In a recent post, I mentioned a paper by David Bachman  claiming to prove Gordon’s conjecture that a connect sum of unstabilized Heegaard splittings is unstabilized. Unfortunately this paper is a bit hard to read and certain topologists have suggested that parts of the paper seem far to optimistic. Thus in its current form the proof has not been verified, though I understand Dave is currently working with a dedicated referee to sew up all the loose ends. Regardless of the status of the proof, the main tool (sequences of generalized Heegaard splittings or SOGs) are interesting and potentially useful in other contexts. (They also come up in Dave’s examples related to the stabilization problem, which he’s currently writing up.) So I wanted to discuss what I understand about SOGs in order to see how much of Dave’s proof I believe and maybe look for other applications of SOGs. As the title suggests, I plan on writing a few entries along these lines.
In this post I will discuss the rough ideas and motivations (as I understand them) for the proof. In later entries I’ll try to flesh out the details. I want to make it clear that I am not the referee for this paper. I’m only trying to understand it in order to see if the ideas in the proof might be useful for other things.
We should start a few decades before Dave’s definition of SOGs with the paper “Reducing Heegaard splittings” by Casson and Gordon, which showed that every weakly reducible Heegaard splitting is either reducible or can be compressed (on both sides) down to an incompressible surface. Scharlemann and Thompson noticed that Casson and Gordon’s proof can be reinterpreted through an analogy to bridge position and thin position for knots. Though I’m not sure if they had this in mind when writing their paper, their notion for thin position for 3-manifolds can be derived from Morse functions.
Recall that a Morse function is a smooth function such that all the singular points are non-degenerate and are in distinct levels. For a 3-manifold, the critical points have index 0, 1, 2 or 3. At an index 0 and 1 critical point the complexity of the level surfaces increases, while at the index 2 and 3 critical point, the complexity decreases. (I won’t define complexity precisely, but you can look it up or come up with an appropriate definition as an exercise.) If we take a sequence of regular levels that are local maxima and minima (in terms of complexity) then these cut the 3-manifold into compression bodies in a very precise way that determines the structure that Scharlemann and Thompson defined, and which has come to be known as a generalized Heegaard splitting.
The set of Morse functions has a very nice structure as a subset of the (vector) space of smooth functions. The space of Morse functions is not connected, but if you throw in near-Morse functions, those where there is a single degenerate critical point, or exactly two critical points in the same level, then the space becomes connected. A path that passes through near-Morse functions corresponds to a sequence of isotopies and moves in which one of two things happen: two critical points may pass each other, or two critical points whose indices differ by one may cancel with each other or may be produced (uncancelled?). On a 3-manifold, one can work out what these two moves do to the generalized Heegaard splitting induced by the Morse function. A sequence of generalized Heegaard splittings is a sequence in which consecutive generalized Heegaard splittings are related by such moves.
One defines a complexity on generalized Heegaard splittings by a lexicographic ordering on the thick surfaces (the surfaces whose complexities are local maxima). In this ordering, one first compares the largest surfaces in each generalized splitting, then the second most complex in each, and so on. With this ordering, a generalized splittings with more surfaces of lower complexity will be less complex than one with fewer, higher complexity surfaces. Scharlemann and Thompson showed that given a minimal complexity generalized Heegaard splitting, the thin surfaces (those whose complexities are local minima) are incompressible. (Also, the thick surfaces are strongly irreducible Heegaard splittings for the complements of the thin surfaces.)
Dave’s idea is to now apply a lexicographic ordering to all the sequences of generalized Heegaard splittings between two fixed generalized Heegaard splittings and choose the path of minimal complexity. Remember, this ordering favors paths that pass through less complex generalized splittings, even if these paths are longer. The claim is that just as the minimally complex generalized Heegaard splittings picked out topological features (incompressible surfaces), the minimal complexity SOGs will do this. More precisely, there should be some feature that is preserved at each step in a minimal SOG, implying that this feature is the same in the initial and final splitting. I’ll be more precise about what is meant by this in my next entry.