The topology of spaces of polygons

Let E_d(ℓ) denote the space of all closed n-gons in R^d (where d ≥ 2) with sides of length ℓ ₁,..., 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 ngons) 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 ℓare 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.