Sequential parametrized topological complexity and related invariants

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 (Formula Presented) is a measure of the complexity of such algorithms. It was studied by Cohen, Farber and Weinberger (2021, 2022) for r = 2 and by Farber and Paul (2022) for r ≥ 2. We analyze the dependence of the complexity (Formula Presented) on an initial bundle with structure group G and on its fibre X viewed as a G–space. Our main results estimate (Formula Presented) 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 (Formula Presented) which plays an important role in the study of sectional category of towers of fibrations.