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

TL;DR

This paper investigates the limitations of desuspending certain homotopy equivalences in algebraic geometry using Galois actions, revealing that a natural equivalence does not lift to a simpler map.

Contribution

It demonstrates that a specific homotopy equivalence derived from purity cannot be desuspended to a map between the underlying spaces, using Galois action techniques.

Findings

Homotopy equivalence from purity does not desuspend

Galois action obstructs desuspension

Limits of algebraic homotopy lifting

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