Fibrewise construction applied to Ljusternik-Schnirelman category
type: Item
instance of: scholarly article
title: Fibrewise construction applied to Ljusternik-Schnirelman category
zbMATH Open document ID: 1011.55003
DOI: 10.1007/BF02785865
author: Donald Stanley
author: Hans Scheerer
author: Daniel Tanré
published in: Israel Journal of Mathematics
publication date: 2003-01-26T00:00:00Z
review text: The Ljusternik-Schnirelman category, \(\text{cat}(X)\), of a space \(X\) is the least integer \(n\) for which \(S\) can be covered by \(n+1\) open sets contractible in \(X\). The Ljusternik-Schnirelman category is a homotopy type invariant; however, it is not easy to compute. The authors define a variant of the LS-category denoted by \(Qcat\). It is obtained by applying a base point version of the functor \(Q=\Omega^{\infty}\Sigma^{\infty}\) fibrewise to the Ganea fibrations. \textit{Y. B. Rudyak} in [Topology 38, 37-55 (1999; Zbl 0927.55005)] defined an invariant, denoted by \(\sigma cat\), which approximates \(cat\). The authors show that \(\text{cat}(X) \geq Q\text{cat}(X) \geq \sigma \text{cat}(X)\). The advantage of \(Qcat\) is that it approximates \(cat\) better: in the case of a rational space \(X\), \(Q\text{cat}(X)=\text{cat}(X)\) and \(\sigma \text{cat}(X)\) equals the Toomer invariant. Futhermore, the authors show that \(Q\text{cat}(X \times Y) \leq Q\text{cat}(X)+Q\text{cat}(Y)\). As the authors say, the invariant \(Qcat\) measures the failure of the formula \(\text{cat}(X \times S^r)=\text{cat}(X)+1\). For any complex \(X\) with two cells, \(Q\text{cat}(X)<\text{cat}(X)\) if and only if \(\text{cat}(X \times S^r)=\text{cat}(X)\) for some \(r \geq 1\).
reviewed by: Jan Jaworowski
Mathematics Subject Classification ID: 55M30
Mathematics Subject Classification ID: 55P62
zbMATH DE Number: 1850617
zbMATH Keywords: Ljusternik-Schnirelman category
zbMATH Keywords: rational space
zbMATH Keywords: Ganea construction
zbMATH Keywords: Hopf invariant
zbMATH Keywords: Toomer invariant
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02785865
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description: scientific article; zbMATH DE number 1850617