  Let $E_{d}(\ell)$ denote the space of all closed $n$-gons in $\R^{d}$ (where
$d\ge 2$) with sides of length $\ell_1,..., \ell_n$, viewed up to translations.
The spaces $E_d(\ell)$ are parameterized by their length vectors
$\ell=(\ell_1,..., \ell_n)\in \R^n_{>}$ encoding the length parameters.
Generically, $E_{d}(\ell)$ 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 $E_{d}(\ell)/{O}(d)$ (the moduli space of shapes of
$n$-gons) has singularities for a generic $\ell$, 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 $d\ge 3$ and $n\ge 3$, the
diffeomorphism types of the manifolds $E_{d}(\ell)$ for varying generic vectors
$\ell$ 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 $d=2$.
