## Algebraic & Geometric Topology

### Joins of DGA modules and sectional category

#### 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 $p∗:H∗(B;ℚ)→H∗(E;ℚ)$. 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 6, Number 1 (2006), 119-144.

Dates
Revised: 11 January 2006
Accepted: 20 January 2006
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796508

Digital Object Identifier
doi:10.2140/agt.2006.6.119

Mathematical Reviews number (MathSciNet)
MR2199456

Zentralblatt MATH identifier
1097.55006

#### Citation

Fernández Suárez, Lucía; Ghienne, Pierre; Kahl, Thomas; Vandembroucq, Lucile. Joins of DGA modules and sectional category. Algebr. Geom. Topol. 6 (2006), no. 1, 119--144. doi:10.2140/agt.2006.6.119. https://projecteuclid.org/euclid.agt/1513796508

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