\(A_{\infty }\)-spaces and L-S category in the category of fibrewise spaces
type: Item
instance of: scholarly article
title: \(A_{\infty }\)-spaces and L-S category in the category of fibrewise spaces
zbMATH Open document ID: 1198.55002
author: Michihiro Sakai
published in: Topology and its Applications
publication date: 2010-08-27T00:00:00Z
review text: Fibrewise homotopy theory has been introduced by James and was developed by \textit{M. Crabb} and \textit{I. James} in their book [Fibrewise homotopy theory. London: Springer (1998; Zbl 0905.55001)]. Then, in [Proc. R. Soc. Edinb., Sect. A 119, No.~1--2, 177--190 (1991; Zbl 0738.55005)], \textit{I. M. James} and \textit{J. R. Morris} defined the LS category for fibrewise pointed spaces. In the present paper the author continues the study of the fibrewise LS category. He constructs a fibrewise version of the projective spaces for A\(_\infty\)-spaces, i.e., fibrewise versions of the Ganea fibrations associated to a space, and he proves the fibrewise analogue of the classical theorem of Ganea: The LS category of a space \(X\) is \(\leq n\) if and only if the \((n+1)^{th}\) Ganea fibration associated to \(X\) has a section.
reviewed by: Yves Félix
Mathematics Subject Classification ID: 55M30
Mathematics Subject Classification ID: 55R70
zbMATH DE Number: 5775730
zbMATH Keywords: Fibrewise space
zbMATH Keywords: Lusternik-Schnirelmann category, \(A_\infty\)-space
MaRDI profile type: Publication
full work available at URL: j.topol.2010.05.007
OpenAlex ID: W2079257003
cites work: Q4212657
cites work: Partitions of unity in the theory of fibrations
cites work: Q5184155
cites work: Q4896541
cites work: Ganea's Conjecture on Lusternik-Schnirelmann Category
cites work: Functors on the category of quasi-fibrations
cites work: Topological complexity is a fibrewise L-S category
cites work: Ex-homotopy theory. I
cites work: Q3675146
cites work: Q4896540
cites work: Fibrewise category
cites work: Fibrewise Hopf construction and Hoo formula for pairings
cites work: On the Hopf-James Homomorphism over B
cites work: A proof of the homotopy push-out and pull-back lemma
cites work: Lectures on the Eilenberg-Moore spectral sequence
cites work: Homotopy Associativity of H-Spaces. I
DOI: 10.1016/J.TOPOL.2010.05.007
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description: scientific article; zbMATH DE number 5775730