Joins of DGA modules and sectional category

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.

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Keywords

Lusternik–Schnirelmann category, sectional category, topological complexity, Sullivan models

Mathematical Subject Classification 2000

Primary: 55M30

Secondary: 55P62

References

Forward citations

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)

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

Abstract