Before I get back to train tracks (as I had promised in my last post), I wanted to point out some interesting recent work on topologically minimal surfaces. The definition of topologically minimal surfaces was introduced by Dave Bachman  as a topological analogue of higher index geometrically minimal surfaces, suggested by work of Hyam Rubinstein. I discussed these in detail in my series of posts on axiomatic thin position, but here’s the rough idea: An incompressible surface has topological index zero because there is no way to compress it, so it’s similar to a local minimum, i.e. an index-zero critical point of a Morse function. A strongly irreducible Heegaard surface has topological index one because there are two distinct ways to compress it, similar to how there are two distinct ways to descend from an index-one critical point (a saddle) in a Morse function. An index two surface will be weakly reducible, but there will be an essential loop of compressions, in the sense that consecutive compressing disks will be disjoint, but the loop is homotopy non-trivial in the complex of compressing disks. This should remind you of an index-two critical point in a Morse function, in which there is a loop of directions in which to descend. Then index-three surfaces have an essential sphere of compressions and so on. Initially, it was unclear how common higher index surfaces would be. I would have guessed that they weren’t very common, and I think Dave felt the same. But a number of recent results indicate quite the opposite.
The first surprise was the three-sphere. Dave had conjectured that the three-sphere should have no topoligically incompressible surfaces. Depending on the details of the definition, the genus one Heegaard surface may or may not count as index one. (It is the only Heegaard surface that is both stabilized and strongly irreducible.) But the idea was that the higher genus surfaces should be too “floppy” to have index. Well, recently an undergraduate student, Dan Appel, working with Dave Gabai discovered that in fact every genus Heegaard splitting of the three-sphere has index . There are apparently no plans to publish this result, but perhaps I can write about it in a future post.
For index two, Dave’s original work showed that minimal common stabilizations of Heegaard surfaces have index two. (He originally called these critical surfaces and this was a large part of the motivation for studying them.) However, Jung Hoon Lee (whose work I’ve written about previously) has shown that many other unstabilized (but weakly reducible) Heegaard surfaces are index-two, including the minimal genus Heegaard splitting of for a closed, orientable surface, and many boundary stabilized Heegaard surfaces . Qiang E and Fengchung Lei have further generalized Lee’s results, showing that one can find many index-two Heegaard surfaces that are amalgamated from Heegaard splittings of two manifolds glued along incompressible boundary surfaces . Note that not every Heegaard splitting that results from amalgamation will be topologically minimal. For example, I recently that the minimal Heegaard splitting of any high distance surface bundle (which is amalgamated from two copies of ) has exactly one pair of weak reducing disks and therefore does not have a well defined topological index .
So index two surfaces are pretty common, but what about higher index? Dave and I found examples of surfaces with arbitrarily high index back in 2009  by gluing together copies of two-bridge link complements in chains along their boundary tori. Each resulting manifold has one index- surface, where is equal to the number of link complements used in the construction. Jung Hoon Lee has now generalized this result to show that every manifold that contains an incompressible surface contains infinitely many topologically minimal surfaces of arbitrarily high genus . These surfaces come from attaching tubes between parallel copies of the incompressible surface. Most likely their topological indices are also arbitrarily high, though Lee is unable to show this. (This is a good open problem – the method that Dave and I used to bound the indices of our surfaces from below cannot be applied to Lee’s examples.)
Note that Tao Li showed  that a non-Haken manifold contains only finitely many index-one Heegaard surfaces. The proof uses branched surfaces which, as I’ll explain after I get back to posting about train tracks, are closely related to normal surface theory. Using Dave Bachman’s results on normalizing higher index surfaces, one could probably apply Li’s techniques to show that a non-Haken manifold has only finitely many higher index topologically minimal surfaces of a fixed index. However, by a direct generalization of Li’s techniques, one probably can’t rule out infinitely many topologically minimal surfaces of arbitrary index. For example, as noted above, the three-sphere has infinitely many topologically minimal surfaces, but only one for each index. So this begs the question: Does every three-manifold contain infinitely many different topologically minimal surfaces?