Let $H$ be a handlebody with $\Sigma = H$. 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 $\psi([f])$?

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 $\psi([f])$?