2018/12/18 by Shelley Kandola, Kandola, Shelley
Computer Science · Mathematics · #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.AT
paper · pdf · doi:10.48550/arxiv.1812.07604
10 pages, 1 figure
arxiv created 2018/12/18 · openalex publication_date 2018/12/18 · arxiv updated 2018/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we examine how topological complexity, simplicial complexity, discrete topological complexity, and combinatorial complexity compare when applied to models of S1. We prove that the topological complexity of non-minimal finite models of S1 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 ≥ 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 T0 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.