Abstract

The fundamental group of the 2-dimensional Linial–Meshulam random simplicial complex $$Y_2(n,p)$$ was first studied by Babson, Hoffman, and Kahle. They proved that the threshold probability for simple connectivity of $$Y_2(n,p)$$ is about $$p\approx n^{-1/2}$$ . In this paper, we show that this threshold probability is at most $$p\le (\gamma n)^{-1/2}$$ , where $$\gamma =4^4/3^3$$ , and conjecture that this threshold is sharp. In fact, we show that $$p=(\gamma n)^{-1/2}$$ is a sharp threshold probability for the stronger property that every cycle of length 3 is the boundary of a subcomplex of $$Y_2(n,p)$$ 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.

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