Topological complexity of Khalimsky circles

TL;DR

This paper calculates the topological complexity of certain finite spaces resembling circles, providing counterexamples to existing conjectures and advancing understanding of topological invariants in finite spaces.

Contribution

It determines the topological complexity of finite spaces homotopy equivalent to a circle and constructs a finite space where tc(X) is less than cat(X×X), addressing conjectures by Tanaka.

Findings

Computed topological complexity for specific finite spaces

Provided a counterexample to a conjecture relating tc and cat

Resolved two open conjectures in finite space topology

Abstract

We determine topological complexity of a series of finite spaces which is weakly homotopy equivalent to a circle $S^1$, and give a finite space $X$ satisfying the inequality tc$(X) <$ cat$(X {\times} X)$. This answers two conjectures on topological complexity for finite spaces raised by K. Tanaka in 2018.