Vanishing of cohomology groups of random simplicial complexes

We consider k-dimensional random simplicial complexes generated from the binomial random (k + 1)-uniform hypergraph by taking the downward-closure. For 1 <= j <= k - 1, we determine when all cohomology groups with coefficients in F2 from dimension one up to j vanish and the zero-th cohomology group is isomorphic to F2. This property is not deterministically monotone for this model, but nevertheless we show that it has a single sharp threshold. Moreover we prove a hitting time result, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction. We also study the asymptotic distribution of the dimension of the j-th cohomology group inside the critical window. As a corollary, we deduce a hitting time result for a different model of random simplicial complexes introduced by Linial and Meshulam, previously only known for dimension two.