# On discrete homology of a free pro-$p$-group

## TL;DR

This paper proves that the second discrete homology group of a finitely generated free pro-$p$-group is uncountably infinite-dimensional over $Z/p$, resolving a question posed by Bousfield.

## Contribution

It establishes the uncountability of the second discrete homology group for free pro-$p$-groups, providing a significant insight into their homological properties.

## Key findings

- $H_2(hat_p,Z/p)$ is uncountable
- Answers Bousfield's problem
- Advances understanding of pro-$p$-group homology

## 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.