The cobordism class of the moduli space of polygons in $\mathbb{R}^3$

TL;DR

This paper characterizes the cobordism class of the moduli space of polygons in three-dimensional space, showing it depends solely on the side lengths, thus linking geometric configurations to topological invariants.

Contribution

It provides an explicit description of the cobordism class of polygon moduli spaces in terms of side lengths, a novel topological classification.

Findings

The cobordism class depends uniquely on the length vector r.

Explicit formulas for the cobordism class are derived.

The classification links geometric parameters to topological invariants.

Abstract

For any vector $r=(r_1,..., r_n)$, let $M_r$ denote the moduli space (under rigid motions) of polygons in $\mathbb{R}^3$ with $n$-sides whose lengths are $r_1,...,r_n$. We give an explicit characterization of the oriented $S^1$-cobordism class of $M_r$ which depends uniquely on the length vector $r$.