Title: On discrete homology of a free pro-$p$-group,
Abstract: For a prime $p$, let $\hat F_p$ be a finitely generated free pro-$p$-group of
rank $\geq 2$. We show that the second discrete homology group $H_2(\hat
F_p,\mathbb Z/p)$ is an uncountable $\mathbb Z/p$-vector space. This answers a
problem of A.K. Bousfield.