A Finite Equivariant Generalization of Motion Planning and Topological Complexity

TL;DR

This paper introduces an equivariant version of topological complexity for finite spaces, establishing its equivalence with combinatorial and simplicial complexities, and explores bounds and relationships with existing invariants.

Contribution

It defines and studies an equivariant combinatorial complexity, linking it with topological and simplicial complexities in finite G-spaces.

Findings

Equivariant topological complexity equals equivariant combinatorial complexity.

Bounds are established for equivariant combinatorial complexity.

Equivariant simplicial complexity matches the topological complexity of realizations.

Abstract

This paper explores topological complexity in the finite equivariant setting. We first define and study an equivariant version of Tanaka's combinatorial complexity for finite topological spaces. We explore the relationships between this invariant and several others already discussed in the literature: Farber's topological complexity, Tanaka's combinatorial complexity, and Colman-Grant's equivariant Lusternik-Schnirelmann category. We find bounds for equivariant combinatorial complexity and for the necessary lengths of equivariant combinatorial motion plannings. We show that the equivariant topological complexity of any finite $G$-space is equal to its equivariant combinatorial complexity. We then adapt Gonz\`{a}lez's simplicial complexity to ordered and unordered $G$-simplicial complexes and explore its first properties. Lastly, we show that the equivariant topological complexity of the realization of any ordered $G$-simplicial complex is equal to the equivariant simplicial complexity.