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

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 $\psi([f])$?

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.