A minimum dimensional counterexample to Ganea's conjecture
type: Item
instance of: scholarly article
title: A minimum dimensional counterexample to Ganea's conjecture
zbMATH Open document ID: 1198.55003
published in: Topology and its Applications
publication date: 2010-08-27T00:00:00Z
review text: Let us recall that the Lusternik-Schnirelmann category of a space \(X\), \(\text{cat}\,X\), is one less than the minimal number of open subspaces of \(X\) contractible in \(X\) that are necessary to cover \(X\). In 1971, Ganea conjectured that \(\text{cat}(X\times S^n)=\text{cat}\,X + 1\) for any finite CW-complex and \(n\geq 1\). By results of Jessup and Hess this is true for rational spaces, but not in general. The first counterexample has been given by Iwase in 1998. The lowest dimension of all the known counterexamples was 10, while Vandembroucq proved that there is no counterexample in dimension \(\leq 5\). The main result of this paper is the production of a counterexample \(T\) in dimension 7. The space \(T\) is quite simple, \(T = S^2\vee S^6\cup_f e^7\) for some particular \(f\). We have \(\text{cat}\,T=2\) and \(\text{cat}(T\times S^r)= 2\) for \(r\geq 2\). The proof is based on the possibility of factoring a Hopf invariant stably.
reviewed by: Yves Félix
Mathematics Subject Classification ID: 55M30
zbMATH DE Number: 5775747
zbMATH Keywords: Ganea's conjecture
zbMATH Keywords: Lusternik-Schnirelmann category
Wikidata QID: Q123251661
author: Donald Stanley
author: Hugo Rodríguez-Ordóñez
MaRDI profile type: Publication
full work available at URL: j.topol.2010.06.009
OpenAlex ID: W2046063260
cites work: Category and generalized Hopf invariants
cites work: Q4421932
cites work: Cellular spaces, null spaces and homotopy localization
cites work: The rational Lusternik-Schnirelmann category of products and of Poincaré duality complexes
cites work: On the Lusternik-Schnirelmann category
cites work: A generalization of the homology and homotopy suspension
cites work: Lusternik-Schnirelmann category and strong category
cites work: Q5647632
cites work: A proof of Ganea's conjecture for rational spaces
cites work: Ganea's Conjecture on Lusternik-Schnirelmann Category
cites work: A\(_{\infty}\)-method in Lusternik-Schnirelmann category
cites work: On category, in the sense of Lusternik-Schnirelmann
cites work: Q4896563
cites work: Rational Approximations to L-S Category and a Conjecture of Ganea
cites work: On category weight and its applications
cites work: Fibrewise construction applied to Ljusternik-Schnirelman category
cites work: Minimal coverings of manifolds with balls
cites work: Spaces with Lusternik-Schnirelmann category \(n\) and cone length \(n + 1\)
cites work: On the Lusternik-Schnirelmann Category of Maps
cites work: Two special cases of Ganea’s conjecture
cites work: The genus of a fiber space
cites work: Algebraic topology -- homology and homotopy.
cites work: Compositional Methods in Homotopy Groups of Spheres. (AM-49)
cites work: Suspension of Ganea fibrations and a Hopf invariant
cites work: Fibrewise suspension and Lusternik-Schnirelmann category
DOI: 10.1016/J.TOPOL.2010.06.009
description: scientific article; zbMATH DE number 5775747