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Topological complexity of spatial polygon spaces

Author: Donald M. Davis
Journal: Proc. Amer. Math. Soc. 144 (2016), 3643-3645
MSC (2010): Primary 55M30, 58D29, 55R80
Published electronically: February 1, 2016
MathSciNet review: 3503733
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Abstract: Let $ \overline {\ell }=(\ell _1,\ldots ,\ell _n)$ be an $ n$-tuple of positive real numbers, and let $ N(\overline {\ell })$ denote the space of equivalence classes of oriented $ n$-gons in $ \mathbb{R}^3$ with consecutive sides of lengths $ \ell _1,\ldots ,\ell _n$, identified under translation and rotation of $ \mathbb{R}^3$. Using known results about the integral cohomology ring, we prove that its topological complexity satisfies $ \operatorname {TC}(N(\overline {\ell }))= 2n-5$, provided that $ N(\overline {\ell })$ is nonempty and contains no straight-line polygons.

References [Enhancements On Off] (What's this?)

  • [1] D. M. Davis, Topological complexity of some planar polygon spaces, on arXiv.
  • [2] -, Topological complexity of planar polygon spaces with small genetic code, on arXiv.
  • [3] Michael Farber, Invitation to topological robotics, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2455573 (2010a:55018)
  • [4] Jean-Claude Hausmann, Mod two homology and cohomology, Universitext, Springer, Cham, 2014. MR 3308717
  • [5] J.-C. Hausmann and A. Knutson, The cohomology ring of polygon spaces, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 1, 281-321 (English, with English and French summaries). MR 1614965 (99a:58027)
  • [6] Alexander A. Klyachko, Spatial polygons and stable configurations of points in the projective line, Algebraic geometry and its applications (Yaroslavl, 1992) Aspects Math., E25, Vieweg, Braunschweig, 1994, pp. 67-84. MR 1282021 (95k:14015)
  • [7] G. Panina and D. Siersma, Motion planning and control of a planar polygonal linkage, on arXiv.

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Additional Information

Donald M. Davis
Affiliation: Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015

Keywords: Topological complexity, spatial polygon spaces
Received by editor(s): July 9, 2015
Received by editor(s) in revised form: October 8, 2015
Published electronically: February 1, 2016
Communicated by: Michael A. Mandell
Article copyright: © Copyright 2016 American Mathematical Society

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