We construct an explicit semifree model for the fiber join of two fibrations and from semifree models of and . Using this model, we introduce a lower bound of the sectional category of a fibration which can be calculated from any Sullivan model of and which is closer to the sectional category of than the classical cohomological lower bound given by the nilpotency of the kernel of . In the special case of the evaluation fibration we obtain a computable lower bound of Farber’s topological complexity . We show that the difference between this lower bound and the classical cohomological lower bound can be arbitrarily large.
"Joins of DGA modules and sectional category." Algebr. Geom. Topol. 6 (1) 119 - 144, 2006. https://doi.org/10.2140/agt.2006.6.119