On Simple Connectivity of Random 2-Complexes

The fundamental group of the 2-dimensional Linial–Meshulam random simplicial complex Y₂(n, p) was first studied by Babson, Hoffman, and Kahle. They proved that the threshold probability for simple connectivity of Y₂(n, p) is about p≈ n^{- 1 / 2}. In this paper, we show that this threshold probability is at most p≤ (γn) ^{- 1 / 2}, where γ= 4 ⁴/ 3 ³, and conjecture that this threshold is sharp. In fact, we show that p= (γn) ^{- 1 / 2} is a sharp threshold probability for the stronger property that every cycle of length 3 is the boundary of a subcomplex of Y₂(n, p) that is homeomorphic to a disk. Our proof uses the Poisson paradigm, and relies on a classical result of Tutte on the enumeration of planar triangulations.