Let $H$ be a handlebody with $\Sigma = \partial H$. Given an automorphism $f : \Sigma \to \Sigma$ we can glue to obtain a closed 3-manifold $M = H \cup\_f H$ and in fact all such 3-manifolds are obtained this way. Since only the isotopy class of $f$ is necessary in order to specify the homeomorphism type of the resulting 3-manifold, we have a map from the mapping class group $\mathop{MCG}(\Sigma)$ to the set of homeomrphism types of closed 3-manifolds. By looking at the action on $H\_1(\Sigma;\mathbb{Z})$ which has a symplectic structure via the cup product, we obtain a surjection
$$
\psi: \mathop{MCG}(\Sigma) \to \mathop{Sp}(H\_1(\Sigma; \mathbb{Z}))
$$ 
What information about $M$ can be obtained from $\psi([f])$?



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](https://www3.nd.edu/~andyp/papers/). 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).


