#### Volume 6, issue 1 (2006)

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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
 arXiv: 0902.4823
##### Abstract

We construct an explicit semifree model for the fiber join of two fibrations $p:\phantom{\rule{0.3em}{0ex}}E\to B$ and ${p}^{\prime }:\phantom{\rule{0.3em}{0ex}}{E}^{\prime }\to B$ from semifree models of $p$ and ${p}^{\prime }$. 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 ${p}^{\ast }:\phantom{\rule{0.3em}{0ex}}{H}^{\ast }\left(B;ℚ\right)\to {H}^{\ast }\left(E;ℚ\right)$. In the special case of the evaluation fibration ${X}^{I}\to X×X$ we obtain a computable lower bound of Farber’s topological complexity $TC\left(X\right)$. 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
Primary: 55M30
Secondary: 55P62