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The Topological Complexity of Finite Models of Spheres
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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.
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Cited by 1 Pith paper
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The Topological Complexity of Spaces of Digital Jordan Curves
The space of digital Jordan curves in a Khalimsky digital plane is path-connected, and among the topologies studied, only the Khalimsky topology makes this space path-connected.
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