An uncountable homology group, where each element is an infinite product of commutators
type: Item
instance of: scholarly article
title: An uncountable homology group, where each element is an infinite product of commutators
zbMATH Open document ID: 1329.57001
author: Andreas Zastrow
author: Oleg Bogopolski
published in: Topology and its Applications
publication date: 2015-12-08T00:00:00Z
review text: Obtaining homotopy invariants of wild topological spaces (i.e. spaces, which are not locally contractible) is hard due to the absence of standard tools of algebraic topology. Even in the simplest case, the Hawaiian Earring, the fundamental group and the first homology group are hard to obtain and to describe. Another example of such a space was introduced by \textit{U. Kh. Karimov} [Sov. Math., Dokl. 33, 113--117 (1986); translation from Dokl. Akad. Nauk SSSR 286, 531--534 (1986; Zbl 0606.55003)]. He constructed \(X\) as the one-point compactification of a certain simplicial complex, so that each homotopy class of loops can be represented as an infinite product of commutators. In this paper the authors prove that the first homology group of Karimov's space \(X\) is uncountable. The proof uses infinite words and relates them to concrete cycles in \(X\).
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20K45
Mathematics Subject Classification ID: 55N10
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 6517548
zbMATH Keywords: singular homology group of a non-triangulable space
zbMATH Keywords: infinite product of commutators
zbMATH Keywords: infinite van-Kampen diagrams
reviewed by: Ziga Virk
MaRDI profile type: Publication
full work available at URL: j.topol.2015.10.017
OpenAlex ID: W2126955886
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DOI: 10.1016/J.TOPOL.2015.10.017
description: scientific article; zbMATH DE number 6517548