Some remarks on Betti numbers of random polygon spaces

TL;DR

This paper investigates the average Betti numbers and Poincaré polynomials of random polygon spaces, revealing their asymptotic behavior and differences between two and three dimensions as the number of sides grows large.

Contribution

It provides new asymptotic results for Betti numbers and Poincaré polynomials of random polygon spaces in high dimensions, highlighting differences between 2D and 3D cases.

Findings

Mean total Betti number in 2D matches equilateral case asymptotically.

In 3D, Betti numbers differ asymptotically from the equilateral case.

Asymptotic formulas for Poincaré polynomials are derived.

Abstract

Polygon spaces like $M_\ell=\{(u_1,...,u_n)\in S^1\times... S^1 ;\ \sum_{i=1}^n l_iu_i=0\}/SO(2)$ or they three dimensional analogues $N_\ell$ play an important r\^ole in geometry and topology, and are also of interest in robotics where the $l_i$ model the lengths of robot arms. When $n$ is large, one can assume that each $l_i$ is a positive real valued random variable, leading to a random manifold. The complexity of such manifolds can be approached by computing Betti numbers, the Euler characteristics, or the related Poincar\'e polynomial. We study the average values of Betti numbers of dimension $p_n$ when $p_n\to\infty$ as $n\to\infty$. We also focus on the limiting mean Poincar\'e polynomial, in two and three dimensions. We show that in two dimensions, the mean total Betti number behaves as the total Betti number associated with the equilateral manifold where $l_i\equiv \bar l$. In three dimensions, these two quantities are not any more asymptotically equivalent. We also provide asymptotics for the Poincar\'e polynomials