Homology of planar polygon spaces
  In this paper we study topology of the variety of closed planar polygons with given side lengths. We describe the Betti numbers of the moduli spaces as functions of the length vector. We also find sharp upper bounds on the sum of Betti numbers of the moduli space 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. 
Michael Farber and Dirk Schuetz