Follow up to More on mapping class groups

I now have an answer to the question on mapping class groups of Heegaard splitting I asked a few weeks ago. It appears that for any Heegaard splitting of the 3-torus, the exact sequence from the kernel to the MCG of the Heegaard splitting to the MCG of the 3-manifold does not split (i.e. there’s no homomorphism back from the MCG of the 3-manifold). In fact, there is no injection from the mapping class group of the 3-torus to and surface mapping class group by a Theorem of Farb and Masur [1]. (Thanks go to Yair Minsky for telling me where to find the result.) Specifically, they prove that any homomorphism from an irreducible lattice in a semi-simple Lie group of rank at least two into a mapping class group has finite image. The mapping class group of the 3-torus is SL(3,Z), which is of this form, so such a homomorphism cannot be an injection.