Geometriae Dedicata

, Volume 125, Issue 1, pp 75–92

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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Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DurhamDurhamUK

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