Symmetric topological complexity for finite spaces and classifying spaces

Kohei Tanaka

DOI: http://dx.doi.org/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-Gonz\'alez-Rudyak-Tamaki's symmetrized topological complexity $\mathrm{TC}^{\Sigma}(\mathcal{B}(P))$ as $k$ becomes larger.

Keywords


Finite space; symmetric topological complexity; configuration space; order complex; classifying space

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References


A. Abrams, Configuration Spaces and Braid Groups of Graphs, Ph.D. thesis, University of California at Berkeley, 2000, home.wlu.edu/~abramsa/publications/thesis.ps.

I. Basabe, J. González, Y.B. Rudyak and D. Tamaki, Higher topological complexity and its symmetrization, Algebr. Geom. Topol. 14 (2014), no. 4, 2103–2124.

E. Babson and D.N. Kozlov, Group actions on posets, J. Algebra 285 (2005), no. 2, 439–450.

J.A. Barmak and E.G. Minian, Simple homotopy types and finite spaces, Adv. Math. 218 (2008), no. 1, 87–104.

J.A. Barmak and E.G. Minian, Strong homotopy types, nerves and collapses, Discrete Comput. Geom. 47 (2012), no. 2, 301–328.

C. Cibils and E.N. Marcos, Skew category, Galois covering and smash product of a kcategory, Proc. Amer. Math. Soc. 134 (2006), no. 1, 39–50.

S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton University Press, Princeton, New Jersey, 1952. xv+328 pp.

M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), no. 2, 211–221.

M. Farber and M. Grant, Symmetric motion planning, Topology and Robotics, Contemp. Math., vol. 438 , Amer. Math. Soc., Providence, RI, 2007, 85–104.

D. Fernández-Ternero, E. Macı́as-Virgós, E. Minuz and J.A. Vilches, Discrete topological complexity, Proc. Amer. Math. Soc. 146 (2018), no. 10, 4535–4548.

M. Furuse, T. Mukouyama and D. Tamaki, Totally normal cellular stratified spaces and applications to the configuration space of graphs, Topol. Methods Nonlinear Anal. 45 (2015), no. 1, 169–214.

J. González, Simplicial complexity: piecewise linear motion planning in robotics, New York J. Math. 24 (2018), 279–292.

D.N. Kozlov, Combinatorial Algebraic Topology, Algorithms and Computation in Mathematics, vol 21, Springer, Berlin, 2008, xx+389 pp.

M.C. McCord, Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J. 33 (1966) 465–474.

D. Quillen, Higher algebraic K-theory I, (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math., Vol. 341, Springer, Berlin 1973, pp. 85–147.

A.S. Schwarz, The genus of a fiber space, Tr. Mosk. Mat. Obs. 10 (1961), 217–272.

E.H. Spanier, Algebraic topology, Corrected reprint of the 1966 original. Springer–Verlag, New York, xvi+528 pp.

R.E. Stong, Finite topological spaces, Trans. Amer. Math. Soc. 123 (1966) 325–340.

K. Tanaka, A model structure on the category of small categories for coverings, Math. J. Okayama Univ. 55 (2013), 95–116.

K. Tanaka, A combinatorial description of topological complexity for finite spaces, Algebr. Geom. Topol. 18 (2018), no. 2, 779–796.


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