Equivariant topological complexity

TL;DR

This paper introduces an equivariant version of topological complexity for spaces with group actions, exploring its properties, relationships with other invariants, and providing examples and computational methods.

Contribution

It defines the equivariant topological complexity and sectional category, establishing their relationship with equivariant Lusternik-Schnirelmann category and illustrating differences from the classical case.

Findings

Provides formulas for equivariant topological complexity

Shows how to estimate non-equivariant topological complexity using equivariant invariants

Includes examples and computations demonstrating the concepts

Abstract

We define and study an equivariant version of Farber's topological complexity for spaces with a given compact group action. This is a special case of the equivariant sectional category of an equivariant map, also defined in this paper. The relationship of these invariants with the equivariant Lusternik-Schnirelmann category is given. Several examples and computations serve to highlight the similarities and differences with the non-equivariant case. We also indicate how the equivariant topological complexity can be used to give estimates of the non-equivariant topological complexity.