n-Nilpotent Obstructions to pi_1 Sections of P^1-{0,1,infty} and Massey Products

TL;DR

This paper explores obstructions to sections of the projective line minus three points using Massey products, revealing new conditions on Galois cohomology related to rational points and fundamental group sections.

Contribution

It introduces a novel connection between Massey products and obstructions to sections of P^1 minus three points, extending previous work to more general group actions.

Findings

Boundary maps are Massey products under certain actions.

Sections from rational points satisfy specific Massey product vanishing conditions.

New obstructions relate to Galois cohomology and rational points.

Abstract

Let pi be a pro-l completion of a free group, and let G be a profinite group acting continuously on pi. First suppose the action is given by a character. Then the boundary maps delta_n: H^1(G, pi/[pi]_n) -> H^2(G, [pi]_n/[pi]_{n+1}) are Massey products. When the action is more general, we partially compute these boundary maps. Via obstructions of Jordan Ellenberg, this implies that pi_1 sections of P^1_k-{0,1,infty} satisfy the condition that associated nth order Massey products in Galois cohomology vanish. For the pi_1 sections coming from rational points, these conditions imply that < (1-x)^{-1}, x^{-1}, x^{-1}, ..., x^{-1} > = 0 where x in H^1(Gal_k, Z_l(chi)) is the image of an element of k^* under the Kummer map.