  Let $f$ be the gluing map of a Heegaard splitting of a 3-manifold $W$. The
goal of this paper is to determine the information about $W$ contained in the
image of $f$ under the symplectic representation of the mapping class group. We
prove three main results. First, we show that the first homology group of the
three manifold together with Seifert's linking form provides a complete set of
stable invariants. Second, we give a complete, computable set of invariants for
these linking forms. Third, we show that a slight augmentation of Birman's
determinantal invariant for a Heegaard splitting gives a complete set of
unstable invariants.
