Lusternik-Schnirelmann category and based topological complexities of motion planning

TL;DR

This paper explores various topological complexities related to motion planning, introduces new classes of complexities motivated by robotic tasks, and compares these notions through computations on familiar spaces.

Contribution

It defines three new classes of topological complexities inspired by motion planning scenarios and compares them with existing complexities, providing computations for common spaces.

Findings

Introduction of three new topological complexities $ extbf{LTC}_n(X)$, $ extbf{ltc}_n(X)$, $ extbf{tc}_n(X)$.

Comparison of these new complexities with existing $ extbf{TC}_n(X)$.

Explicit computations of these complexities for familiar topological spaces.

Abstract

Farber and Rudyak introduced topological complexity $\mathbf{TC}(X)$ of motion planning and its higher analogs $\mathbf{TC}_n(X)$ to measure the complexity of assigning paths to point tuples. Motivated by motion planning where a robotic system starts at the home configuration and possibly comes back after passing through a list of locations, we define three other classes of topological complexities $\mathbf{LTC}_n(X)$, $\mathbf{ltc}_n(X)$ and $\mathbf{tc}_n(X)$. We will compare these notions and compute the latter for some familiar classes of spaces.