Università di Roma ”La Sapienza”

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 Γ̂g,n be its profinite completion. There is a natural homomorphism from the Galois cohomology of Γ̂g,n, with coefficients in a discrete torsion Γ̂g,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. Mathematics Subject Classifications (2000): 14G32, 14H10, 32G15, 14F35, 11R34. Introduction Let Mg,n, for 2g−2+n > 0, be the functor of n-pointed, genus g, smooth algebraic curves over C. This functor is represented by a smooth connected Deligne-Mumford stack (briefly D–M stack) over C of dimension 3g − 3 + n, also denoted by Mg,n. Let Mg,n denote also the underlying topological étale groupoid. In the category of topological étale groupoids, there are natural definitions of fundamental group (see, for instance, §4, Ch. I, [M]) and of general homotopy groups (see §1.2, Ch. V, in [M]). The fundamental group then turns out to be naturally isomorphic to the first homotopy group (Theorem V.1.6, [M]). For topological étale groupoids like Mg,n, the theory is simpler, since they admit connected finite Galois étale coverings in the subcategory of topological manifolds. They are given by the level structures Mg,n such that λ dominates an abelian level of order at least 3 (see Proposition 1.2 in [B]). The fundamental group of Mg,n then fits in the short exact sequence (see Proposition I.4.3, [M]): 1 → π1(M λ g,n, a) → π1(Mg,n, a) → G λ → 1,