By the word sideshow in the title of this post, I’m thinking of one of those old circus side shows (that I’m too young to have ever seen) with the bearded lady and all the other strange examples of people and things. I wanted to put together a list of intersting examples among Heegaard splittings, since it’s becoming harder and harder for me to keep track of them all. I’m not going to write about Seifert fibered spaces because their Heegaard splittings are pretty well understood and they belong in the main tent.

We’ll start with 3-manifolds admitting infinitely many irreducible splittings. There’s two ways this can happen: First, there are the examples by Casson and Gordon in which there are strongly irreducible Heegaard splittings of arbitrarily high genus (all in the same 3-manifold.) The original examples were never published, but Kobayashi [1] generalized the construction to produce 3-manifolds where not only are there irreducible Heegaard splittings of arbitrarily high genus, but the number of splittings at each genus grows with genus. Lustig and Moriah [2] generalized Kobayahi’s construction to an even larger class of examples.

The other way one can get infinitely many irreducible Heegaard splittings is by taking one splitting and spinning it around an incompressible torus. (This is like the 3-dimensional version of a Dehn twist.) If you set things up right, the new Heegaard splittings will not be isotopic to the original. Morimoto and Sakuma [3] showed that this happens for all genus two Heegaard splittings of toroidal knot complements. Sakuma also wrote a preprint demonstrating the same behavior for a closed 3-manifold with a genus two Heegaard splitting, but it doesn’t seem to have gotten published ever. Sakuma also has a student, Yeonhee Jang, who has used similar methods to find a three-bridge knot that admits infinitely many non-isotopic three-bridge surfaces, but this paper doesn’t appear to be publically available yet.

The examples based on Casson and Gordon’s construction all start off with fairly high genus Heegaard splittings, so one might ask whether a closed 3-manifold can have both a genus two Heegaard splitting and a higher genus irreducible splitting. This was demonstrated recently by Jung Hoon Lee [4]. (I mentioned this a few weeks ago.) His 3-manifold has both a genus two Heegaard splitting and an irreducible genus three splitting.

Next there are questions of weakly reducible vs. strongly irreducible splittings. Weakly reducible ones come from cutting the 3-manifold along incompressible surfaces, taking Heegaard splittings for the complementary pieces, then constructing a Heegaard splitting for the original 3-manifold from these pieces. Strongly irreducible splittings, on the other hand, are in some sense independent of any incompressible surfaces. Schultens and Weidmann [5] showed that even if the Heegaard splittings of the pieces are irreducible, the result of the amalgamation can be highly reducible. In their examples, which are amalgamations along a torus, the Heegaard genus of the whole manifold is half the sum of the genera of the pieces.

In Seifert Fibered spaces, the weakly reducible (vertical) Heegaard splittings generally have much lower genus than the strongly irreducible (horizontal) splittings. However, this does not seem to be the case in general. Kobayashi and Reick [6] recently showed that there are 3-manifolds containing both a strongly irreducible splitting and a weakly reducible splitting that are both minimal genus over all splittings. To my knowledge, it is still open whether there is a manifold with a minimal genus strongly irreducible splitting and a non-minimal genus weakly reducible splitting. Lee’s example from above would be a candidate for such an example if one could show that the genus three splitting he constructs is weakly reducible.

Next there’s the problem of flipping a Heegaard surface, i.e. isotoping it off itself and then back in a way that interchanges the handlebodies. Hass, Thompson and Thurston [7] showed that there are splittings that not only can’t be flipped, but you can stabilize them (i.e. add trivial handles) and they will remain unflippable until you double their genus. (I also mentioned these examples previously.) Bachman has also announced similar examples [8], using different techniques.

Speaking of stabilization, one can also ask how many stabilizations are needed, given two Heegaard surface for a 3-manifold, to turn one into the other. Hass-Thompson-Thurston address this problem if you consider oriented Heegaard surfaces and want to turn one of their non-flippable surfaces into itself with the opposite orientation. If you want to ignore orientation, then I have examples [9], for every integer *k*, of 3-manifolds with Heegaard splittings of genera 2*k*-1 and 2*k* such that one needs to stabilize up to genus 3*k*-1 to turn one into the other. Bachman has also announded examples of this sort, again using different techniques.

So, those are the examples I can think of but, as I mentioned, I’m having trouble keeping track. Did I miss any? Any thoughts on questions or conjectures that might be inspired by this sideshow?

[1] T. Kobayashi; *A construction of 3-manifolds whose homeomorphism classes of Heegaard splittings have polynomial growth*, Osaka J. Math. 29, (1992), 653 – 674.

[3] M. Sakuma, K. Morimoto; *On unknotting tunnels for knots*, Math. Ann. 289 (1991) 143-167.

Regarding the statement: “Morimoto also wrote a preprint demonstrating the same behavior for a closed 3-manifold with a genus two Heegaard splitting, but it doesn’t seem to have gotten published ever.”

Actually, the preprint was written by Sakuma (not Morimoto) for a conference proceedings in 1988, and called “Manifolds with infinitely many non-isotopic Heegaard splittings of minimal genus”. Right after Sakuma finished the paper, he heard that Johannson had announced a better result. Thus he didn’t publish the paper, even though it later turned out that Johannson had not obtained such a result. I have a copy of Sakuma’s preprint, the techniques used are almost totally algebraic, different in spirit from the examples later found.

“Morimoto also wrote a preprint demonstrating the same behavior for a closed 3-manifold with a genus two Heegaard splitting, but it doesn’t seem to have gotten published ever.”

Comment by Ryan DT — February 18, 2009 @ 9:44 am |

That’s a good point! (I just fixed it, but for the record, the original version of this post had Morimoto instead of Sakuma.) I’m also glad to hear an explanation of why Sakuma never published the preprint.

Comment by Jesse Johnson — February 18, 2009 @ 9:52 am |