Title: A finite Q-bad space,
Abstract: We prove that for a free noncyclic group $F$, $H_2(\hat F_\mathbb Q, \mathbb
Q)$ is an uncountable $\mathbb Q$-vector space. Here $\hat F_\mathbb Q$ is the
$\mathbb Q$-completion of $F$. This answers 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 circles is $\mathbb Q$-bad in the sense of
Bousfield-Kan. The same methods as used in the proof of the above results allow
to show that, the homology $H_2(\hat F_\mathbb Z,\mathbb Z)$ is not divisible
group, where $\hat F_\mathbb Z$ is the integral pronilpotent completion of $F$.