Desuspensions of S^1 /\ P^1_Q-{0,1,infty}

Abstract

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 $\mathbb{G}_{m,\mathbb{Q}} \vee \mathbb{G}_{m,\mathbb{Q}} \to \mathbb{P}_{\mathbb{Q}}^1 - \{0,1,\infty \}$.