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 $F$.

References in corpus (1)

• On discrete homology of a free pro-$p$-group

Cited by in corpus (4)

• Homological properties of parafree Lie algebras
• Formality and finiteness in rational homotopy theory
• Higher structures in rational homotopy theory
• Parafree augmented algebras and Gröbner-Shirshov bases for complete augmented algebras