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

Abstract

We use the Galois action on ${\pi }_1^{\text{et}}({\mathbb{P}}_{\overline{\mathbb{Q}}}^1-\{0,1,\infty \})$ 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 ${\mathbb{G}}_{m,\mathbb{Q}}\vee {\mathbb{G}}_{m,\mathbb{Q}}\to {\mathbb{P}}_{\mathbb{Q}}^1-\{0,1,\infty \}$.