The moment polytope of the abelian polygon space

TL;DR

This paper characterizes the moment polytope of the moduli space of planar chains with fixed lengths using combinatorial data and classifies aspherical chain spaces, linking geometric and combinatorial properties.

Contribution

It provides a complete combinatorial description of the moment polytope for the moduli space of chains and classifies aspherical chain spaces.

Findings

Moment polytope characterized by short code of length vector

Classification of aspherical chain spaces

Connection between geometric moduli space and combinatorial data

Abstract

The moduli space of $n$ chains in the plane with generic side lengths that terminate on a fixed line is a smooth, closed manifold of dimension $n-1$. This manifold is also equipped with a locally standard action of $\mathbb{Z}_2^{n-1}$. The orbit space of this action is a simple polytope called the moment polytope. Interestingly, this manifold is also the fixed point set of an involution on a toric manifold known as the abelian polygon space. In this article we show that the moment polytope of the moduli space of chains is completely characterized by the combinatorial data, called the \emph{short code} of the length vector. We also classify aspherical chain spaces using a result of Davis, Januszkiewicz and Scott.