#### 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 $F\phantom{\rule{0.3em}{0ex}}$.

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##### Keywords
homology, nilpotent completion, Bousfield–Kan completion, R–good space, R–bad space
##### Mathematical Subject Classification 2010
Primary: 14F35, 16W60, 55P60
##### Publication
Received: 27 July 2017
Revised: 29 July 2018
Accepted: 29 September 2018
Published: 28 May 2019
Proposed: Mark Behrens
Seconded: Haynes R Miller, Stefan Schwede
##### Authors
 Sergei O Ivanov Laboratory of Modern Algebra and Applications St. Petersburg State University Saint Petersburg Russia Roman Mikhailov Laboratory of Modern Algebra and Applications St. Petersburg State University Saint Petersburg Russia St. Petersburg Department of Steklov Mathematical Institute Saint Petersburg Russia