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In addition to the first homology group, it also determines the Seifert pairing on the torsion in the first homology group. What is more, in an appropriate sense this is all it determines. This is all contained in my paper "Symplectic Heegaard splittings and linked abelian groups" with Joan Birman and Dennis Johnson, which can be downloaded from my webpage here. This paper also shows that the symplectic matrix also determines a certain subtle invariant of the Heegaard splitting (that alas vanishes after a single stabilization).