I’ve been looking through the proceedings of the Heegaard splittings conference at the Technion in 2005, in particular at a list of open problems Cameron Gordon compiled based on the talks there. Lots of good problems are discussed, but I wanted to mention a couple that were suggested by Yair Minsky. Recall that given a Heegaard surface in a 3-manifold, for each handlebody in the complement one can consider the set of vertices in the curve complex for the surface corresponding to loops that bound disks in the handlebody. This is called a handlebody set and every Heegaard splitting determines two such sets.
The self-homeomorphisms of a handlebody act on the curve complex in a way that preserves the corresponding handlebody set. For the two handlebodies in a Heegaard splitting, the intersection of their mapping class groups (as subgroups of the mapping class group of the surface) is precisely the mapping class group of the Heegaard splitting (the group of automorphisms of the ambient manifold that take each handlebody onto itself.) In addition to the intersection, one can also consider the subgroup of the mapping class group for the surface generated by the mapping class groups of the handlebodies. Yair asks whether this group is a free product with amalgamation of the mapping class groups of the handlebodies, amalgamated along their intersection. It seems like a reasonable problem.
Second, Yair suggests looking at the set of all loops in the curve complex for the Heegaard surface that are homotopy trivial in the ambient manifold. The loops in the handlebody sets are all homotopy trivial, and in fact are all unknots. Loops outside the handlebody set may also be homotopy trivial, whether or not they’re unknotted. For example, in a Heegaard splitting of the 3-sphere, every loop in the curve complex is homotopy trivial. Yair asks when the set of homotopy trivial loops is equal to the image of the handlebody sets under the action of the group generated by the mapping class groups of the handlebodies. My guess is that this is not true except in a few simple cases, though it seems more reasonable that the first set is always contained in the second. Given a 3-manifold with a homogeneous metric, one might also ask what the set of loops that are isotopic to geodesics looks like. This is probably a much harder problem.