Topological complexity of configuration spaces

TL;DR

This paper completes the calculation of the topological complexity for configuration spaces of n points in Euclidean m-space, providing a comprehensive understanding of motion planning complexity in these spaces.

Contribution

It extends previous results by computing the topological complexity for all m>1 and n>1, and offers general results on bounds for this complexity.

Findings

Complete computation of topological complexity for all m>1 and n>1

New bounds and sharpness results for topological complexity

Clarification of motion planning complexity in configuration spaces

Abstract

The topological complexity TC(X) is a homotopy invariant which reflects the complexity of the problem of constructing a motion planning algorithm in the space X, viewed as configuration space of a mechanical system. In this paper we complete the computation of the topological complexity of the configuration space of n distinct points in Euclidean m-space for all m>1$ and n>1; the answer was previously known in the cases m=2 and m odd. We also give several useful general results concerning sharpness of upper bounds for the topological complexity.