J. Birman had observed that the homological information about a given Heegaard splitting of genus g is contained in a double coset in the group of symplectic 2g×2g integer matrices with respect to a suitable subgroup, and found a determinant invariant of this double coset. We obtain complete invariants of these double cosets by characterizing it in terms of the linking form of the manifold lifted to a handlebody of the Heegaard splitting and then finding complete invariants of this lifted form
