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2019 Symmetric topological complexity for finite spaces and classifying spaces
Kohei Tanaka
Topol. Methods Nonlinear Anal. 54(2A): 477-493 (2019). DOI: 10.12775/TMNA.2019.048

Abstract

We present a combinatorial approach to the symmetric motion planning in polyhedra using finite spaces. For a finite space $P$ and a positive integer $k$, we introduce two types of combinatorial invariants, $\mathrm{CC}^{S}_k(P)$ and $\mathrm{CC}^{\Sigma}_k(P)$, that are closely related to the design of symmetric robotic motions in the $k$-iterated barycentric subdivision of the associated simplicial complex $\mathcal{K}(P)$. For the geometric realization $\mathcal{B}(P)=|\mathcal{K}(P)|$, we show that the first $\mathrm{CC}^{S}_k(P)$ converges to Farber-Grant's symmetric topological complexity $\mathrm{TC}^{S}(\mathcal{B}(P))$ and the second $\mathrm{CC}^{\Sigma}_k(P)$ converges to Basabe-González-Rudyak-Tamaki's symmetrized topological complexity $\mathrm{TC}^{\Sigma}(\mathcal{B}(P))$ as $k$ becomes larger.

Citation

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Kohei Tanaka. "Symmetric topological complexity for finite spaces and classifying spaces." Topol. Methods Nonlinear Anal. 54 (2A) 477 - 493, 2019. https://doi.org/10.12775/TMNA.2019.048

Information

Published: 2019
First available in Project Euclid: 29 July 2019

zbMATH: 07198793
MathSciNet: MR4061306
Digital Object Identifier: 10.12775/TMNA.2019.048

Rights: Copyright © 2019 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.54 • No. 2A • 2019
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