Last week was a very Heegaard-intensive week for the Topology/Geometry seminar here at Yale. On Thursday, Yoav Moriah (who is visiting for the year) gave a talk on some recent work with Martin Lustig : Given a Heegaard splitting, they have a way of finding a loop in the Heegaard surface such that all but two integral Dehn surgeries on that loop produce a high distance Heegaard splitting (of a new 3-manifold.) Moreover, they can show that such loops are in some sense “generic” in the surface. (Yoav suggested that there should really be only one slope that doesn’t produce a high distance splitting, but they’re still working on that.) This is a very powerful tool for constructing high distance Heegaard splittings, as previously known methods all involved composing the gluing map with high powers of pseudo-Anosov homeomorphisms, rather than just a simple Dehn twist. The idea seems to be that by rotating the curve complex are around a loop that is far from both handlebody sets, you can pull the two sets apart from each other.
The previous day, Wednesday, Dave Bachman stopped by on his way to Columbia and gave a talk about finding a definition of topological index. This is part of a (perhaps unoffical) program that was started by Hyam Rubinstein and of which Dave has been working to flesh out the details. (The sequences of generalized Heegaard splittings that I discussed in a series of posts previously also fall under this program.)
This program, which I will call the Rubinstein program, involves creating a dictionary between topological properties of topological surfaces, geometric properties of minimal surfaces and combinatorial properties of normal surfaces. An incompressible surface is always isotopic to a minimal surface in a reasonable metric, and to a normal surface with respect to any triangulation. Pitts and Rubinstein showed that, modulo a few caveats, a strongly irreducible Heegaard splitting is isotopic to an index-one minimal surface. Rubinstein and Stocking showed independently that strongly irreducible Heegaard splittings are isotopic to almost normal surfaces.
There is a very nice analogy between strongly irreducible, index-one minimal, and almost normal surfaces: In each case, it is possible (in some suitably vague sense) to “push” the surface in two independent directions, each of which reduces the complexity of the surface. Strongly irreducible surfaces, can be compressed on either side, but once you compress on one side, you can’t compress on the other. Index-one minimal surfaces can be isotoped in one of two directions to reduce their area, but not both. Almost normal surfaces can be isotoped in two different directions to reduce their intersection with the 1-skeleton of a triangulation, but not both.
For minimal surfaces, there is a clear definition of higher index surfaces (which I think is called the Morse index). There is also a definition of higher index normal surfaces which was proposed by Rubinstein. (He didn’t get it quite right at first, but that’s a different story…) The idea that Dave discussed in his talk was a definition of index in the topological context, i.e. how to define a topological index for surfaces. The long term goal would be to show that having topological index n implies that the surface is isotopic to an index-n minimal or normal surface. But lets not worry about that yet.
The definition that Dave proposed for topological index surprised me quite a bit, but after a long, thorough discussion, I think I believe it: First, we define incompressible surfaces to have index zero. For compressible surfaces, consider the curve complex for the surface in question, and the subset consisting of all loops that bound compressing disks on either side of the surface. This is a subcomplex of the curve complex and we can consider its homotopy groups. The topological index of a surface is one plus the dimension of the first non-trivial homotopy group for this subcomplex. In particular, for strongly irreducible surfaces, the set of compression disks is disconnected (one set on each side of the surface) so its 0-th homotopy group is non-trivial, making it index-1.
When I first heard this definition, I thought Dave had gone off the deep end. I couldn’t figure out what homotopy groups had to do with anything, but once I saw how he planned to use this definition in a proof, it started to make sense. A compressing disk should be thought of as a direction in which we can “push” the surface. If the nth homotopy group is non-trivial then there is an n-dimensional sphere of directions that we can “push” it in. This seems to mesh reasonably well with the intuition (or at least my intuition) for what it means to be an index-(n+1) minimal surface, i.e. there is an (n+1)-dimensional subspace of directions (the cone over an n-sphere) to isotope the surface that decrease its area. As far as using this definition in a proof, there still seem to be some details to iron out, but I think the general intuition is right and I’m excited to see where this line or reasoning leads.