#### Volume 23, issue 3 (2019)

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A finite $\mathbb{Q}$–bad space

### Sergei O Ivanov and Roman Mikhailov

Geometry & Topology 23 (2019) 1237–1249
##### Abstract

We prove that, for a free noncyclic group $F\phantom{\rule{0.3em}{0ex}}$, the second homology group ${H}_{2}\left({\stackrel{̂}{F}}_{ℚ},ℚ\right)$ is an uncountable $ℚ$–vector space, where ${\stackrel{̂}{F}}_{ℚ}$ denotes the $ℚ$–completion of $F\phantom{\rule{0.3em}{0ex}}$. 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 $ℚ$–bad in the sense of Bousfield–Kan. The same methods as used in the proof of the above result serve to show that ${H}_{2}\left({\stackrel{̂}{F}}_{ℤ},ℤ\right)$ is not a divisible group, where ${\stackrel{̂}{F}}_{ℤ}$ is the integral pronilpotent completion of $F\phantom{\rule{0.3em}{0ex}}$.

##### Keywords
homology, nilpotent completion, Bousfield–Kan completion, R–good space, R–bad space
##### Mathematical Subject Classification 2010
Primary: 14F35, 16W60, 55P60