A quick proof of Singhof's \(cat(M\times S^ 1)=cat(M)+1\) theorem
type: Item
instance of: scholarly article
title: A quick proof of Singhof's \(cat(M\times S^ 1)=cat(M)+1\) theorem
zbMATH Open document ID: 0639.55001
DOI: 10.1007/BF01171745
author: L. Montejano
published in: Manuscripta Mathematica
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/154921
review text: For a topological space X, the Lyusternik-Shnirel'man category, cat(X), is the smallest number N such that X can be covered by N open subsets each of which is contractible in X. \textit{W. Singhof} [ibid. 29, 385-415 (1979; Zbl 0415.55001)] proved that the minimal number of n-balls which suffice to cover a closed PL n-manifold M coincides with the Lyusternik- Shnirel'man category if the latter is not too small compared with the dimension of M. As a consequence, one obtains in this case that \(cat(M\times S^ 1)=cat(M)+1\), thus establishing a special case of a long-standing conjecture. The purpose of this paper is to give a quick proof of Singhof's results by exploiting the linear structure between the k-skeleton of a polyhedron and its dual skeleton.
Mathematics Subject Classification ID: 55M30
Mathematics Subject Classification ID: 57Q99
zbMATH DE Number: 4042837
zbMATH Keywords: category of the product of a polyhedron with a sphere
zbMATH Keywords: Lyusternik- Shnirel'man category
zbMATH Keywords: k-skeleton of a polyhedron
zbMATH Keywords: dual skeleton
MaRDI profile type: Publication
cites work: On the Lusternik-Schnirelmann category
cites work: Lusternik-Schnirelmann Category and Cocategory
cites work: Q3949684
cites work: On category, in the sense of Lusternik-Schnirelmann
cites work: Minimal coverings of manifolds with balls
description: scientific article; zbMATH DE number 4042837