Topological complexity and rational homotopy

The sectional category of a continuous map between topological spaces is a numerical invariant of the homotopy type. It is of special interest because it generalises important concepts such as Lusternik-Schnirelmann category and Michael Farber's topological complexity. The latter has grown to be widely studied during the last decade since it has important applications in the world of robotics, namely, it estimates the motion planning complexity of a mechanical system through its configuration space. In this work, we study sectional category using classical homotopy tools, abstract homotopy theory and rational homotopy theory, always keeping an eye on consequences on topological complexity. In particular, we give a simple way to compute rational sectional category for a wide class of maps. This method generalises the Félix-Halperin method for computing rational Lusternik-Schnirelmann category and proves as a particular case the Jessup-Murillo-Parent conjecture for computing rational topological complexity. We also study the Doeraene-El Haouari conjecture which relates sectional category to relative category. We show that this conjecture includes the Iwase-Sakai conjecture which says that topological complexity equals monoidal topological complexity, a stronger version of topological complexity which gives the "smart" motion planning complexity of a mechanical system.