Lusternik-Schnirelmann category and Hilbert cube manifolds
type: Item
instance of: scholarly article
title: Lusternik-Schnirelmann category and Hilbert cube manifolds
zbMATH Open document ID: 0646.58013
DOI: 10.1016/0166-8641(87)90055-1
author: L. Montejano
published in: Topology and its Applications
publication date: 1987-01-01T00:00:00Z
review text: For a compact polyhedron P, the Lusternik-Schnirelmann category cat(P) is defined as the smallest integer k such that P can be covered with k subpolyhedra each of which is null homotopic in P. The geometric category gcat(P) is the smallest integer k such that P can be covered with k contractible subpolyhedra. The strong category Cat(P) in the sense of Ganea is the smallest integer k such that \(gcat(K)=k\) for some compact polyhedron K homotopy equivalent to P. In the paper \(Cat(P)=\min_{n\geq 0}gcat(P\times I^ n)\) is proved provided that \(\pi_ 1(P)\) is either free or free abelian. If P is connected then \(cat(P)=smallest\) integer k such that \(P\times Q\) can be covered with \(k+1\) open subsets each of which is homeomorphic to \(Q\times [0,1)\), where Q stands for the Hilbert cube. It is proved that if \(M^ m\) is a compact connected PL m-manifold then \(cat(M^ m)=smallest\) integer k such that \(M^ m\times I^ n\) can be covered with \(k+1\) boundary balls. If \(M^ m\) is a closed connected PL m-manifold then \(cat(M^ m)=smallest\) integer k such that \(M^ m\times S^ n\) can be covered with \(k+1\) PL balls.
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 57N20
zbMATH DE Number: 4054643
zbMATH Keywords: simple homotopy equivalence
zbMATH Keywords: Hilbert cube manifold
zbMATH Keywords: splitting
zbMATH Keywords: Lusternik- Schnirelmann category
MaRDI profile type: Publication
full work available at URL: 0166-8641(87)90055-1
OpenAlex ID: W2076233576
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description: scientific article; zbMATH DE number 4054643