As Professor Birman
indicated in [Bi1] 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. She found in [Bi1] a determinant
invariant of this double coset and we prove in this paper that invariant (strengthened
a bit when the first torsion number is even) is complete. We obtain this result by
characterizing the double coset in terms of the linking form of the manifold lifted to a
handlebody of the Heegaard splitting and by finding complete invariants of this lifted
form. Professor Birman has kindly pointed out to us that the characterization of the
double cosets by means of her invariant is contained in the unpublished manuscript
[Bi-J].
