Sequential parametrized topological complexity and related invariants

TL;DR

This paper investigates the complexity of parametrized motion planning algorithms through the lens of sequential parametrized topological complexity, analyzing its dependence on bundle structures and developing new invariants for sectional categories.

Contribution

It provides estimates of the sequential parametrized topological complexity based on bundle and fiber invariants and introduces a new invariant for sectional categories.

Findings

Estimates of ${ m TC}_r[p:E o B]$ in terms of bundle and fiber invariants.

Development of a calculus of sectional categories with a new invariant.

Analysis of the dependence of complexity on bundle structure and fiber actions.

Abstract

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.