Tao Li recently posted a paper on Heegaard splittings of 3-manifolds that come from gluing together two 3-manifolds along a pair of boundary components [1]. Coincidentally, the question considered in this paper is related to my recent papers on the stabilization problem [2], [3] and Dave Bachman’s (announced but still in being written) program for the stabilization problem [4]. I’d like to talk about these in a future post, but for now let’s focus on Li’s paper. As always, the discussion will be vague so that non-experts can (hopefully) get an idea of what’s going on. For details, you can read the paper.

The setup is that we have a 3-manifold *M* that comes from gluing together 3-manifolds *M_1, M_2* along a pair of boundary components. The image in *M* of the pair of boundary components is a surface *F* and we would like to understand the Heegaard splittings of *M* in relation to *F*. For example, if we take a Heegaard surface for *M_1* and a Heegaard surface for *M_2*, then the union of these surfaces and *F* cut *M* into a collection of compression bodies, defining what’s called a *generalized Heegaard splitting*. A construction called *amalgamation *turns this generalized Heegaard splitting into a (regular) Heegaard splitting for *M*.

This amalgamated Heegaard splitting *S* has a very nice form with respect to *F*; a subsurface of *S* is parallel to *F* (minus a few punctures) and the rest of *S* consists of handles that extend into *M_1* and *M_2*. So this Heegaard surface is in some sense parallel to *F*. Note that such a Heegaard splitting is determined by the initial Heegaard splittings for M_1 and M_2 and regardless of the gluing map between the two boundary components. Because *S* is “parallel” to *F*, changing the gluing map along *F* doesn’t affect the existence of *S*.

Of couse, there may be other Heegaard splittings for *M* that don’t come from amalgamations along *F*. We want to think of these as being transverse to *F*. The idea is that because they’re transverse to *F*, their existence depends entirely on having a gluing map that attaches the intersection of *S* and *M_1* to the intersection of *S* and *M_2*. In particular, there is a strong relationship between the genus of *S* and a certain distance in the curve complex for *F* that measures the complexity of the gluing map. The hard part is to work out what this distance is.

If the glued boundary components of *M_1* and *M_2* are compressible into their respective manifolds then the surface *F* will be compressible in both directions. The genus of a Heegaard surface *S* that is “transverse” to *F* is in this case bounded below in terms of the minimal distance between a compressing disk on one side and a compressing disk on the other. This is a beautiful result of Scharlemann and Tomova [5].

If F is incompressible on one or both sides then finding the right definition of distance is a little tougher. Bachman, Schleimer and Sedgwick [6] did this in the case where *F* is an incompressible torus by considering slopes of the boundaries of properly embedded normal surfaces in the complement of *F* for some fixed triangulation. They define the distance in terms of the boundaries of minimal genus surfaces of this form and relate the genus of a non-amalgamated Heegaard splitting to this distance.

Tao Li generalizes these results to the case where *F* is any separating surface, incompressible or compressible on one or both sides. He defines the distance in terms of either compressing disks in the case when *F* is compressible on a given side or essential properly embedded surfaces when F is incompressible to one side (plus a separate case for interval bundles). He then shows that given *M_1* and *M_2*, for every genus *g*, there is some distance *K* such that if *M* comes from gluing map determining a surface *F* with distance greater than *K* then every Heegaard splitting of genus below *g* is an amalgamation along *F*.

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