Equivariant topological complexities

TL;DR

This paper reviews various equivariant generalizations of topological complexity, examining their relationships with quotient space complexity and fixed point sets, and discusses calculations and open questions in the area.

Contribution

It provides a comprehensive overview of equivariant topological complexities, highlighting differences, relationships, and open problems compared to classical topological complexity.

Findings

Examples of calculations of equivariant topological complexities

Discussion on the relation between equivariant and quotient space complexities

Open questions about free actions and their impact on topological complexity

Abstract

The aim of this article is to review different generalizations of the the notion of topological complexity to the equivariant setting. In particular, we review the relation (or non-relation) between these notions and the topological complexity of the quotient space and the topological complexity of the fixed point sets. We give examples of calculations and stress the question: When the action is free, do we recover the topological complexity of the quotient?