In my last post, I described how a train track on a surface determines a collection of loops in a surface, namely the loops that are carried by the track. Looking at these loops from the perspective of the the Farey graph for the torus, this set consists of the loops corresponding to vertices in one of the components that results from cutting the Farey graph along a certain edge. In the curve complex, train tricks define partitions that are almost as simple, though they are necessarily more complicated because there is no one simplex that separates separates the complex. Still, this type of partition comes in very useful for calculating distances in the curve complex (and was central to my recent preprint with Yoav Moriah) but to see how that works, we need something a bit stronger. In this post, I’ll explain how we can turn the partition defined by a train track into two sets of curves with a buffer between them. By placing these buffers next to each other, we can build larger gaps that imply a lower bound on the distance between certain loops in the curve complex.
March 31, 2014
March 14, 2014
A little over a year ago, I started writing a series of posts on train tracks and normal loops, then got distracted by other things. In the mean time, I wrote a paper with Yoav Moriah involving train tracks and curve complex distances, which gave me a whole new perspective on what train tracks really mean, more in line with much of Masur and Minsky’s work . So, I want to resuscitate the series of posts on train tracks, but in a slightly different direction than where I was headed before. I’ll start by looking at a very simple case: train tracks on a torus. If you need a review of what train tracks are (the mathematical object, not the literal ones), you can reread my earlier post.
November 26, 2013
Mark your calendars now: in June 2014, Cornell University will host “What’s Next? The mathematical legacy of Bill Thurston”. It looks like it will be a very exciting event, see the (lightly edited) announcement from the organizers below the fold.
August 22, 2012
Bill Thurston passed away yesterday at 8pm, succumbing to the cancer that he had been battling for the past two years. I don’t think it’s possible to overstate the revolutionary impact that he had on the study of geometry and topology. Almost everything we blog about here has the imprint of his amazing mathematics. Bill was always very generous with his ideas, and his presence in the community will be horribly missed. Perhaps I will have something more coherent to say later, but for now here are some links to remember him by:
September 29, 2011
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 .
June 15, 2011
The Kakimizu Complex for a knot is what you get by taking the definition of the curve complex for a surface and replacing loops in the surface with minimal genus Seifert surfaces for . It consists of a vertex for each isotopy class of minimal genus Seifert surface for with edges connecting any two vertices with disjoint representatives, and simplices spanning larger collections of pairwise disjoint surfaces. This complex turns out to be contractible  and for atoroidal knots, it’s finite . But something I found quite surprising is an example by Jessica Banks  whose Kakimizu complex is locally infinite. This knot is relatively simple and you can see a picture of it below the fold.
February 26, 2010
Here’s a neat result about mapping class goups of Heegaard splittings that was proved in a recent preprint  by Marion Moore and Matt Rathbun: The mapping class group of a Heegaard splitting is determined by the coarse geometry of the curve complex for the Heegaard surface. In particular, a Heegaard splitting determines two quasi-convex subsets of the complex of curves for the Heegaard surface and one can define the quasi-mapping class group for a Heegaard splitting in terms of the quasi-isometries of the complex that keep each set within a bounded neighborhood of itself. Their result shows that (modulo a technicality in genus two) the quasi-Mapping class group of a Heegaard splitting is isomorphic to its mapping class group. (more…)
January 22, 2009
Given a loop in the boundary of a genus two handlebody, it is occasionally useful to characterize the curve in terms of what you get when you attach a 2-handle along that curve. In particular, such a loop is called primitive if attaching a 2-handle produces a solid torus and is called Seifert if attaching the 2-handles produces a (small) Seifert fibered space.
Given a knot in the 3-sphere that happens to sit in a genus two Heegaard surface, we can characterize that loop in terms of how it sits with respect to the two handlebodies bounded by the Heegaard surface. For example, if it is primitive in both handlebodies (i.e. double primitive) then we call the knot a Berge knot after John Berge, who classified such knots. (A very simple construction shows that every Berge knot has a Dehn filling producing a lens space. The converse of this statement is the Berge conjecture.) Knots that are primitive to one side and Seifert on the other are called Dean knots after John Dean, who studied them in his dissertation at UT Austin. (Also, Michael Williams has shown  that all primitive/Seifert loops satisfy the Berge conjecture.)
What I’d like to know is how many different positions on a genus-two Heegaard surface it is possible for a given knot type to have. Two loops have the same knot type if there is an ambient isotopy of the 3-sphere taking one to the other. We will say that two loops in a genus two surface are equivalent if there is an ambient isotopy of the 3-sphere that takes the genus two surface to itself and takes one of the loops onto the other. So, how many inequivalent loops in a genus two Heegaard surface can we have with the same, given knot type?
November 4, 2008
Here’s a recent preprint that sounds pretty interesting by Behrstock, Drutu and Sapir . The asymptotic cone of a metric space X is a new metric space that one constructs by scaling the metric on X by smaller and smaller numbers (i.e. you define for small s) and taking a limit as the scaling factor goes to zero. (Actually, you take an ultralimit, which is determined by an ultrafilter, which I won’t explain here. But I do plan on trying to use the prefix “ultra” in my own definitions whenever I can.) The asymptotic cone is a popular construction in coarse geometry because when you shrink your metric like this, the coarse features of the space turn into Lipschitz features. Whatever is left in the limit is completely determined by the large scale geomety. For example, the asymptotic cone of a delta-hyperbolic space is always a tree. The asymptotic cone of Euclidean space is Euclidean space. The asymptotic cone of any bounded-diameter space is a point.
Behrstock, Drutu and Sapir look at the asymptotic cone of the mapping class group of a surface. One does this by choosing a finite generating set for this group, then constructing the Cayley graph for the group and the generating set, then setting each edge length equal to one to make the Cayley graph a metric space. The resulting space is somehow very close to being delta-hyperbolic (it’s related to the complex of curves, which is in fact delta hyperbolic) but it it’s not quite delta hyperbolic. It has these large Euclidean subspaces that come, for example, from taking Dehn twists along a collection of disjoint essential simples closed curves in the surface. These Dehn twists commute with each other so the subgroup of the mapping class group is Abelian. In the geometric picture, an Abelian group is Euclidean (since it is of the form ) and triangles in this Euclidean subspace are not delta thin. (Update: See Jason Behrstock’s comment for more on this.) But the philosophy is that if you find a way to ignore this flat regions, the space looks delta-hyperbolic.
Since the geometry of the mapping class group looks delta hyperbolic when you ignore the Euclidean parts, one would expect the asymptotic cone to combine tree-like features with Euclidean features. Behrstock, Drutu and Sapir show that there is a Lipschitz map from the asymptotic cone of the mapping class group into a direct product of trees. So, while the geometry doesn’t necessarily look like such a product, the topology of the asymptotic cone does. The direct product accounts for the large flat regions in the group. while the trees account for the delta-hyperbolic remainder.
October 28, 2008
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.