I’m running out of open questions to write about, so I want to urge all you readers to submit your own open questions via the comments box on the open problems page. For now, though, here’s a problem about mapping class groups of Heegaard splittings that may not be that hard, but I haven’t had time to think about it myself.
The mapping class group of a Heegaard splitting is the group of automorphisms of the ambient manifold that take the Heegaard splitting onto itself, modulo isotopies that keep the Heegaard surface on itself. (This is sometimes referred to as the Goeritz group of the Heegaard splitting, though other times the Goeritz group refers specifically to Heegaard splittings of the 3-sphere.) Since every automorpism of the Heegaard splitting comes from an automorphism of the ambient manifold, there is a canonical homomorphism from the mapping class group of the Heegaard splitting to the mapping class group of the ambient manifold. In general, this homomorphism won’t be one-to-one or onto, though by stabilizing the Heegaard splitting, we can find a splitting for any manifold such that the homomorphism is onto.
Once one has a Heegaard splitting for which this homomorphism is onto, one can ask further whether there is an isomorphic copy of the mapping class group of the 3-manifold contained in the mapping class group of the Heegaard splitting. More generally, given a Heegaard splitting, consider the short exact sequence from the kernel of the canonical homomorphism to the mapping class group of the Heegaard splitting to the mapping class group of the image of the homomorphism in the mapping class group of the 3-manifold. Does this short exact sequence split? (I.e. is there an isomorphism from the mapping class group of the 3-manifold back to the mapping class group of the Heegaard splitting such that the composition of this isomorphism with the canonical homomorphism is the identity?)
If the short exact sequence splits then the mapping class group of the Heegaard splitting is a semi-direct product of the kernel of the canonical homomorphism and its image (which in many cases will be the mapping class group of the 3-manifold). We can also ask the weaker question: Given a 3-manifold, is there always a Heegaard splitting such that the short exact sequence splits? And an even weaker version: can every mapping class group of a 3-manifold be embedded in the mapping class group of a surface? I don’t know if an answer to this last question is known, though it seems like it might be easier to answer through other means. If one can find a 3-manifold whose mapping class group is not a subgroup of a surface mapping class group then no Heegaard splitting of this 3-manifold can have a split short exact sequence.