Geometriae Dedicata

, Volume 125, Issue 1, pp 75–92

Homology of planar polygon spaces

Original Paper

DOI: 10.1007/s10711-007-9139-7

Cite this article as:
Farber, M. & Schütz, D. Geom Dedicata (2007) 125: 75. doi:10.1007/s10711-007-9139-7


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.


Polygon spacesMorse theory of manifolds with involutionsVarieties of linkages

Mathematics Subject Classification (2000)

Primary 58Exx

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DurhamDurhamUK