The Symplectic Nature of Fundamental Groups of Surfaces

A symplectic structure on a manifold is a closed nondegenerate exterior 2form. The most common type of symplectic structure arises on a complex manifold as the imaginary part of a Hermitian metric which is Klhler. Many moduli spaces associated with Riemann surfaces have such Kahler structures: the Jacobi variety, Teichmiiller space, moduli spaces of stable vector bundles and even the first real cohomology group have such structures. In all of these examples the topology of the associated spaces depends, remarkably, only on the topology of the Riemann surface, while often their complex structures vary as the complex structure of the Riemann surface changes. However, the symplectic structure of these spaces depends only on the underlying topological surface. The purpose of this paper is to present a general explanation for this phenomenon. We present a single construction which unifies all of the above examples and interprets their symplectic structures in terms of the intersection pairing on the surface. Our setup is as follows. Consider a closed oriented topological surface S with fundamental group rt and let G be a connected Lie group. The space Hom(n, G) consisting of representations rr + G (given the compact-open topology) is a real analytic variety (which is an algebraic variety if G is an algebraic group). There is a canonical G-action on Hom(rr, G) obtained by composing representations with inner autormorphisms of G. The resulting quotient space Hom(x, G)/G is a space canonically associated with S (or 7~) and G. When G is an abelian group Hom(n, G)/G = Hom(7c, G) = H’(S; G) has a natural (abelian) group structure. The present paper addresses the question of what sort of natural structure Hom(n, G)/G possesses when G is not necessarily abelian. We find that under fairly general conditions on G (e.g., if it is reductive) Hom(r, G)/G admits in a natural way a symplectic structure which generalizes the Kahler forms on all of the spaces mentioned above. Perhaps the main specific new result in this paper is that the Weil-