The topology of spaces of polygons

2011/05/03 by Farber, Michael, Fromm, Viktor · 3 citations
#Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1105.0613

Abstract

Let E_d(ℓ) denote the space of all closed n-gons in \R^d (where d≥ 2) with sides of length ℓ₁,..., ℓ_n, viewed up to translations. The spaces E_d(ℓ) are parameterized by their length vectors ℓ=(ℓ₁,..., ℓ_n)∈ \Rⁿ_> encoding the length parameters. Generically, E_d(ℓ) is a closed smooth manifold of dimension (n-1)(d-1)-1 supporting an obvious action of the orthogonal group O(d). However, the quotient space E_d(ℓ)/O(d) (the moduli space of shapes of n-gons) has singularities for a generic ℓ, assuming that d>3; this quotient is well understood in the low dimensional cases d=2 and d=3. Our main result in this paper states that for fixed d≥ 3 and n≥ 3, the diffeomorphism types of the manifolds E_d(ℓ) for varying generic vectors ℓ are in one-to-one correspondence with some combinatorial objects -- connected components of the complement of a finite collection of hyperplanes. This result is in the spirit of a conjecture of K. Walker who raised a similar problem in the planar case d=2.

Cited by

• Morse theory and moduli spaces of self-avoiding polygonal linkages
• Smooth critical points of eigenvalues on the torus of magnetic perturbations of graphs
• Optimization and the Topology of Spaces of Parseval Frames

Related

• The manifold of polygons degenerated to segments
• Manifold properties of planar polygon spaces
• Homology of planar polygon spaces
• On the conjecture of Kevin Walker
• Moduli space of simple polygons