A finite $\mathbb{Q}$–bad space

Abstract

We prove that, for a free noncyclic group $F$, the second homology group $H_2\left({\widehat{F}}_{\mathbb{Q}},\mathbb{Q}\right)$ is an uncountable $\mathbb{Q}$–vector space, where ${\widehat{F}}_{\mathbb{Q}}$ denotes the $\mathbb{Q}$–completion of $F$. This solves a problem of A K Bousfield for the case of rational coefficients. As a direct consequence of this result, it follows that a wedge of two or more circles is $\mathbb{Q}$–bad in the sense of Bousfield–Kan. The same methods as used in the proof of the above result serve to show that $H_2\left({\widehat{F}}_{\mathbb{Z}},\mathbb{Z}\right)$ is not a divisible group, where ${\widehat{F}}_{\mathbb{Z}}$ is the integral pronilpotent completion of $F$.