I was recently reading Scharlemann’s description [1] of Berge’s construction [2] of a pair of genus two, distance three Heegaard splittings that are non-isotopic in the same manifold. The proof that the splittings are distinct involves showing that each has a unique minimal crossing number diagram, using a version of the rectangle condition. Today, however, I’d like to focus on the construction that turns one of the Heegaard surfaces into the other. Scharlemann describes the construction by thinking of one of the surfaces in a four component link complement and pushing two of the components into the surface from opposite sides, then out on the other side. But there seems to be another description that lends itself better to generalization, and which starts with Figure 4 in his paper. I’ve redrawn this picture below the fold.

While it’s somewhat difficult to see, the Figure shows two loops in the Heegaard surface, each of which separates the surface into two punctured tori. Since both loops are separating, they define a checkerboard coloring of the surface. Following the original picture, I’ve colored one set of regions green. Below it is the same pattern with the other set of regions colored blue. If you look very carefully, you can count two blue octagons and four green quadrilaterals. Both of the separating loops bound disks inside the handlebody. If we thicken the surface, then glue a 2-handle to each side along each of the disks, the result is a 3-manifold with four torus boundary components. To get the manifold we want, we will Dehn fill along these tori by some reasonably complicated surgery.

In the induced genus two Heegaard surface for the resulting 3-manifold, these two loops bound disks on opposite sides. We will use these disks to construct a new Heegaard splitting for the same 3-manifold as follows: Let be the cell complex . (This is a cell complex because the complement in of the boundaries of the disks is a collection of disks.) If we remove the two cells that came from then we get back.

But there are also the two cells that come from those blue octagons in the figure. What happens if we remove those? Well, the complement of consists of four open solid tori. Each of the octagons connects one of these tori to another, and in fact connects them in two pairs. Thus if we remove the two octagons, the complement of the resulting object will be a pair of handlebodies. By looking at the local properties of , one can check that removing the two octagons actually produces a surface. (I’ll leave this as an exercise.) So, removing the blue octagons produces a new Heegaard surface.

This new Heegaard surface is precisely the alternate Heegaard surface constructed by Berge/Scharlemann. To show that the new surface is not isotopic to the original takes a careful argument that I won’t summarize here. Instead, I want to point out that the outline of this new description is easy to generalize: Assume we have a Heegaard splitting with separating disks on opposite sides such that one of the colored regions in their checkerboard coloring consists of exactly two disks. Let and let be the result of removing the two blue disks from . The same local argument (again left to the reader) implies that what’s left is a surface whose complement is a pair of handlebodies, so is a Heegaard surface. If the blue region consists of more than two disks, then you can do the same thing and get a higher genus Heegaard surface.

Of course, this new surface may be highly stabilized or it may be isotopic to the old. But I’m curious how many of the existing examples of non-isotopic Heegaard splittings in the same 3-manifold can be constructed this way. For example, given two 3-manifolds with Heegaard splittings, one can construct a Heegaard splitting for their connect sum by cutting open each of the two Heegaard splittings along a disk, then gluing the handlebodies in pairs along these disks. There are two choices of how we pair up the handlebodies, which potentially give us two distinct Heegaard splitting. (Kazuto Takao [3] has shown that if the original Heegaard splittings had sufficiently high distance, then not only will these two Heegaard splittings be non-isotopic, but their smallest common stabilization will have high genus.)

In this construction, the sphere along which the summands are glued defines a pair of disks on opposite sides of the Heegaard surface. We can make these disks transverse so that their boundaries intersect each other in two points and cut the Heegaard surface into two blue disks and two green pieces of higher genus. The reader can check that performing the construction above using these two disks turns one of the connect sum Heegaard splittings into the other.

Looking at further known examples of manifolds with distinct Heegaard splittings, this construction doesn’t seem to be completely universal. The different vertical Heegaard splittings of Seifert fibered spaces are related by pushing a core loop from one side of the Heegaard surface to the other, so it should be possible to relate the different vertical splittings by this construction. However, it doesn’t seem to be possible to get between the different horizontal Heegaard splittings, or from a horizontal to a vertical Heegaard splitting by this construction. I also can’t find a way to describe the Casson-Gordon-Paris examples by this construction. I am very curious if there is a way to generalize the construction described above to a construction that always produces a new Heegaard splitting and that can be used to describe these other examples of distinct splittings.

The set of alternative genus two Heegaard splitting omitted from the original Rubinstein/Scharlemann paper, of which some are distance three splittings,is much more extensive than those arising from the construction Scharlemann describes, based on Figure 4 of [1]. They all arise, as in Scharlemann’s construction, from Dehn fillings of manifolds with four toroidal boundary components, each obtained by gluing a pair of disks to opposite sides of a closed orientable genus two surface S along a pair of separating simple closed curves in S.

Section 3 of [2] provides a characterization of these alternate Heegaard splittings. In particular, Figure 3b of Section 3 is relevant. In this figure, and are disjoint simple closed curves, such that is disjoint from a separating disk D in one handlebody of a genus two splitting, and is disjoint from a separating disk D’ in the other handlebody of the splitting. Furthermore, if in Figure 3b, then . This, together with the fact that, if these are distance three splittings, then must cut the Heegaard surface S into disks, implies .

It follows that in order to obtain all alternate distance three splittings, one must allow to be arbitrarily large. However, only the separating curves of Scharlemann’s Figure 4, with , have the desired pair of “blue octagons”.

Comment by Berge — June 21, 2010 @ 7:59 am |