Integral homology of random simplicial complexes

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 $n\to \mathrm{\infty }$, the first homology group over $\mathbb{Z}$ vanishes at the very moment when all the edges are covered by triangular faces.