A combinatorial description of topological complexity for finite spaces

TL;DR

This paper introduces a combinatorial approach to topological complexity for finite spaces, showing it matches the traditional topological complexity and bounds the complexity of related structures.

Contribution

It provides a combinatorial analog of topological complexity that aligns with the classical concept and offers bounds for related order complexes.

Findings

Combinatorial topological complexity equals classical topological complexity for finite spaces.

The combinatorial measure bounds the topological complexity of the order complex.

The approach simplifies calculations of topological complexity in finite spaces.

Abstract

This paper presents a combinatorial analog of topological complexity for finite spaces. We demonstrate that this coincides with the genuine topological complexity of the original finite space, and constitutes an upper bound for the topological complexity of its order complex.