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:
October 16, 2011
Recall that the mapping class group of a Heegaard splitting is the group of automorphisms of the ambient 3-manifold that take the Heegaard surface onto itself, modulo isotopies of that keep on itself. The isotopy subgroup is the group of such maps that are isotopy trivial on , when you ignore the Heegaard surface. Hyam Rubinstein and I constructed a number of examples of Heegaard splittings with non-trivial isotopy subgroups , but all of these groups were generated by reducible automorphisms. Before our paper, Darren Long had constructed a strongly irreducible Heegaard splitting with a pseudo-Anosov element in its mapping class group  but the ambient manifold is a Seifert fibered space over a sphere with five singular fibers and Heegaard splittings of Seifert fibered spaces have large mapping class groups, usually including lots of reducible maps. This raises the question: Is there a Heegaard splitting with a non-trivial isotopy subgroup consisting entirely of pseudo-Anosov maps (other than the identity)? I will go a step further and, in the spirit of my last two posts, suggest a reckless conjecture:
The reducible automorphism conjecture: The isotopy subgroup of every Heegaard splitting is generated by reducible automorphisms.
May 17, 2011
In the previous post, we discussed MIST presentations for mapping class groups of closed oriented surfaces of genus 1 and 2, and (in a weaker sense) also for genus 3. How might we set about obtaining good presentations for mapping class groups of surfaces of higher genus? A-priori, this looks as though it might be a difficult problem.
The breakthrough was a paper by Hatcher and Thurston. The background for their idea was a result of Brown about how to deduce a presentation of the group from a finite description of its action on a simply-connected simplicial complex . The mapping class group acts on a surface rather than on a simply-connected complex; but simply-connected complexes can be built out of a choice of curves on the surface. Hatcher and Thurston make such a choice, and construct the cut-system complex, which they show to be simply-connected using Morse-Cerf theory. This gives an algorithm which in principle constructs a finite presentation for a mapping class group of a surface of arbitrarily high genus. All papers about presentations of mapping class groups for surfaces of arbitrary genus seem to factor through these ideas of Hatcher and Thurston.
May 16, 2011
One of low dimensional topology’s most popular groups is the mapping class group of a surface. We care about mapping class groups because of how they relate to Heegaard splittings of 3-manifolds. Number theorists and physicists care about mapping class groups because mapping class groups are orbifold fundamental groups of moduli spaces of Riemann surfaces.
Being an unsophisticated sort of a bloke when it comes to group theory, nothing makes me feel that I understand a group like a good concrete presentation of that group, with generators and relations. In my opinion, a good presentation of a group should have the following properties, which I’ll call MIST for fun.
- Memorable- The presentation should be easy to remember.
- Informative- The generators and the relations should be conceptually meaningful and natural, and should tell me something enlightening about the group.
- Simple- The presentation should be easy to work with, by which I mean mainly that it should be short, with as few generators and relations as possible, and with the relations being as short as possible.
- Typical- It should fit into a bigger picture, by which I mean mainly that it should relate easily to similar presentations of similar groups.
Lots of work has been done to find the best presentation for a mapping class group. But I think there’s still a (relatively short) way to go; and I think that, if the work’s worth doing, then it’s worth doing right. I’ve been thinking a bit about presentations of mapping class groups for a summer course I’m teaching, and I wish I could teach a MIST presentation. But not enough is known, and I don’t know enough about what is known, so in this post I’ll briefly summarize my understanding of the state of the art for mapping class group presentations for genus (higher genus next time), and you can tell me where I’m wrong and where there’s more to be known!
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…)
September 10, 2009
During a recent visit, number theorist Jordan Ellenberg told me about a “time-worn analogy” between
(a) A pseudo-Anosov homeomorphism acting on a surface.
(b) The Frobenius automorphism of a smooth algebraic curve .
Jordan has two very interesting posts on this subject, one on what the dilatation should be in case (b) and a recent one where he discusses the finite field analogue of the following question related to the Virtual Haken Conjecture:
Conjecture: A hyperbolic 3-manifold which fibers over the circle has a finite cover with .
