What does the matrix of a mapping class tell you about the 3-manifold?

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).

Well, from the characteristic polynomial of the matrix, thanks to Casson, one can figure out if the gluing map is pseudo-Anosov (if the characteristic polynomial is irreducible, non-cyclotomic, and does not have the form $P(x) = Q(x^k),$ for some $k>1.$). In principle, there should be a some criterion in terms of the matrix for low-bounding the translation distance in the curve complex, and if that is at least $3,$ and so the manifold will be hyperbolic, but I don't know what the criterion is.