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
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