A note on finite-sheeted covering maps from 2-dimensional compact Abelian groups
type: Item
instance of: scholarly article
title: A note on finite-sheeted covering maps from 2-dimensional compact Abelian groups
zbMATH Open document ID: 1202.57002
author: Vlasta Matijević
published in: Topology and its Applications
publication date: 2010-10-29T00:00:00Z
review text: A (weak) solenoidal space is the limit space of an inverse sequence, where each term is a connected, locally pathwise connected, semilocally 1-connected space and each bounding map is a (covering map) regular covering map [cf. \textit{M. C. McCord}, Trans. Am. Math. Soc. 114, 197--209 (1965; Zbl 0136.43603)]. Each compact connected 2-dimensional abelian group \(X\) is a solenoidal space obtained as the limit of an inverse sequence, where each term is a 2-torus \(\mathbb{T}^2\) and each bounding map is a covering homomorphism; such groups are called toroidal groups. A Klein bottle weak solenoidal space is the limit of an inverse sequence, where each term is a Klein bottle \(K\) and each bounding map is a covering map from \(K\) to \(K\). \textit{C. Tezer} introduced and studied Klein bottle weak solenoidal spaces \(\pmb{\Sigma}(\mathbf{p,q,r})\), where \(p=(p_i),q=(q_i), r=(r_i)\) are sequences of integers, \(p_i\neq 0\) and \(r_i\) odd for each \(i\) in [Q. J. Math., Oxf. II. Ser. 40, No.~158, 225--243 (1989; Zbl 0675.55008)]. Recently, finite-sheeted covering maps over toroidal groups \(X\) were studied by \textit{K. Eda} and \textit{V. Matijević} [Topology Appl. 153, No.~7, 1033--1045 (2006; Zbl 1104.22005)]. It turned out that finite-sheeted covering maps over \(X\) are determined by finite-index torsion free supergroups of the Pontryagin dual \(\widehat{X}\), cf. \textit{K. Eda} and \textit{V. Matijević} [J. Algebra 319, 3567--3587 (2008; Zbl 1225.20046)]. Moreover, using finite index subgroups of \(\widehat{X}\) the authors also presented finite-sheeted covering maps from \(X\) to other compact connected groups. ``The main step in the investigation was the reduction to the case of finite-sheeted covering homomorphisms \(f:X'\rightarrow X\) between two toroidal groups. Since \(T^2\) is a covering space for a Klein bottle \(K\), the next step of investigation was to study finite-sheeted covering maps over Klein bottle weak solenoidal spaces \(\pmb{\Sigma}(\mathbf…
reviewed by: Ioan Pop
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 54B35
Mathematics Subject Classification ID: 54C56
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 22C05
zbMATH DE Number: 5808696
zbMATH Keywords: Covering map
zbMATH Keywords: 2-dimensional torus
zbMATH Keywords: Klein bottle
zbMATH Keywords: compact group
zbMATH Keywords: solenoid
zbMATH Keywords: 2-dimensional compact abelian group
zbMATH Keywords: weak solenoidal space.
MaRDI profile type: Publication
full work available at URL: j.topol.2010.07.026
OpenAlex ID: W2039525585
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DOI: 10.1016/J.TOPOL.2010.07.026
description: scientific article; zbMATH DE number 5808696