The integer homology threshold in $Y_d(n,p)$

We prove that in the $d$-dimensional Linial–Meshulam stochastic process the $(d-1)$st homology group with integer coefficients vanishes exactly when the final isolated $(d-1)$-dimensional face is covered by a top-dimensional face. This generalizes the $d=2$ case proved recently by \L uczak and Peled and establishes that $p=\frac{d\log n}{n}$ is the sharp threshold for homology with integer coefficients to vanish in $Y_d(n,p),$ answering a 2003 question of Linial and Meshulam.