Joins of DGA modules and sectional category

Abstract

We construct an explicit semifree model for the fiber join of two fibrations $p:E\to B$ and $p^{\prime }:E^{\prime }\to B$ 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 $p^{\ast }:H^{\ast }\left(B;\mathbb{Q}\right)\to H^{\ast }\left(E;\mathbb{Q}\right)$. 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 $TC\left(X\right)$. We show that the difference between this lower bound and the classical cohomological lower bound can be arbitrarily large.