Wedges, Rational completion and $\mathbb{Q}$-bad spaces

Let $A$ and $X$ be connected based CW complexes with $H_1(A;\mathbb{Q})\mathrm{\ne }0$, and let $m\ge 1$ be the least degree with ${\tilde{H}}_m(X;\mathbb{Q})\mathrm{\ne }0$. We prove that the Bousfield–Kan rational completion map of $A\vee X$ induces a map on $H_{2m}(-;\mathbb{Q})$ whose cokernel has dimension at least $2^{{\mathrm{\aleph }}_0}$. In particular, $A\vee X$ is $\mathbb{Q}$-bad.