Some remarks on Betti numbers of random polygon spaces

Polygon spaces such as M = {(u1, . . . , un) ∈ S1 × . . . S1,∑ni=1 liui = 0}/SO(2), or the three-dimensional analogs N play an important rôle 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 randommanifold. The complexity of such manifolds can be approached by computing Betti numbers, the Euler characteristics, or the related Poincaré polynomial. We study the average values of Betti numbers of dimension pn when pn → ∞ as n → ∞. We also focus on the limiting mean Poincaré 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 ≡ l̄. In three dimensions, these two quantities are not any more asymptotically equivalent. We also provide asymptotics for the Poincaré polynomials. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 37, 67–84, 2010