Topological Methods in Nonlinear Analysis

Higher topological complexity of subcomplexes of products of spheres and related polyhedral product spaces

Jesús González, Bárbara Gutiérrez, and Sergey Yuzvinsky

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Abstract

We construct "higher" motion planners for automated systems whose spaces of states are homotopy equivalent to a polyhedral product space $Z(K,\{(S^{k_i},\star)\})$, e.g. robot arms with restrictions on the possible combinations of simultaneously moving nodes. Our construction is shown to be optimal by explicit cohomology calculations. The higher topological complexity of other families of polyhedral product spaces is also determined.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 48, Number 2 (2016), 419-451.

Dates
First available in Project Euclid: 21 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1482289222

Digital Object Identifier
doi:10.12775/TMNA.2016.051

Mathematical Reviews number (MathSciNet)
MR3642766

Zentralblatt MATH identifier
06712726

Citation

González, Jesús; Gutiérrez, Bárbara; Yuzvinsky, Sergey. Higher topological complexity of subcomplexes of products of spheres and related polyhedral product spaces. Topol. Methods Nonlinear Anal. 48 (2016), no. 2, 419--451. doi:10.12775/TMNA.2016.051. https://projecteuclid.org/euclid.tmna/1482289222


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References

  • A. Bahri, M. Bendersky, F. R. Cohen and S. Gitler, The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces, Adv. Math. 225 (2010), 1634–1668.
  • I. Basabe, J. González, Y.B. Rudyak and D. Tamaki, Higher topological complexity and its symmetrization, Algebr. Geom. Topol. 14 (2014), 223–244.
  • J.G. Carrasquel-Vera, Computations in rational sectional category, Bull. Belg. Math. Soc. Simon Stevi 22 (2015) (to appear).
  • D.C. Cohen, F.R. Cohen and M. Xicoténcatl, Lie algebras associated to fiber-type arrangements, Int. Math. Res. Not. 29 (2003), 1591–1621.
  • D.C. Cohen and M. Farber, Topological complexity of collision-free motion planning on surfaces, Compos. Math. 147 (2011), 649–660.
  • D.C. Cohen and G. Pruidze, Motion planning in tori, Bull. Lond. Math. Soc. 40 (2008), 249–262.
  • D.M. Davis, A strong nonimmersion theorem for real projective spaces, Ann. of Math. (2) 120 (1984), 517–528.
  • A. Dold, Lectures on Algebraic Topology, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 200, Springer–Verlag, Berlin-New York, second ed., 1980.
  • A. Dranishnikov, Topological complexity of wedges and covering maps, Proc. Amer. Math. Soc. 142 (2014), 4365–4376.
  • M. Farber, Topology of Robot Motion Planning, Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, NATO Sci. Ser. II Math. Phys. Chem., vol. 21, Springer, Dordrecht, 2006, 185–230.
  • M. Farber, S. Tabachnikov and S. Yuzvinsky, Topological robotics: motion planning in projective spaces, Int. Math. Res. Not. 34 (2003), 1853–1870.
  • Y. Félix and D. Tanré, \i Rational homotopy of the polyhedral product functor, Proc. Amer. Math. Soc. 137 (2009), 891–898.
  • J. González, M. Grant, E. Torres-Giese and M. Xicoténcatl, Topological complexity of motion planning in projective product spaces, Algebr. Geom. Topol. 13 (2013), 1027–1047.
  • J. González, M. Grant and L. Vandembroucq, Hopf invariants for sectional category with an application to topological robotics (submitted).
  • J. González, B. Gutiérrez, D. Gutiérrez and A. Lara, Motion planning in real flag manifolds, Homology, Homotopy Appl. (accepted for publication).
  • J. González, B. Gutiérrez, A. Guzmán, C. Hidber, M.L. Mendoza and Ch. Roque, Motion planning in tori revisited, Morfismos 19, No. 1 (2015), 7–18.
  • M. Grant, Topological complexity of motion planning and Massey products, Banach Center Publ. 85 (2009), 193–203.
  • M. Grant, G. Lupton and J. Oprea, Spaces of topological complexity one, Homology Homotopy Appl. 15 (2013), 73–81.
  • A. Hattori, Topology of $C^{n}$ minus a finite number of affine hyperplanes in general position, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), 205–219.
  • P. Orlik and H. Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer–Verlag, Berlin, 1992.
  • Y.B. Rudyak, On higher analogs of topological complexity, Topology Appl. 157 (2010), 916–920.
  • A. Schwarz, The genus of a fiber space, Amer. Math. Soc. Transl. (2) 55 (1966), 49–140.
  • S. Yuzvinsky, Higher topological complexity of Artin type groups, arXiv:1411.1778v1 [math.AT].
  • ––––, Topological complexity of generic hyperplane complements, Topology and Robotics, Contemp. Math., vol. 438, Amer. Math. Soc., Providence, RI, 2007, 115–119.