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Our goal in this article is to give an expository account of some recent work on the classification of topological field theories. More specifically, we will outline the proof of a version of the cobordism hypothesis conjectured by Baez and Dolan in [2].

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Let G be a compact Lie group and BG a classifying space for G. Then a class in H n 1 ÔBG; ZÕ leads to an n-dimensional topological quantum field theory (TQFT), at least for n 1, 2, 3. The theory for n 1 is trivial, but we include it for completeness. The theory for n 2 has some infinities if G is not a finite group; it is a topological limit of 2-dimensional Yang-Mills theory. The most direct analog for n 3 is an L 2 version of the topological quantum field theory based on the classical Chern-Simons invariant, which is only partially defined. The TQFT constructed by Witten and Reshetikhin-Turaev which goes by the name 'Chern-Simons theory' (sometimes 'holomorphic Chern-Simons theory' to distinguish it from the L 2 theory) is completely finite.The theories we construct here are extended, or multi-tiered, TQFTs which go all the way down to points. For the n 3 Chern-Simons theory, which we term a '0-1-2-3 theory' to emphasize the extension down to points, we only treat the cases where G is finite or G is a torus, the latter being one of the main novelties in this paper. In other words, for toral theories we provide an answer to the longstanding question: What does Chern-Simons theory attach to a point? The answer is a bit subtle as Chern-Simons is an anomalous field theory of oriented manifolds.1 This framing anomaly was already flagged in Witten's seminal paper [Wi]. Here we interpret the anomaly as an invertible 4-dimensional topological field theory A , defined on oriented manifolds. The Chern-Simons theory is a "truncated morphism" Z : 1 A from the trivial theory to the anomaly theory. For example, on a closed oriented 3-manifold X the anomaly theory produces a complex line A ÔXÕ and the Chern-Simons invariant ZÔXÕ is a (possibly zero) element of that line. This is the standard vision of an anomalous quantum field theory in general; here we use this description down to points. The invariant of a 4-manifold in the theory A involves its signature and Euler characteristic. It was first discovered in a combinatorial description [CY] and Walker [W] also uses A in his description of Chern-Simons (for a more general class of gauge groups). Since a torus is an abelian group, the classical Chern-Simons action is quadratic in the connection and so the theory is in some sense "free". Indeed, one expects that the semi-classical approximation is exact. This is the point of view taken by Manoliu [Ma], who constructs Chern-Simons for circle groups as a 2-3 theory. The invariant this theory assigns to a closed oriented 3-manifold is made from classical invariants of 3-manifold topology: is the integral over the space of flat connections of the square root of the Reidemeister torsion times a spectral flow phase. (There is an overall

Let X be an algebraic curve defined over a finite field Fq and let G be a smooth affine group scheme over X with connected fibers whose generic fiber is semisimple and simply connected. In this paper, we affirm a conjecture of Weil by establishing that the Tamagawa number of G is equal to 1.