Homology of planar polygon spaces

Abstract

In this paper, we study topology of the variety of closed planar n-gons with given side lengths l 1 , . . . , l n . The moduli space M where = (l 1 , . . . , l n ), encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces M as functions of the length vector = (l 1 , . . . , l n ). We also find sharp upper bounds on the sum of Betti numbers of M 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 index of a cyclic polygon. II

“…The linkages with nonsmooth moduli spaces are described by nonzero linear equations in the parameter space R n + (see [3]). From the set of all polygonal linkages, each of the above conditions "cuts out" a closed set with empty interior.…”

“…In general, the moduli space is a smooth manifold, with dimension equal to (n − 3) (see [3]). We denote it by M(L).…”

“…For n = 5, 6 it was explicitly described by Zvonkine in [11]. In [3], Farber and Shuetz computed homology groups of M(L) for arbitrary number of sides.…”

Topology of random linkages

“…J -Cl Hausmann and A Knutson [7] applied methods of symplectic topology (toric varieties) to study the cohomology algebras H .N`/. Betti numbers of planar polygon spaces M`as functions of the length vector`were found in Farber and Schütz [6]; this result uses techniques of Morse theory of manifolds with involution. The recent preprint of the author, Hausmann and Schütz [4] gives a general classification of diffeomorphism types of polygon spaces M`and N`in terms of combinatorics of chambers and the action of the symmetric group † n .…”

“…First we recall the result of [6] describing Betti numbers b p .M`/ of planar polygon spaces as functions of the length vector`. Fix an index 1 Ä i Ä n such that l i is maximal among l 1 ; : : : ; l n .…”