Homological connectedness of random hypergraphs

We consider simplicial complexes that are generated from the binomial random 3-uniform hypergraph by taking the downward-closure. We determine when this simplicial complex is homologically connected, meaning that its first homology group with coefficients in F2 vanishes and the zero-th homology group is isomorphic to F2. Although this is not intrinsically a monotone property, we show that it has a single sharp threshold, and indeed prove a hitting time result relating the connectedness to the disappearance of the last minimal obstruction.