Volume 6, issue 1 (2006)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Joins of DGA modules and sectional category

Lucía Fernández Suárez, Pierre Ghienne, Thomas Kahl and Lucile Vandembroucq

Algebraic & Geometric Topology 6 (2006) 119–144

arXiv: 0902.4823

Abstract

We construct an explicit semifree model for the fiber join of two fibrations p: E B and p: E B from semifree models of p and p. Using this model, we introduce a lower bound of the sectional category of a fibration p which can be calculated from any Sullivan model of p and which is closer to the sectional category of p than the classical cohomological lower bound given by the nilpotency of the kernel of p: H(B; ) H(E; ). In the special case of the evaluation fibration XI X × X we obtain a computable lower bound of Farber’s topological complexity TC(X). We show that the difference between this lower bound and the classical cohomological lower bound can be arbitrarily large.

Keywords
Lusternik–Schnirelmann category, sectional category, topological complexity, Sullivan models
Mathematical Subject Classification 2000
Primary: 55M30
Secondary: 55P62
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Publication
Received: 1 November 2005
Revised: 11 January 2006
Accepted: 20 January 2006
Published: 24 February 2006
Authors
Lucía Fernández Suárez
Universidade do Minho
Centro de Matemática
Campus de Gualtar
4710-057 Braga
Portugal
Pierre Ghienne
Université d’Artois
Faculté des Sciences Jean Perrin
Laboratoire de Mathématiques
Rue Jean Souvraz - SP 18
62307 Lens Cedex
France
Thomas Kahl
Universidade do Minho
Centro de Matemática
Campus de Gualtar
4710-057 Braga
Portugal
Lucile Vandembroucq
Universidade do Minho
Centro de Matemática
Campus de Gualtar
4710-057 Braga
Portugal