The Borsuk-Ulam theorem for planar polygon spaces

TL;DR

This paper explores the topological properties of moduli spaces of planar polygons, computing invariants like index and coindex, and establishing conditions for generalized Borsuk-Ulam theorems based on side lengths.

Contribution

It introduces a formula for the Stiefel-Whitney height using genetic code data and determines when generalized Borsuk-Ulam theorems apply to these polygon spaces.

Findings

Computed index and coindex for certain polygon moduli spaces

Derived a formula for Stiefel-Whitney height from genetic code

Identified conditions for generalized Borsuk-Ulam theorems to hold

Abstract

The moduli space of planar polygons with generic side lengths is a closed, smooth manifold. Mapping a polygon to its reflected image across the $X$-axis defines a fixed-point-free involution on these moduli spaces, making them into free $\mathbb{Z}_2$-spaces. There are some important numerical parameters associated with free $\mathbb{Z}_2$-spaces, like index and coindex. In this paper, we compute these parameters for some moduli spaces of polygons. We also determine for which of these spaces a generalized version of the Borsuk-Ulam theorem hold. Moreover, we obtain a formula for the Stiefel-Whitney height in terms of the the genetic code, a combinatorial data associated with side lengths.