Some remarks on Betti numbers of random polygon spaces

Polygon spaces like Mℓ = {(u1, · · · , un) ∈ S1 × · · · S1 ; n i=1 liui = 0}/SO(2) or they three dimensional analogues Nℓ play an important rle in geometry and topology, and are also of interest in robotics where the li model the lengths of robot arms. When n is large, one can assume that each li 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 pn when pn → ∞ as n → ∞. 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 li ≡ ¯l. In three dimensions, these two quantities are not any more asymptotically equivalent. We also provide asymptotics for the Poincar´e polynom...

• Type: Journal
• Year: 2010
• Where: RSA
• Authors: Clément Dombry, Christian Mazza
• Keywords: Betti Number, Poincar´e Polynomial, RSA 2010, Total Betti Number