### Abstract

We prove that, for a free noncyclic group F, the second homology group H_{2}(FQ;Q/ is an uncountable Q-vector space, where FQ denotes the Q-completion of F. This solves a problem of AK 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} (FZ, Z) is not a divisible group, where FZ is the integral pronilpotent completion of F.

Original language | English |
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Pages (from-to) | 1237-1249 |

Number of pages | 13 |

Journal | Geometry and Topology |

Volume | 23 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

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### Scopus subject areas

- Geometry and Topology

### Cite this

*Geometry and Topology*,

*23*(3), 1237-1249. https://doi.org/10.2140/gt.2019.23.1237

}

*Geometry and Topology*, vol. 23, no. 3, pp. 1237-1249. https://doi.org/10.2140/gt.2019.23.1237

**A finite Q-bad space.** / Ivanov, Sergei O.; Mikhailov, Roman.

Research output

TY - JOUR

T1 - A finite Q-bad space

AU - Ivanov, Sergei O.

AU - Mikhailov, Roman

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We prove that, for a free noncyclic group F, the second homology group H2(FQ;Q/ is an uncountable Q-vector space, where FQ denotes the Q-completion of F. This solves a problem of AK 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 H2 (FZ, Z) is not a divisible group, where FZ is the integral pronilpotent completion of F.

AB - We prove that, for a free noncyclic group F, the second homology group H2(FQ;Q/ is an uncountable Q-vector space, where FQ denotes the Q-completion of F. This solves a problem of AK 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 H2 (FZ, Z) is not a divisible group, where FZ is the integral pronilpotent completion of F.

UR - http://www.scopus.com/inward/record.url?scp=85068852470&partnerID=8YFLogxK

U2 - 10.2140/gt.2019.23.1237

DO - 10.2140/gt.2019.23.1237

M3 - Article

AN - SCOPUS:85068852470

VL - 23

SP - 1237

EP - 1249

JO - Geometry and Topology

JF - Geometry and Topology

SN - 1465-3060

IS - 3

ER -