  We use the Galois action on
$\pi_1^{\textrm{et}}(\mathbb{P}_{\overline{\mathbb{Q}}}^1 - \{0,1,\infty \})$
to show that the homotopy equivalence $S^1 \wedge (\mathbb{G}_{m,\mathbb{Q}}
\vee \mathbb{G}_{m,\mathbb{Q}}) \cong S^1 \wedge (\mathbb{P}_{\mathbb{Q}}^1 -
\{0,1,\infty \}) $ coming from purity does not desuspend to a map
$\mathbb{G}_{m,\mathbb{Q}} \vee \mathbb{G}_{m,\mathbb{Q}} \to
\mathbb{P}_{\mathbb{Q}}^1 - \{0,1,\infty \}$.
