Sequential parametrized topological complexity and related invariants

Abstract

Parametrized motion planning algorithms have a high degree of universality and flexibility; they generate the motion of a robotic system under a variety of external conditions.The latter are viewed as parameters and constitute part of the input of the algorithm.The concept of sequential parametrized topological complexity TC r OEp W E !B is a measure of the complexity of such algorithms.It was studied by Cohen, Farber and Weinberger (2021, 2022) for r D 2 and by Farber and Paul (2022) for r 2. We analyze the dependence of the complexity TC r OEp W E !B on an initial bundle with structure group G and on its fibre X viewed as a G-space.Our main results estimate TC r OEp W E !B in terms of certain invariants of the bundle and the action on the fibre.Moreover, we also obtain estimates depending on the base and the fibre.Finally, we develop a calculus of sectional categories featuring a new invariant secat f OEp W E !B which plays an important role in the study of sectional category of towers of fibrations.55M30 1. Introduction 1755 2. The concept of sequential parametrized topological complexity 1758 3. Relation with the equivariant sequential topological complexity 1759 4. Calculus of sectional categories 1763 5. Sectional category of towers of fibrations 1770 6. Product inequalities 1771 7. Weak equivariant topological complexity TC w r;G .X / 1774 8. Bounds for the sequential parametrized topological complexity 1777