On Simple Connectivity of Random 2-Complexes

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 threshold probability for simple connectivity is about $$p\approx n^{-1/2}$$ . In this paper, we show at most $$p\le (\gamma n)^{-1/2}$$ , where $$\gamma =4^4/3^3$$ conjecture sharp. fact, $$p=(\gamma a sharp stronger property every cycle length 3 boundary subcomplex homeomorphic to disk. Our proof uses Poisson paradigm, relies on classical result Tutte enumeration planar triangulations.