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

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.