Relative pro-l completions of Teichmüller groups

Let Γg,n, for 2g − 2 + n > 0, be the Teichmüller group of an n-punctured genus g compact Riemann surface Sg,n and let Γ(l), for a prime l ≥ 2, be the abelian level defined by the kernel of the natural representation Γg,n → Sp2g(Z/l). The pro-l completion Γ(l)(l) of Γ(l) defines a profinite completion of Γg,n, which we call the relative pro-l completion and denote by Γ (l) g,n. There is a natural map from the Galois cohomology of Γ (l) g,n with coefficients in a discrete l-primary torsion Γ (l) g,n-module to the cohomology of the discrete group Γg,n with induced coefficients. The main result of this paper is that this map is an isomorphism. From previous results by Hain, Kawazumi, Looijenga and Morita, it follows that, the component of pure Hodge type of the rational cohomology of the moduli space of curves Mg,n, for n = 0, 1, in degree less than g, is generated by tautological classes. Since, by a result of Pikaart, the stable rational cohomology is of pure Hodge type, this implies Mumford’s conjecture, whose proof was recently announced by Madsen and Weiss. Mathematics Subject Classifications (2000): 14H10, 14H15, 14F35, 11R34.