A new approach to the equivariant topological complexity

TL;DR

This paper introduces a symmetric topological complexity that better captures the impact of symmetry on motion planning algorithms, providing bounds and relations to existing invariants, and aligning with Farber's complexity in free actions.

Contribution

It proposes a new symmetric topological complexity, offering improved analysis of symmetry effects and establishing its relation to known invariants and orbit space complexities.

Findings

Provides bounds for symmetric topological complexity

Shows equivalence to Farber's complexity in free actions

Defines Whitehead version of the invariant

Abstract

We present a new approach to equivariant version of the topological complexity, called a symmetric topological complexity. It seems that the presented approach is more adequate for the analysis of an impact of symmetry on the the motion planning algoritm than the one introduced and studied by Colman and Grant. We show many bounds for the symmetric topological complexity comparing it with already known invariants and prove that in the case of a free action it is equal to the Farber's topological complexity of the orbit space. We define the Whitehead version of it.