As I noted earlier, this is known when the fiber has genus two, or more broadly if the monodromy is hyperelliptic. Intriguingly, Jordan explains the analogous conjecture in the context of (b) is also known in exactly this case…
September 6, 2009
I don’t think it would be too controversial to assert that the Kirby Theorem is an important theorem in low dimensional topology. Given a 3-manifold and an framed link (by “framed” let’s mean “integer framed”), let denote the 3-manifold obtained from by surgery around . The Kirby Theorem states that, given two framed links , the 3-manifolds and are homeomorphic if and only if and are related by a finite sequence of the following local moves:
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 15, 2008
The mapping class group of a Heegaard splitting is the group of automorphisms of the ambient 3-manifold that take the Heegaard surface onto itself, modulo ambient isotopies that keep the Heegaard surface on itself. A few things are known about mapping class groups of irreducible Heegaard splittings (in particular, those with high Hempel distance), but for stabilized Heegaard splittings, with one exception, essentially nothing is known. (The exception is the genus two Heegaard splitting of the 3-sphere, about which almost everything is known. But come on, genus two is easy, right?)
Recall that a Heegaard splitting is stabilized if it is the result of attaching a trivial handle to a lower genus splitting. I’m really interested in the subgroup of the mapping class group coming from isotopy automorphisms of the ambient 3-manifold. For the rest of the entry, this is what I will mean by the mapping class group.
The mapping class group of a stabilized Heegaard splitting should be related to the group of the lower genus splitting that it comes from, but it’s unclear how the two groups should be related. In fact, I don’t think there is even a conjecture for what the mapping class group of a once-stabilized Heegaard splitting should be. Well, to fix this absence, I’m going to state one. It should probably be stated as a question rather than a conjecture, but conjectures are more dramatic (plus I think I can prove it for certain cases.)
First consider a once-stabilized Heegaard splitting of a surface cross an interval S x I. The initial Heegaard surface for S x I is a horizontal surface parallel to S that cuts the manifold into two surface cross interval pieces, each of which is a (trivial) compression body. When we stabilize this splitting, we attach a one handle to each compression body. The resulting Heegaard splitting has exactly one non-separating meridian disk on each side of the Heegaard surface, so any mapping class element must take these two meridian disks onto themselves. This makes the mapping class group really easy to understand, and it turns out to be a semi-direcy product of the findamental group of S (coming from draggin the stabilization around S) and the infinite cyclic group (coming from spinning the stabilization around a reducing sphere). (The proof of this is left as an exercise for the reader.)
If S is a Heegaard surface then a stabilization of S is contained in a regular neighborhood of S and the above group is a subgroup of the stabilization’s mapping class group, which I’ll call the middle subgroup. The stabilized surface can also be pushed into each of the handlebodies of the original Heegaard splitting, forming a stabilized Heegaard surface for this handlebody. This Heegaard splitting of the handlebody has a unique meridian disk on one side (but not the other), making a description of its mapping class group not too hard (though I won’t describe it here). This defines two more subgroups of the stabilized surface’s mapping class group, one for each of the original handlebodies, which I’ll call the left and right subgroups. (There are probably better terms for these. Leave your suggestions in the comments!) The middle subgroup is contained in each of the left/right subgroups and the three groups generate a subgroup of the mapping class group of the stabilized splitting. (I think it’s an amalgamated free product of the left and right subgroups.)
Finally, the original Heegaard splitting has its own mapping class group. We can extend an automorphism of the original splitting to the stabilized surface by tweaking the original automorphism to be the identity in a disk neighborhood of the area where the stabilization is added. Then it extends to the stabilized surface by making it the identity on the new handle. Of course, there are lots of different ways that we can tweak the original map. In fact, this construction is only defined up to composition by the middle subgroup. If we look at all the ways of tweaking the original map we get a larger subgroup, which I’ll call the naive subgroup (again, feel free to suggest a better term) and there appears to be a short exact sequence from the middle subgroup into the naive subgroup into the original mapping class group.
Ok, so we now have four subgroups of the mapping class group of the stabilized splitting and they seem to have reasonably nice structures. I’ll call the subgroup generated by all these subgroups the induced subgroup. This subgroup is always persent in the mapping class group of the stabilized splitting. But is it the whole group? That’s the conjecture:
Conjecture: The induced subgroup (described above) should be the entire mapping class group of the stabilized Heegaard splitting.