The threshold for integer homology in random d-complexes

Let $Y \sim Y_d(n,p)$ denote the Bernoulli random $d$-dimensional simplicial complex. We answer a question of Linial and Meshulam from 2003, showing that the threshold for vanishing of homology $H_{d-1}(Y; \mathbb{Z})$ is less than $80d \log n / n$. This bound is tight, up to a constant factor.