Sequential parametrized topological complexity of sphere bundles

TL;DR

This paper investigates the sequential parametrized topological complexity of sphere bundles, providing lower bounds using characteristic classes and illustrating results through explicit examples.

Contribution

It introduces new bounds on the topological complexity of sphere bundles using Euler and Stiefel-Whitney classes, extending prior work on related cases.

Findings

Lower bounds on topological complexity derived from characteristic classes.

Explicit examples demonstrating the bounds and their applications.

Extension of previous results for specific cases like r=2.

Abstract

Autonomous motion of a system (robot) is controlled by a motion planning algorithm. A sequential parametrized motion planning algorithm \cite{FP22} works under variable external conditions and generates continuous motions of the system to attain the prescribed sequence of states at prescribed moments of time. Topological complexity of such algorithms characterises their structure and discontinuities. Information about states of the system consistent with states of the external conditions is described by a fibration $p: E\to B$ where the base $B$ parametrises the external conditions and each fibre $p^{-1}(b)$ is the configuration space of the system constrained by external conditions $b\in B$; more detail on this approach is given below. Our main goal in this paper is to study the sequential topological complexity of sphere bundles $\dot \xi: \dot E\to B$; in other words we study {\it \lq\lq parametrized families of spheres\rq\rq} and sequential parametrized motion planning algorithms for such bundles. We use the Euler and Stiefel - Whitney characteristic classes to obtain lower bounds on the topological complexity. We illustrate our results by many explicit examples. Some related results for the special case $r=2$ were described earlier in \cite{FW23}.