  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 zero-th and
first homology groups with coefficients in $\mathbb{F}_2$ vanish. Although this
is not intrinsically a monotone property, we show that it nevertheless has a
single sharp threshold, and indeed prove a hitting time result relating the
connectedness to the disappearance of the last minimal obstruction.
