Homological connectivity of random k‐dimensional complexes

Abstract

Let ∆ n−1 denote the (n − 1)-dimensional simplex. Let Y be a random k-dimensional subcomplex of ∆ n−1 obtained by starting with the full (k − 1)-dimensional skeleton of ∆ n−1 and then adding each k-simplex independently with probability p. Let H k−1 (Y ; R) denote the (k − 1)-dimensional reduced homology group of Y with coefficients in a finite abelian group R. It is shown that for any fixed R and k ≥ 1 and for any function ω(n) that tends to infinity

The fundamental group of random 2-complexes

“…If H 1 (X, Z/pZ) = 0 for every p, then H 1 (X, Z) = 0 as well [7]. By the Linial-Meshulam-Wallach results ( [10,11]), H 1 (Y, Z) is finite and has no p-torsion for any fixed p. So once p 2 log n/n, either H 1 (Y, Z) is trivial, or it is a finite generated abelian group with torsion approaching infinity. The first scenario might seem more plausible, but as far as we know, nothing is proved either way.…”

The threshold function for vanishing of the top homology group of random 𝑑-complexes

“…More recently, the two-dimensional analog Y n,p,2 of the Erdős-Rényi model was considered by Linial-Meshulam in [11], and, further, the d-dimensional model Y n,p,d , for d ≥ 3, was considered by Meshulam-Wallach in [13].…”

“…Theorem 1.1 (Erdős-Rényi Theorem, [4]). Assume that w(n) is any function w : N → R, such that lim n→∞ w(n) = ∞, and p = p(n) is the probability depending on n. Then we have More recently, the two-dimensional analog Y n,p,2 of the Erdős-Rényi model was considered by Linial-Meshulam in [11], and, further, the d-dimensional model Y n,p,d , for d ≥ 3, was considered by Meshulam-Wallach in [13].…”

“…Curiously, the methods of [11,13] do not easily extend to the case of integer coefficients, and finding the threshold functions for the vanishing of…”

Large random simplicial complexes, I

“…One may mention random surfaces [18], random 3-dimensional manifolds [9], random configuration spaces of linkages [11]. Linial, Meshulam and Wallach [15], [17] studied an important analogue of the classical Erdős-Rényi [10] model of random graphs in the situation of high-dimensional simplicial complexes. The random simplicial complexes of [15], [17] are d-dimensional, have the complete (d − 1)-skeleton and their randomness shows only in the top dimension.…”