Homology of planar polygon spaces

TL;DR

This paper investigates the topological properties of planar polygon spaces, providing formulas for Betti numbers and bounds on their sums, using Morse theory and involutions.

Contribution

It introduces a novel approach linking Morse functions and involutions to analyze the topology of polygon spaces, deriving explicit Betti number formulas and bounds.

Findings

Betti numbers of polygon spaces are explicitly described as functions of side lengths.

Sharp upper bounds on the sum of Betti numbers depending only on the number of links n.

Method reveals interaction between Morse functions and involutions in topological analysis.

Abstract

In this paper we study topology of the variety of closed planar polygons with given side lengths. We describe the Betti numbers of the moduli spaces as functions of the length vector. We also find sharp upper bounds on the sum of Betti numbers of the moduli space depending only on the number of links n. Our method is based on an observation of a remarkable interaction between Morse functions and involutions under the condition that the fixed points of the involution coincide with the critical points of the Morse function.