Geometriae Dedicata

, Volume 125, Issue 1, pp 75–92 | Cite as

Homology of planar polygon spaces

Original Paper

Abstract

In this paper, we study topology of the variety of closed planar n-gons with given side lengths \(l_1, \dots, l_n\). The moduli space \(M_\ell\) where \(\ell =(l_1, \dots, l_n)\), encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces \(M_\ell\) as functions of the length vector \(\ell=(l_1, \dots, l_n)\). We also find sharp upper bounds on the sum of Betti numbers of \(M_\ell\) depending only on the number of links n. Our method is based on an observation of a remarkable interaction between Morse functions and involutions under the condition that the fixed points of the involution coincide with the critical points of the Morse function.

Keywords

Polygon spaces Morse theory of manifolds with involutions Varieties of linkages 

Mathematics Subject Classification (2000)

Primary 58Exx 

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References

  1. 1.
    Hausmann, J.-Cl.: Sur la topologie des bras articulés. In: Algebraic Topology, Poznan. Lecture Notes, vol. 1474, pp. 146 – 159. Springer, Heidelberg (1989)Google Scholar
  2. 2.
    Hausmann J.-Cl. and Knutson A. (1998). Cohomology rings of polygon spaces. Ann. Inst. Fourier (Grenoble) 48: 281–321 MATHMathSciNetGoogle Scholar
  3. 3.
    Hausmann J.-C. and Rodriguez E. (2004). The space of clouds in an Euclidean space. Exp. Math. 13: 31–47 MATHMathSciNetGoogle Scholar
  4. 4.
    Kamiyama Y., Tezuka M. and Toma T. (1998). Homology of the configuration spaces of quasi-equilateral polygon linkages. Trans. AMS 350: 4869–4896 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Kamiyama Y. and Tezuka M. (1999). Topology and geometry of equilateral polygon linkages in the Euclidean plane. Q. J. Math. 50: 463–470 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kapovich M. and Millson J.L. (1995). On the moduli space of polygons in the Euclidean plane. J. Differ. Geometry 42: 133–164 MATHMathSciNetGoogle Scholar
  7. 7.
    Kapovich M. and Millson J.L. (1996). The symplectic geometry of polygons in Euclidean space. J. Differ. Geometry 44: 479–513 MATHMathSciNetGoogle Scholar
  8. 8.
    Klyachko, A.A.: Spatial polygons and stable configurations of points in the projective line. Algebraic geometry and its applications. Aspects Math., E25, pp 67–84. Vieweg, Braunschweig (1994)Google Scholar
  9. 9.
    Milgram R.J. and Trinkle J.C. (2004). The geometry of configuration spaces for closed chains in two and three dimensions. Homol. Homot. Appl. 6: 237–267 MATHMathSciNetGoogle Scholar
  10. 10.
    Milnor J. (1969). Morse theory. Princeton University Press, Princeton Google Scholar
  11. 11.
    Milnor J. (1966). Lectures on the h-cobordism theorem. Princeton University Press, Princeton Google Scholar
  12. 12.
    Thurston W. and Weeks J. (1984). The mathematics of three-dimensional manifolds. Sci. Am. 251: 94–106 CrossRefGoogle Scholar
  13. 13.
    Vardi I. (1991). Computational recreations in mathematics. Addison-Wesley, Redwood City, CA Google Scholar
  14. 14.
    Walker, K.: Configuration spaces of linkages. Undergraduate thesis, Princeton (1985)Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DurhamDurhamUK

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