  The random $2$-dimensional simplicial complex process starts with a complete
graph on $n$ vertices, and in every step a new $2$-dimensional face, chosen
uniformly at random, is added. We prove that with probability tending to $1$ as
$n\to\infty$, the first homology group over $\mathbb Z$ vanishes at the very
moment when all the edges are covered by triangular faces.
