The proper Lusternik-Schnirelmann category of semistable one-ended 3-manifolds
type: Item
instance of: scholarly article
title: The proper Lusternik-Schnirelmann category of semistable one-ended 3-manifolds
zbMATH Open document ID: 1269.55001
author: Francisco F. Lasheras
author: Manuel Cárdenas
author: Antonio Quintero
published in: Topology and its Applications
publication date: 2013-04-05T00:00:00Z
review text: Let \(X\) be a topological space. The proper Lusternik--Schnirelmann category of \(X\), denoted \(\mathrm{p-cat}(X)\), is the least integer \(n\) such that \(X\) admits a categorical cover \(\{U_j\}_{j=1}^n\) with the property that for every \(j\) the inclusion of the closure of \(U_j\) \(i_j: \overline{U_j}\to X\) factors up to proper homotopy through \(\mathbb{R}^+\) or \(\{x_0\}\). The study of \(\mathrm{p-cat}(M)\), where \(M\) is a \(3\)-manifold with additional properties, is the main focus of the paper under review. The authors use a Ganea fibration definition of proper category to prove the following conjecture from [\textit{M. Cárdenas}, \textit{F. F. Lasheras} and \textit{A. Quintero}, Math. Proc. Camb. Philos. Soc. 152, No. 2, 223--249 (2012; Zbl 1245.55002)]: If \(M\) is an orientable one-ended semistable open \(3\)-manifold, then \(\mathrm{p-cat}(M)\leq 3\) if and only if \(\pi_1(M)\) is free. Part of the proof utilizes the \(2\)-dimensional Berstein-Svarc theorem due to \textit{A. Dranishnikov} and \textit{Y. Rudyak} [Math. Proc. Camb. Philos. Soc. 146, No. 2, 407--413 (2009; Zbl 1171.55002)].
reviewed by: Nicholas A. Scoville
Mathematics Subject Classification ID: 55M30
Mathematics Subject Classification ID: 55P57
Mathematics Subject Classification ID: 57N65
zbMATH DE Number: 6150727
zbMATH Keywords: proper homotopy theory
zbMATH Keywords: Lusternik-Schnirelmann category
zbMATH Keywords: category weight
zbMATH Keywords: 3-manifolds
MaRDI profile type: Publication
full work available at URL: j.topol.2013.01.012
OpenAlex ID: W1965533090
DOI: 10.1016/J.TOPOL.2013.01.012
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description: scientific article; zbMATH DE number 6150727