A finite Q-bad space

TL;DR

This paper demonstrates that the second homology of the rational completion of a free noncyclic group is uncountably dimensional, revealing new insights into the structure of $Q$-bad spaces and their homological properties.

Contribution

It proves that the second homology group of the rational completion of a free noncyclic group is uncountably dimensional, answering a longstanding problem and analyzing $Q$-bad spaces.

Findings

$H_2(\hat F_Q, Q)$ is uncountable-dimensional.

A wedge of circles is $Q$-bad.

$H_2(\hat F_Z, Z)$ is not divisible.

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 $F$.