology (cohomology, respectively) locally connected if for each point \(x \in X\) and neighborhood \(N\) of \(x\), there exists a neighborhood \(M \subset N\) of \(x\) such that the homomorphism induced by the inclusion \(H_{\ast}^s(M,\{x\}) \to H_{\ast}^s(N,\{x\})\) (\(\check{H}^{\ast}(N,\{x\}) \to \check{H}^{\ast}(M,\{x\})\), respectively) is trivial, and we call the space an \textit{HLC}-space (\textit{clc}-space, respectively). Here \(H_{\ast}^s(Y)\) and \(\check{H}^{\ast}(Y)\) are singular homology and Čech cohomology groups of a topological space \(Y\) with integer coefficients. The authors show that there exists a two-dimensional compact metric space \(X\) such that (1) \(X\) is acyclic in Čech cohomology; (2) \(X\) is a \textit{clc}-space, and (3) \(X\) is not an \textit{HLC}-space. The result is an extension of \textit{G. E. Bredon's} example in [Sheaf Theory (GTM vol.170, 131--132, Springer-Verlag) (1997; Zbl 0874.55001)], and an alternative construction by \textit{H. G. Griffith} [Proc. Lond. Math. Soc., III. Ser. 6, 455--480 (1956; Zbl 0071.01902)].
reviewed by: Akira Koyama
Mathematics Subject Classification ID: 55N99
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 55N05
Mathematics Subject Classification ID: 55N10
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 55N40
zbMATH DE Number: 1787506
zbMATH Keywords: Čech cohomology
zbMATH Keywords: singular homology
zbMATH Keywords: (co)homology local connectedness
zbMATH Keywords: commutator length
MaRDI profile type: Publication
cites work: A certain class of non-aspherical 2-complexes
cites work: Q3125206
cites work: Free \(\sigma\)-products and noncommutatively slender groups
cites work: Q5832080
cites work: Infinite Products of Semi-Groups and Local Connectivity
cites work: A Finitely Generated Infinite Simple Group
cites work: Q5510561
cites work: On suspensions of noncontractible compacta of trivial shape
cites work: Comparison of singular and Čech homology in locally connected spaces
cites work: Q4748811
full work available at URL: s0166-8641(01)00087-6
OpenAlex ID: W2077144272
description: scientific article; zbMATH DE number 1787506