Low Dimensional Topology

July 23, 2008

Amalgamations

Filed under: 3-manifolds,Heegaard splittings — Jesse Johnson @ 11:15 am

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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