  We consider $k$-dimensional random simplicial complexes that are generated
from the binomial random $(k+1)$-uniform hypergraph by taking the
downward-closure, where $k\geq 2$. For each $1\leq j \leq k-1$, we determine
when all cohomology groups with coefficients in $\mathbb{F}_2$ from dimension
one up to $j$ vanish and the zero-th cohomology group is isomorphic to
$\mathbb{F}_2$. This property is not deterministically monotone for this model
of random complexes, 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 in [Linial and Meshulam, Combinatorica, 2006], a result which was
previously only known for dimension two [Kahle and Pittel, Random Structures
Algorithms, 2016].
