Joins of DGA modules and sectional category

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

doi:10.2140/agt.2006.6.119

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Joins of DGA modules and sectional category

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

Algebraic & Geometric Topology 6 (2006) 119–144

DOI: 10.2140/agt.2006.6.119

arXiv: 0902.4823

Abstract

We construct an explicit semifree model for the fiber join of two fibrations $p : E \to B$ and $p^{'} : E^{'} \to B$ 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 $p^{*} : H^{*} (B; ℚ) \to H^{*} (E; ℚ)$ . In the special case of the evaluation fibration $X^{I} \to X \times X$ 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

Volume 6, issue 1 (2006)