The topology of spaces of polygons

Abstract:

Let $E_{d}(\ell )$ denote the space of all closed $n$-gons in $\mathbb {R}^{d}$ (where $d\ge 2$) with sides of length $\ell _1, \dots , \ell _n$, viewed up to translations. The spaces $E_d(\ell )$ are parameterized by their length vectors $\ell =(\ell _1, \dots , \ell _n)\in \mathbb {R}^n_{>}$ 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 $d=2$.