Jeff Weeks, the author of SnapPea, has written some new educational games this year that are highly worth trying out:
Daniel Moskovich suggested that, in light of the recent advances, I summarize the state of the Virtual Haken Conjecture.
So let M be a closed hyperbolic 3-manifold. Then one has the following sequence of increasingly strong conjectures.
Conjecture 1: 
Conjecture 2: M has a finite cover N which is Haken, i.e. contains a closed incompressible surface.
Conjecture 3: M has a finite cover N with 
Conjecture 4: For each 
Conjecture 5: M has a finite cover N which is large, i.e. 
Here Conjecture 2 is the classical form of the VHC. Conjecture 3 could also be stated as N contains a finite cover which contains a non-separating incompressible surface. In addition one has the question about virtual fibering
Nicolas Bergeron and Dani Wise posted an interesting preprint “A boundary criterion for cubulation” on the arXiv this morning. Using Kahn and Markovic, and work of Ian Agol, they show
Theorem 6.5. If quasi-fuchsian surface subgroups of the fundamental group of a hyperbolic 3-manifold M are separable, then M is virtually fibered.
This has also been proved by Ian using the same approach, see his Georgia slides. The idea is to apply a construction of Sageev to build a CAT(0)-cube complex from a large collection of quasi-fuchsian surface subgroups, and then apply a theorem of Haglund and Wise to get enough residual control over the fundamental group to apply Ian’s earlier work on virtual fibering.
Another aspect of [BW] was the implicit announcement by Wise of the following result
Theorem. Suppose M is a hyperbolic 3-manifold containing an embedded incompressible quasi-fuchsian surface. Then the fundamental group of M is subgroup separable.
In particular, [BW] refers to a 175 page(?!) preprint by Wise showing this, though it doesn’t seem to be on his webpage yet. If correct, this would be another major breakthrough in the study of the Virtual Haken Conjecture.
This post is so Marc Culler and I can announce SnapPy, a computer program for studying hyperbolic structures on 3-manifolds. It is based on Jeff Weeks’ SnapPea kernel from his very influential program of the same name, written in the early 1990s for Macintosh computers. While Jeff’s program doesn’t work (except in emulation) on any computer you can buy today, SnapPy runs on Mac OS X, Linux, and Windows. SnapPy combines a link editor and 3D-graphics for Dirichlet domains and cusp neighborhoods with a powerful command-line interface based on the Python programming language.
You can download it here, and we put some effort in making it trivial to install for OS X and Windows, and reasonably easy on Linux. There are screenshots of it in action, or you can watch an 11-minute tutorial on YouTube. Unlike previous Python interfaces to the SnapPea, this one has decent documentation and also useful graphics. SnapPy can also be used within my favorite general purpose mathematical software package Sage and has some extra features there for dealing with finite covers.
SnapPy was written by Marc Culler and myself, using Jeff’s kernel code, and today we released version 1.0 (superseding 1.0a and 1.0b). If you successfully install SnapPy, or installed it earlier this summer, leave a note in the comments mentioning what type of system you’re using. (We test it on seven or eight different setups, but you never know what happens out there in the wild.) Enjoy!
The basic tenet of Thurston’s approach to 3-manifolds is the idea that the topology of a 3-manifold should determine a fairly canonical geometric structure on the manifold (modulo some technicalities). This suggests that there should be a dictionary for translating topological features of the manifold into geometric features, an idea that has been called WYSIWYG (what you see is what you get) topology. (I think Steve Kerckhoff coined the term.) While the Theorems relating the topology of Heegaard splittings to geometry are still a bit coarse, they give a very nice intuitive picture that I think is useful, and I would like to describe this picture below.
Hyperbolic 3-manifolds with a single cusp have canonical ideal triangulations constructed by Epstein and Penner via a certain convex hull construction involving the light-cone in the hyperboloid model of hyperbolic space. (Occasionally, these “triangulations” are really cellulations more complicated cells.) When such a 3-manifold fibers over the circle, there is another type of natural triangulation, called a layered triangulation. Roughly, one starts with a certain ideal triangulation of the fiber surface, looks at the image of this triangulation under the bundle monodromy, interpolates between these by a series of Pachner moves which can then be realized geometrically by layering on tetrahedra.
When the fiber is a once-punctured torus, these two type of triangulations coincide. This was shown by Marc Lackenby using a remarkably soft and elegant argument. It’s natural to wonder whether this phenomena occurs more broadly. For instance Sakuma suggested considering the following:
Conjecture: Canonical triangulations of punctured surface bundles are always layered.
Saul Schleimer and I have discovered that this is false in general. In particular, the manifold v1348 from the SnapPea census is fibered by a once-punctured surface of genus 5 yet has a canonical triangulation which is not layered. Precisely, the canonical triangulation does not admit one of Lackenby’s taut structures so that the resulting branched surface carries something with positive weights. Note that v1349 is in fact the complement of a certain knot in the 3-sphere [CFP].
Technical details: The canonical triangulation of v1348 is in fact just the triangulation encoded in the SnapPea census (and it is a triangulation, not a celluation). It’s easy to check that it fibers using the BNS invariant (cf. [DT]), and compute the genus of the fiber from the Alexander polynomial. One then checks that there are no taut structures of this type using Marc Culler and I’s t3m Python package.
This post has comes with a PDF version, which contains more details.
In my opinion, Habiro and Goussarov’s theory of claspers is a milestone in low dimensional topology [1][2][3]. Using this theory, they were able to solve important outstanding problems in quantum topology, and to build a bridge to connect the “quantum” with the “classical”.
In this series of posts, I would like to introduce claspers to readers of this blog. I think that claspers are an handy tool for low-dimensional topologists, as a natural extension (or refinement) of Dehn surgery which (in a sense to be explained in future posts) is compatible with the lower central series of the Torelli group of a Heegaard surface. Claspers are already fairly mainstream in some circles (used or referenced in around 300 papers according to MathSciNet), but I think that they should be more popular with a wider audience (if gropes are familiar to you- in dimension 3 claspers and gropes are essentially equivalent).
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Hajimemashite. My name is Daniel Moskovich, and I’m a JSPS postdoc at RIMS in Kyoto University. I’m originally from Jerusalem, Israel. I moved to Japan for graduate school, where I finished my PhD in 2006 under the tutelage of Tomotada Ohtsuki.
I’m a quantum topologist, but most things I actually work on are pretty classical. I think that the major barrier to really understanding quantum invariants is that the classical aspects aren’t sufficiently well understood. Most topics I would like to post on are related to this divide- I’ll explain classical ideas and recent developments which I hope will eventually clarify the nature of quantum invariants. I’ll try to choose topics which are relevant in other contexts as well.
Hello all. Jesse has invited me to blog occasionally here at the premier geometric topology blog. I’m also a 3-dimensional topologist, and while I usually describe myself as a geometer, it’s probably more accurate to say I’m an algebraist at heart. Or at least I use a lot of algebra in my work, not to mention number theory. I also do a lot of computation, and I’m planning on doing some posts on experiments that I’ve run that don’t merit formal publication but are still (I hope!) interesting.
One of my main interests has been the Virtual Haken Conjecture and related questions, so I’m very excited by the recent announcement of Kahn and Markovic. Danny Calegari has posted a detailed summary of the argument that Jeremy sent him, which is definitely worth reading.
It looks like Danny Calegari recently started a blog, Geometry and the Imagination. His most recent post gives an outline of Kahn and Morkovich’s proposed proof that every hyperbolic 3-manifold contains an immersed injective surface (based on Ian Agol’s summary in the comments of the last post on this blog.) And there’s a lot more good stuff on there too.
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