The Topological Complexity of Finite Models of Spheres

TL;DR

This paper investigates the topological complexity of finite models of spheres, establishing bounds and exact values for different models and introducing methods to compare various complexity measures.

Contribution

It provides new results on the topological complexity of finite models of spheres, including bounds and exact values, and compares multiple notions of complexity.

Findings

The TC of non-minimal finite models of S^1 can be ≤ 3.

The TC of the minimal finite model of any n-sphere is exactly 4 for n ≥ 1.

Spaces weakly homotopy equivalent to a wedge of circles can have arbitrarily high TC.

Abstract

In this paper, we examine how topological complexity, simplicial complexity, discrete topological complexity, and combinatorial complexity compare when applied to models of $S^1$. We prove that the topological complexity of non-minimal finite models of $S^1$ can be less-than-or-equal-to 3, and that the TC of the minimal finite model of any $n$-sphere is equal to 4 for $n \geq 1$. We show the former using properties of the LS-category, and we show the latter by proving that the TC of the non-Hausdorff suspension of any finite connected $T_0$ space is equal to 4. We also prove a result about the topological complexity of non-Hausdorff joins of discrete finite spaces, allowing us to exhibit spaces weakly homotopy equivalent to a wedge of circles with arbitrarily high TC.