Today, I will continue on my quest to find the most interesting conjectures about Heegaard splittings. (Most of these conjectures, including this one, fail criteria one and two in Daniel’s recent post, but strive to satisfy criteria three.) Here’s the latest:

**The minimal genus Heegaard splitting conjecture:** For every positive integer , there is a constant such that if is a hyperbolic 3-manifold with Heegaard genus then has at most isotopy classes of (minimal) genus Heegaard splittings.

Note that the hyperbolic condition is necessary because there are toroidal manifolds, particularly Seifert fibered, spaces with infinitely many minimal genus Heegaard splittings [1]. Minimal genus (rather than just irreducible) is also necessary since there are hyperbolic manifolds with infinitely many distinct irreducible splittings [2]. On the other hand, Lustig and Moriah have constructed manifolds with arbitrarily many minimal genus Heegaard splittings [3], but in order to increase the number of splittings, they have to increase the genus of the the Heegaard splittings. Note that Lustig and Moriah’s examples show that if exists, it grows at least exponentially with . I’ve been toying lately with constructions that produce many distinct irreducible splittings, but again I need to increase the genus in order to increase the number of minimal genus splittings.

My intuition is that for a relatively simple hyperbolic 3-manifold, there are relatively few ways to (efficiently) cut up its topology. In order to find a manifold that can be cut up in more ways, you need to make it more complicated, which increases its genus. However, “more complicated” in this case can’t mean simply higher volume because there are hyperbolic 3-manifolds of arbitrarily high volume with bounded Heegaard genus, though their minimal genus Heegaard splittings will, generally speaking, be unique.

Instead, the picture of what a 3-manifold with lots of minimal genus Heegaard splittings should look like seems to be related to the conjecture that Ian Biringer wrote about a few years ago here. The conjecture is, roughly, that all hyperbolic manifolds with bounded rank and injectivity radius are made up of a finitely many types of pieces glued together along their (incompressible) boundaries. More “complcated” gluing maps should correspond to higher volume manifolds, but once you choose the types of pieces and pair up their boundary components, there will be a bound on the number of minimal genus Heegaard splittings. To increase this number, you would need to either choose more blocks (which increases the genus) or choose more complicated blocks by allowing higher rank (and higher genus) or lower injectivity radius.

[2] Unpublished preprint by Andrew Casson and Cameron Gordon. See this paper for a generalization of the same construction.

[3] Lustig, Martin and Moriah, Yoav. *On the complexity of the Heegaard structure of hyperbolic 3-manifolds.* Math. Z. 226 (1997), no. 3, 349–358.

I think this is definitely true if there is a lower bound on injectivity radius (and likely true in general).

Let’s also assume the manifold is non-Haken, to avoid working with generalized Heegaard splitings. By Pitts-Rubinstein, one can find a minimal surface representative of each genus g Heegaard splitting. By a result of Lackenby (Prop. 6.1 of http://www.ams.org/mathscinet-getitem?mr=2218779), a genus g minimal surface in the manifold has bounded diameter. A refinement of Pitts-Rubinstein shows that you may find minimal surface representatives of all genus g Heegaard surfaces intersecting a given Heegaard surface (this involves a barrier argument). Thus, all of the surfaces intersect a subset of the manifold of bounded diameter. Then a standard compactness argument shows that there are only finitely many Heegaard splittings (universally only depending on g and the lower bound on Heegaard genus, e.g. using pointed Gromov-Hausdorff limits and limits of minimal surfaces).

To see that there must be minimal Heegaard splitting surfaces which intersect, consider a fixed minimal Heegaard surface S, and consider another Heegaard surface R. Take a maximal collection of disjoint minimal surfaces R_1, R_2, …, R_k isotopic to R, and suppose that none of them intersects S. One may assume that the surfaces R_i are linearly ordered, and the regions between them are product regions. Moreover, they alternate between stable and unstable minimal surfaces, with the ends being unstable. Then S is sandwiched between a stable minimal surfaces R_i and unstable R_{i+1} (R_i might be empty). One may push off R_{i+1} in the direction of R_i, and use S as a barrier to find another minimal surface R’ between S and R_{i+1}, which gives a contradiction (cf the work of Colding-Gabai on foliations of manifolds by Heegaard surfaces via mean curvature flow).

I think one ought to be able to generalize this to consider non-Haken manifolds by using generalized Heegaard splittings instead, but I haven’t thought it through. Also, it ought to work with no lower bound on injectivity radius, but will be trickier. Notice though, for g bounded, the short curves must be unlinked http://www.math.ubc.ca/~jsouto/papers/knotting.pdf. Minimal surfaces will have bounded diameter outside of the thin part, but this is not enough to get finiteness, since the Margulis tubes might have arbitrarily large volume. But I think a careful analysis of models ought to yield the result.

Comment by Ian Agol — November 29, 2011 @ 8:59 pm |