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 $\ge 2$. 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.

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