Sequential motion planning assisted by group actions

TL;DR

This paper introduces higher analogues of topological complexity for spaces with group actions, providing new invariants that assist in understanding motion planning problems influenced by external symmetries.

Contribution

It defines higher effective and effectual topological complexities and introduces orbital topological complexity as new invariants for G-spaces, with applications to actions on surfaces and spheres.

Findings

Orbital topological complexity bounds the quotient space complexity.

New invariants are G-homotopy invariant and applicable to symmetric spaces.

Applications to group actions on orientable surfaces and spheres.

Abstract

We study higher analogues of effective and effectual topological complexity of spaces equipped with a group action. These are $G$-homotopy invariant and are motivated by the (higher) motion planning problem of $G$-spaces for which their group action is thought of as an external system assisting the motion planning. Related to this interpretation we define what we call orbital topological complexity, which is also a $G$-homotopy invariant that provides an upper bound for the topological complexity of the quotient space by the group action. We apply these concepts to actions of the group of order two on orientable surfaces and spheres.