In this paper, we study topology of the variety of closed planar n-gons with given side lengths l1,...,ln. The moduli space M l where l =(l1,...,ln), encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces M l as functions of the length vector l = (l1,...,l n). We also find sharp upper bounds on the sum of Betti numbers of Ml 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.
- Morse theory of manifolds with involutions
- Polygon spaces
- Varieties of linkages