So, we have a sequence of generalized Heegaard splittings that we have chosen to be minimal with respect to the lexicographic ordering on the elements of the sequence. We’re looking at the locally maximal splittings in the sequence. Last time, I explained why all but one of the thick surfaces in this generalized Heegaard splitting should be strongly irreducible, implying that we can just focus on the one weakly reducible thick surface.
It might be useful at this point to picture a graph whose vertices are all generalized Heegaard splittings and whose edges connect splittings that are related by a weak reduction or a destabilization. The splittings right before and after the local maximum are each connected to the local max. by an edge, but any other (possibly longer) edge path from one to the other must at some point pass through a more complicated splitting. Thus we can think of this graph as a mountain range in which the local maximum is the lowest mountain pass from the preceding splitting to the following splitting.
In this graph, each edge down from the mountain pass corresponds to a pair of disks in the one weakly reducible thick surface whose boundaries are either disjoint (defining a weak reduction) or isotopic (defining a destabilization). These correspond to pairs of points in the curve complex where the handlebody sets for the Heegaard splitting defined by the thick surface either intersect or are connected by an edge. We want to divide the set of downward edges (and the corresponding pairs of vertices in the curve comples) into sets such that two edges are in the same set if and only if they are connected by a path that never goes above the original mountain pass. Since we started with a local maximum, the down edges get divided into at least two such sets.
I should mention that in Dave’s exposition, he always divides the vertices into exactly two sets, and he throws in all the other loops in the handlebody sets as well, though it doesn’t matter which set they get tossed into. I think this is distracting, so I’m not going to do it.
So, we have cut the set of weak reducing pairs for the thick surface into a number of sets such that each set determines a different side of the mountain. Dave now claims that if two such sets contain a common loop then both pairs are in the same subset i.e. on the same side of the mountain. I’m pretty sure I believe this, and moreover that proving it is a reasonable exercise once one understands generalized Heegaard splittings. However, I don’t know if I can explain the proof well enough without going into more detail than I think a blog entry should contain. So, I’ll just give a quick sketch of the idea, which you might be better off skipping and trying to work it out on your own.
Let’s focus on the case when the two pairs define weak reductions and the weak reducing pairs are D_1, D_2 and D_1, D’_2. The first weak reduction corresponds to pushing the handle corresponding to D_1 past the handle corresponding to D_2. After doing this, we’ll further push D_1 past all the other handles on the same side as D_2 that we can. The resulting generalized Heegaard splitting has a thin surface that comes from compressing the original thick surface along D_1, then compressing it along D_2, then as much as possible on the same side as D_2. If D’_2 intersects this final surface then it defines a further compression. Thus by doing all possible compressions on the side opposite D_1, we’ve compressed D’_2 out as well as D_2. If we had started by compressiong along D_1 then D’_2, we would end up with the same thin surface. Thus there is a series of weak reductions starting with D_1, D_2 and a series of weak reductions starting with D_1, D’_2 that lead to the same generalized splitting, so we didn’t need to go over the pass.
The point of all this is that we now have a description of the local maxima of our minimized SOG in terms of the curve complex of the one weakly reducible thick surface: The intersection of one of the handlebody sets with a 1-neighborhood of the other handlebody set contains more than one component. Dave calls such a thick surface critical, and we will say that a generalized Heegaard splitting is critical if it has one critical thick surface and the remaining thick surfaces are strongly irreducible. Next time I’ll discuss how this fits into the larger picture of Dave’s two proposed proofs.