Topology of Random 2-Complexes

Abstract

We study the Linial-Meshulam model of random two-dimensional simplicial complexes. One of our main results states that for p n −1 a random 2-complex Y collapses simplicially to a graph and, in particular, the fundamental group π 1 (Y ) is free and H 2 (Y ) = 0, asymptotically almost surely. Our other main result gives a precise threshold for collapsibility of a random 2-complex to a graph in a prescribed number of steps. We also prove that, if the probability parameter p satisfies p n −1/2+ , where > 0, then an arbitrary finite two-dimensional simplicial complex admits a topological embedding into a random 2-complex, with probability tending to one as n → ∞. We also establish several related results; for example, we show that for p < c/n with c < 3 the fundamental group of a random 2-complex contains a non-abelian free subgroup. Our method is based on exploiting explicit thresholds (established in the paper) for the existence of simplicial embeddings and immersions of 2-complexes into a random 2-complex.

Large random simplicial complexes, I

“…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. Some interesting results about the topology of random 2-complexes in the LinialMeshulam model were obtained in [1], [3], [5].…”

Minors in random and expanding hypergraphs

“…Brown, Erdős and Sós [6] showed that there are 2-dimensional complexes on n vertices with as many as Ω(n 5/2 ) triangles that do not contain a subcomplex homeomorphic to the 2-sphere S 2 , i.e., that do not contain K 2 4 as a topological minor. 7 Thus, the analogue of Conjecture 2 fails for topological minors.…”

“…By contracting the edge uv, we obtain the complex Y on Figure 1: Closing a hole by contracting an edge. 7 A simple alternative proof of this is due to Linial (personal communication) and uses a random complex X 2 (n, cn −1/2 ) with small alterations the right, which is a subdivided triangle. Now consider the complete 2-complex K 2 7 on seven vertices.…”

“…This model has received a lot of attention, and the threshold probabilities for a number of topological properties of X k (n, p) have been determined, see [27,36,3,2,7,8,23].…”

Correction to Our Article “Topology of Random 2-Complexes” Published in DCG 47 (2012), pp. 117–149

“…

Theorem 27 from our article [1] states that any closed triangulated surface S is balanced, i.e. μ(X ) ≥ μ(S) for any subcomplex X ⊂ S. Here the notation μ(Y ) stands for the ratio v/ f where v and f are the numbers of vertices and faces (i.e.

…”