Sequential parametrized topological complexity and related invariants

Parametrized motion planning algorithms \cite{CFW} 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 ${\sf TC}_r[p:E\to B]$ is a measure of the complexity of such algorithms. It was studied in \cite{CFW, CFW2} for $r=2$ and in \cite{FP} for $r\ge 2$. In this paper we analyse the dependence of the complexity ${\sf TC}_r[p:E\to B]$ on an initial bundle with structure group $G$ and on its fibre $X$ viewed as a $G$-space. Our main results estimate ${\sf TC}_r[p:E\to B]$ 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 ${\sf secat}_f[p:E\to B]$ which plays an important role in the study of sectional category of towers of fibrations.