Homology of planar polygon spaces

By Michael Farber, Dirk Schuetz. 2006

73 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9aea240ba4ff18c906d154e3d2736dfe80484b2d