Asymptotic Topology of Random Subcomplexes in a Finite Simplicial Complex

Abstract

We consider a finite simplicial complex K together with its successive barycentric subdivisions Sd d (K), d ≥ 0, and study the expected topology of a random subcomplex in Sd d (K), d 0. We get asymptotic upper and lower bounds for the expected Betti numbers of those subcomplexes, together with the average Morse inequalities and expected Euler characteristic.

Asymptotic Measures and Links in Simplicial Complexes

“…Remark 10 The first part of Corollary 9 was independently (not as a corollary of Theorem 8) observed by T. Akita [1]. In [8], we provide a probabilistic proof of it.…”

“…From these theorems we see that asymptotically, the complexity of the link and the dual block is almost everywhere constant with respect to dvol K . In [8], we study the asymptotic topology of a random subcomplex in a finite simplicial complex K and its successive barycentric subdivisions. It turns out that the Betti numbers of such a subcomplex get controlled by the measures given in Theorem 6.…”

“…Remark 26 In [8], we study the expected topology of a random subcomplex in a finite simplicial complex K and its barycentric subdivisions. The Betti numbers of such a subcomplex turn out to be asymptotically controlled by the measure given by Theorem 25.…”

“…Having spheres in mind for instance, Theorem 1 and Theorem 2 exhibit a striking behavior of simplicial structures compared to cellular structures. In [8], we also provide a probabilistic proof of the first part of Theorem 2.…”