Effective topological complexity of spaces with symmetries

TL;DR

This paper introduces a new version of topological complexity tailored for symmetric configuration spaces in mechanical systems, offering distinct properties and an upper bound related to Farber's original concept.

Contribution

The authors develop a novel invariant of topological complexity that accounts for symmetries, differing significantly from existing approaches and bounded by Farber's topological complexity.

Findings

The new invariant has different properties compared to previous methods.

It is bounded from above by Farber's topological complexity.

The approach is applicable to symmetric configuration spaces in mechanical systems.

Abstract

We introduce a version of Farber's topological complexity suitable for investigating mechanical systems whose configuration spaces exhibit symmetries. Our invariant has vastly different properties to the previous approaches of Colman-Grant, Dranishnikov and Lubawski-Marzantowicz. In particular, it is bounded from above by Farber's topological complexity.