The topology of spaces of polygons

TL;DR

This paper classifies the topological types of spaces of closed polygons in higher dimensions, showing they correspond to combinatorial objects related to hyperplane arrangements, extending known results from low dimensions.

Contribution

It establishes a complete classification of the diffeomorphism types of polygon spaces in dimensions three and higher, linking them to hyperplane complement components.

Findings

Diffeomorphism types correspond to hyperplane arrangement components

Classification extends low-dimensional results to higher dimensions

Provides a combinatorial description of polygon space topology

Abstract

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 $d=2$.