An algebraic proof for the symplectic structure of moduli space

Goldman has constructed a symplectic form on the moduli space Hom(π, G)/G, of flat G-bundles over a Riemann surface S whose fundamental group is π. The construction is in terms of the group cohomology of π. The proof that the form is closed, though, uses de Rham cohomology of the surface S, with local coefficients. This symplectic form is shown here to be the restriction of a tensor, that is defined on the infinite product space G^π. This point of view leads to a direct proof of the closedness of the form, within the language of group cohomology. The result applies to all finitely generated groups π whose cohomology satisfies certain conditions. Among these are the fundamental groups of compact Kahler manifolds.