The topology of spaces of polygons

Abstract

Let E d ( ℓ ) E_{d}(\ell ) denote the space of all closed n n -gons in R d \mathbb {R}^{d} (where d ≥ 2 d\ge 2 ) with sides of length ℓ 1 , … , ℓ n \ell _1, \dots , \ell _n , viewed up to translations. The spaces E d ( ℓ ) E_d(\ell ) are parameterized by their length vectors ℓ = ( ℓ 1 , … , ℓ n ) ∈ R > n \ell =(\ell _1, \dots , \ell _n)\in \mathbb {R}^n_{>} encoding the length parameters. Generically, E d ( ℓ ) E_{d}(\ell ) is a closed smooth manifold of dimension ( n − 1 ) ( d − 1 ) − 1 (n-1)(d-1)-1 supporting an obvious action of the orthogonal group O ( d ) { {O}}(d) . However, the quotient space E d ( ℓ ) / O ( d ) E_{d}(\ell )/{{O}}(d) (the moduli space of shapes of n n -gons) has singularities for a generic ℓ \ell , assuming that d > 3 d>3 ; this quotient is well understood in the low-dimensional cases d = 2 d=2 and d = 3 d=3 . Our main result in this paper states that for fixed d ≥ 3 d\ge 3 and n ≥ 3 n\ge 3 , the diffeomorphism types of the manifolds E d ( ℓ ) E_{d}(\ell ) for varying generic vectors ℓ \ell 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 d=2 .