Building planar polygon spaces from the projective braid arrangement

TL;DR

This paper demonstrates that planar polygon spaces can be constructed from the projective Coxeter complex of type A through iterative cellular surgery guided by the genetic code, providing a new perspective on their topological structure.

Contribution

It introduces a novel method of constructing polygon spaces via cellular surgery, expanding the understanding of their relationship with Coxeter complexes and the genetic code.

Findings

Polygon spaces can be obtained by cellular surgery on the Coxeter complex.

The sub-collection of the minimal building set is determined by the genetic code.

This approach offers a new perspective on the topology of polygon spaces.

Abstract

The moduli space of planar polygons with generic side lengths is a smooth, closed manifold. It is known that these manifolds contain the real points of the moduli space of distinct points on the projective line as an open dense subset. Hence, such a polygon space is a compactification of this real moduli space. Kapranov showed that the real points of the Deligne-Mumford-Knudson compactification can be obtained from the projective Coxeter complex of type $A$ (equivalently, the projective braid arrangement) by iteratively blowing up along the minimal building set. In this paper we show that these planar polygon spaces can also be obtained from the projective Coxeter complex of type $A$ by performing an iterative cellular surgery along a sub-collection of the minimal building set. Interestingly, this sub-collection is determined by the combinatorial data associated with the length vector called the genetic code.