The mapping class group of a Heegaard splitting is the group of automorphisms of the ambient 3-manifold that take the Heegaard surface onto itself, modulo ambient isotopies that keep the Heegaard surface on itself. A few things are known about mapping class groups of irreducible Heegaard splittings (in particular, those with high Hempel distance), but for stabilized Heegaard splittings, with one exception, essentially nothing is known. (The exception is the genus two Heegaard splitting of the 3-sphere, about which almost everything is known. But come on, genus two is easy, right?)
Recall that a Heegaard splitting is stabilized if it is the result of attaching a trivial handle to a lower genus splitting. I’m really interested in the subgroup of the mapping class group coming from isotopy automorphisms of the ambient 3-manifold. For the rest of the entry, this is what I will mean by the mapping class group.
The mapping class group of a stabilized Heegaard splitting should be related to the group of the lower genus splitting that it comes from, but it’s unclear how the two groups should be related. In fact, I don’t think there is even a conjecture for what the mapping class group of a once-stabilized Heegaard splitting should be. Well, to fix this absence, I’m going to state one. It should probably be stated as a question rather than a conjecture, but conjectures are more dramatic (plus I think I can prove it for certain cases.)
First consider a once-stabilized Heegaard splitting of a surface cross an interval S x I. The initial Heegaard surface for S x I is a horizontal surface parallel to S that cuts the manifold into two surface cross interval pieces, each of which is a (trivial) compression body. When we stabilize this splitting, we attach a one handle to each compression body. The resulting Heegaard splitting has exactly one non-separating meridian disk on each side of the Heegaard surface, so any mapping class element must take these two meridian disks onto themselves. This makes the mapping class group really easy to understand, and it turns out to be a semi-direcy product of the findamental group of S (coming from draggin the stabilization around S) and the infinite cyclic group (coming from spinning the stabilization around a reducing sphere). (The proof of this is left as an exercise for the reader.)
If S is a Heegaard surface then a stabilization of S is contained in a regular neighborhood of S and the above group is a subgroup of the stabilization’s mapping class group, which I’ll call the middle subgroup. The stabilized surface can also be pushed into each of the handlebodies of the original Heegaard splitting, forming a stabilized Heegaard surface for this handlebody. This Heegaard splitting of the handlebody has a unique meridian disk on one side (but not the other), making a description of its mapping class group not too hard (though I won’t describe it here). This defines two more subgroups of the stabilized surface’s mapping class group, one for each of the original handlebodies, which I’ll call the left and right subgroups. (There are probably better terms for these. Leave your suggestions in the comments!) The middle subgroup is contained in each of the left/right subgroups and the three groups generate a subgroup of the mapping class group of the stabilized splitting. (I think it’s an amalgamated free product of the left and right subgroups.)
Finally, the original Heegaard splitting has its own mapping class group. We can extend an automorphism of the original splitting to the stabilized surface by tweaking the original automorphism to be the identity in a disk neighborhood of the area where the stabilization is added. Then it extends to the stabilized surface by making it the identity on the new handle. Of course, there are lots of different ways that we can tweak the original map. In fact, this construction is only defined up to composition by the middle subgroup. If we look at all the ways of tweaking the original map we get a larger subgroup, which I’ll call the naive subgroup (again, feel free to suggest a better term) and there appears to be a short exact sequence from the middle subgroup into the naive subgroup into the original mapping class group.
Ok, so we now have four subgroups of the mapping class group of the stabilized splitting and they seem to have reasonably nice structures. I’ll call the subgroup generated by all these subgroups the induced subgroup. This subgroup is always persent in the mapping class group of the stabilized splitting. But is it the whole group? That’s the conjecture:
Conjecture: The induced subgroup (described above) should be the entire mapping class group of the stabilized Heegaard splitting.