Morse theory and moduli spaces of self-avoiding polygonal linkages

TL;DR

This paper proves that certain moduli spaces of self-avoiding polygonal linkages are diffeomorphic to Euclidean spaces using Lyapunov-Reeb functions, resolving the Refined Carpenter's Rule Problem and confirming a conjecture.

Contribution

It introduces a method to show moduli spaces are Euclidean by constructing Lyapunov-Reeb functions, solving longstanding geometric problems.

Findings

Moduli spaces are diffeomorphic to Euclidean spaces.

Confirmed the Refined Carpenter's Rule Problem.

Described foliation structures and developed algorithms.

Abstract

We show that a smooth $d$-manifold $M$ is diffeomorphic to $\mathbb R^d$ if it admits a Lyapunov-Reeb function, i.e., a smooth map $f:M\to\mathbb R$ that is proper, lower-bounded, and has a unique critical point. By constructing such functions, we prove that the moduli spaces of self-avoiding polygonal linkages and configurations are diffeomorphic to Euclidean spaces. This resolves the Refined Carpenter's Rule Problem and confirms a conjecture proposed by Gonz\'{a}lez and Sedano-Mendoza. Furthermore, we describe foliation structures of these moduli spaces via level sets of Lyapunov-Reeb functions and develop algorithms for related problems.