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- W2950749807 abstract "Polygon spaces like $M_ell={(u_1,...,u_n)in S^1times... S^1 ; sum_{i=1}^n l_iu_i=0}/SO(2)$ or they three dimensional analogues $N_ell$ play an important role 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 Poincare polynomial. We study the average values of Betti numbers of dimension $p_n$ when $p_ntoinfty$ as $ntoinfty$. We also focus on the limiting mean Poincare 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 $l_iequiv bar l$. In three dimensions, these two quantities are not any more asymptotically equivalent. We also provide asymptotics for the Poincare polynomials" @default.
- W2950749807 created "2019-06-27" @default.
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- W2950749807 date "2008-09-12" @default.
- W2950749807 modified "2023-10-17" @default.
- W2950749807 title "Some remarks on Betti numbers of random polygon spaces" @default.
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