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.
November 26, 2013
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.
April 29, 2008
Here’s an interesting example that Sangbum Cho (a student of Daryl McCullough at Oklahoma) showed me: If I did things correctly, there should be a picture below of a genus to handlebody with some simple closed curves drawn in its boundary surface. The two blue loops form a Heegaard diagram for the lens space L(3,1). The red, green and orange loops are the boundaries of disks in the handlebody.
A disk in a handlebody of a Heegaard splitting is called primitive if there is a disk in the other handlebody such that the two boundary loops intersect in a single point. The thing about a primitive disk is that compressing the Heegaard surface across a primitive disk produces a new, lower genus Heegaard splitting for the same 3-manifold. (The original Heegaard splitting is a stabilization of the new one.) Notice that each of red, green and orange loops intersects one of the blue loops in a single point (and the other blue loop in possible more points).
In the curve complex, one can consider the subset consisting of boundaries of primitive disks for each of the handlebodies in a Heegaard splitting. This comes up, for example, in Cho and McCullough’s work on the tree of unknotting tunnels and Cho’s work on the Goeritz group. In the example above, the three primitive disks form a pair of pants decomposition for the surface, corresponding to a maximal (two) dimensional simplex in the curve complex.
The interesting thing is that the dimension of the set of primitive disks for genus two Heegaard splittings of lens spaces depends on the lens space. If you try to generalize the diagram above, you can find a family of lens spaces (with criteria in terms of the continued fraction expansion of p/q) that have two dimensional primitive sets. Sangbum has a nice proof (though I can’t reproduce it here) that these lens spaces are the only ones that have a two dimensional primitive set. All other lens spaces have a one dimensional primitive set.
Although the primitive set has important connections (especially for someone like me who’s obsessed with stabilization), I don’t know of any direct applications of knowing the dimension of the set. But it is pretty interesting that it can vary within a class of such similar seeming manifolds. I think it would be interesting to see what this set can look like for general 3-manifolds. For example, does every 3-manifold have a Heegaard splitting for which the primitive set has maximal dimension?
January 16, 2008
John Hempel has suggested a simple method for finding reasonably efficient paths between loops in the curve complex. Recall that the curve complex is the simplicial complex whose vertices are isotopy classes of essential simple closed curves in a given surface and whose simplices bound collections of pairwise disjoint (pairwise non-isotopic) loops. A path in this complex is a sequence of loops in the given surface such that consecutive loops are disjoint. It’s not too hard to show that this complex is connected. A geodesic between two vertices is a path of minimal length.
This algorithm might generalize to surfaces with boundary, but I’m going to assume we have a closed surface. We start with two loops, say a and z that have been isotoped to intersect minimally and whose complement is a collection of disks. (If a component of the complement contains a non-trivial loop then they are distance one or two and finding a geodesic is trivial.) Each component of the complement is a polygon such that the edges of the polygon alternate between arcs of a and arcs of z. Put a vertex at the center of each such polygon and draw an edge from the vertex to each edge of the polygon that comes from an arc of z. We connect the edges along arcs of z to form a graph.
The vertices in squares in the complement are valence two so we can erase these vertices and think of only the vertices in the larger polygons. The resulting graph turns out to be a spine for the complement in our surface of the loop a. Any simple closed edge path determines an essential loop in the complement of a and Hempel suggests we let b be any such loop. I want to be a little more cautious. Since there are a finite number of simple closed edge paths, we might as well pick one that intersects z minimally. In fact, if we do this then we will have chosen a loop in the complement of a that minimizes the intersection number with z over all loops in the complement of a. (This is a reasonably straightforward exercise.) We can then repeat the process with b and z and so on until we get to a loop that is distance two from z.
On principle, this process should give us a very efficient path. The question is, does it yield a geodesic? Hempel has shown that it does for very short paths (I think he said it works for distance three or four). Now, there is already an algorithm for computing geodesics in the curve complex, due to work of Kenneth Shakleton. However, Hempel’s algorithm is extremely simple. The algorithm points in the direction that locally seems best, always choosing a loop that is distance one from a and intersects z minimally. It would be very interesting to see if such a simple idea could be effective for understanding an object as complex and unforgiving as the curve complex.