Simple connectivity in random 2-complexes

The random Linial-Meshulam complex Y_d(n,p) is the higher-dimensional analogue of the random graph model G(n,p). In recent years, the properties of random hypergraphs and simplicial complexes have been studied intensively. In this talk, we consider the fundamental group of a random 2-dimensional simplicial complex.

Babson, Hoffman and Kahle proved that the threshold for the fundamental group to vanish, i.e. simple connectivity, is approximately p=n^(-1/2). We show that in fact this threshold is at most (c n)^(-1/2) for c=4^4/3^3, and conjecture that this is the true threshold. In fact, we prove a sharp threshold for the stronger property that every cycle is the boundary of a subcomplex of Y_2(n,p) that is homeomorphic to a sphere.

The talk is based on a joint work with Yuval Peled.