Soon after John Hempel introduced the notion of (curve complex) distance for Heegaard splittings, Kevin Hartshorn showed that the existence of an incompressible surface implies a bound on the distance for any Heegaard splitting of the same 3-manifold. Scharlemann and Tomova noted that a strongly irreducible Heegaard surface behaves much like an incompressible surface, and generalized Hartshorn’s Theorem as follows: If admits a distance Heegaard surface then every other genus Heegaard surface with is a stabilization of . This is a great theorem and has had huge consequences for the field, but there is one thing that has always bothered me about it: It leaves open the possibility that, for example, there may a 3-manifold with a genus three, distance 100 Heegaard surface and a second Heegaard surface of genus 201 that is unrelated to the first one. This has always seemed very unlikely to me, so I propose the following conjecture:
The generalized Scharlemann-Tomova conjecture: For every genus , there is a constant such that if is a genus , distance Heegaard surface then every Heegaard surface for is a stabilization of .
I should mention that this conjecture is in the spirit of a comment Cameron Gordon made in his talk at Hyamfest last summer, along the lines of “If you’re going to ask a question, you may as well make a conjecture because people love to prove you wrong.” (As proof of this he noted, that someone had once found a counter example to a “question of Gordon” which he hadn’t even stated as a conjecture.) In this case, I have no particular reason to believe that the conjecture is true and no ideas for how to prove it. Moreover, in the last few weeks, I’ve been more inclined to look for counter examples.
The equivalent generalization of Hartshorn’s Theorem is not true, as Saul Schleimer pointed out to me a while ago: Take any 3-manifold with a genus Heegaard splitting and infinite first homology (such as a connect sum of copies of where the first homology has rank ). Choose a pseudo-Anosov map on the Heegaard surface that acts trivially on its first homology (a Torelli map) and whose stable and unstable laminations are not limits of disks in either handlebody. Hempel showed that the second condition implies that cutting the manifold along the Heegaard surface and regluing by composing with high powers of this map produces Heegaard splittings of arbitrarily high genus. The homology condition implies that the first homology group of each new manifold will be isomorphic to the first homology of the original manifold. Infinite first homology implies infinite second homology (by duality) so these manifolds with arbitrarily high distance Heegaard splitting are all Haken.
The generalized S-T conjecture doesn’t necessarily have weighty and far reaching consequences, but it seems to me like a good conjecture to help motivate progress in the field. Much of the recent work on Heegaard surfaces and bridge surfaces has been fueled by the ability to generalize the ideas in Scharlemann-Tomova’s proof to other situations. However, they all include the caveat that they only restrict the existence of Heegaard splittings below some genus bound. To keep the progress from stalling, we will need an injection of fundamentally new ideas to determine whether or not this caveat is necessary.