The next big thing in quantum topology?

The place to be in May for a quantum topologist is Vietnam. After some wonderful-sounding mini-courses in Hanoi, the party with move to Nha Trang (dream place to visit) for a quantum topology conference.

I’d like to tell you very briefly about some exciting developments which I expect will be at the centre of the Nha Trang conference, and which I expect may significantly effect the landscape in quantum topology. The preprint in question is -Efficient triangulations and the index of a cusped hyperbolic -manifold by Garoufalidis, Hodgson, Rubinstein, and Segerman (with a list of authors like that, you know it’s got to be good!).

Manolescu refutes the Triangulation Conjecture

This past week, Ciprian Manolescu posted a preprint on ArXiv proving (allegedly- I haven’t read the paper beyond the introduction) that the Triangulation Conjecture is false.

-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture.

This is big news. I feel it’s the last nail in the coffin of the Hauptvermutung. I’d like to tell you a little bit about the conjecture, and about Manolescu’s strategy, and what it has to do with low dimensional topology.

Train tracks

A few posts back, I defined normal loops in the triangulation of a surface and said I would use this idea to define train tracks on a surface. The key property of normal loops is that the normal arcs form parallel families and we can encode the topology of the curve by keeping track of how many parallel arcs are in each family. Train tracks encode loops in a surface in a very similar way. A train track is a union of bands in the surface (disks parameterized as ) with disjoint interiors, but that fit together along their horizontal sides. In other words, the top and bottom edges of each band are contained in the union of the horizontal edges of other bands. A picture of this is shown below the fold.

Normal loops in surfaces

I plan to write a post or two about normal surfaces and branched surfaces in three-dimensional manifolds, but I want to warm up first, with two posts about the two-dimensional analogues of these objects. Train tracks play a huge role in the approach to the topology of surfaces initiated by Nielsen and Thurston, for understanding mapping class groups, Teichmuller space, laminations, etc. They organize the set of isotopy classes of simple closed curves in a surface in a way that allows one to take limits of infinite sequences of loops. (The limits are called projective measured laminations.)  In this post and the next, I will discuss train tracks from a rather unusual perspective, via normal loops in a triangulation of the given surface.

SnapPy 1.7: Ptolemy and reps to PSL(n, C).

SnapPy 1.7 is out. The main new feature is the ptolemy module for studying representations into PSL(n, C). This code was contributed by Mattias Görner, and is based on the the following two very interesting papers:

1. Stavros Garoufalidis, Matthias Goerner, Christian K. Zickert: Gluing equations for PGL(n,C)-representations of 3-manifolds.
2. Stavros Garoufalidis, Dylan P. Thurston, Christian K. Zickert: The complex volume of SL(n,C)-representations of 3-manifolds.

You can get the latest version of SnapPy at the usual place.

Bill Thurston is dead at age 65.

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:

• Wikipedia.
• 2010 lecture on The mystery of 3-manifolds.
• On proof and progress in mathematics.

Regina on Windows and MacOS

For those non-Linux users who’ve wanted to tinker with Regina:

Regina 4.92 came out a couple of weeks ago, and has some big portability improvements.  Mac users now have a simple drag-and-drop install (no need for fink), and for the first time there is an installer for MS Windows.  As always, there are also ready-made packages for several GNU/Linux distributions.

The new version adds features such as fundamental normal surfaces and boundary slopes for spun-normal surfaces, and the user interface is cleaner.  For more information or to have a play, hop over to regina.sourceforge.net.

Rethinking the basics

Some nights, one gazes up at the stars, and thinks about philosophy. Who are we? What is the meaning of life? What is reality? What are manifolds really?

This morning, I looked at Poincaré’s original definition in Papers on Topology: Analysis Situs and Its Five Supplements, translated by John Stillwell. His original definition was pretty-much that a manifold is a quotient of by a properly discontinuous group action, that group being his original fundamental group. Implicitly, his smooth, PL, and topological categories were all the same thing (indeed true for 1-manifolds, and for dimensions 2 and 3 PL and smooth categories still “coincide” in a sense that can be made fully precise); nowadays we understand that the situation is more subtle. But I’m still not sure that I understand what a manifold is- what it really is.

In some non-mathematical, philosophical (theological?) sense, I believe that both smooth and PL manifolds actually exist, in the sense that natural numbers exist, and tangles exist. Our clumsy formal definitions are attempts at describing something that is actually out there, as the Peano axioms describe the natural numbers. I also believe that Physics is a guide to Mathematics, because things that really exist might also be observed… so ideas from Physics (topological invariants defined by means of path integrals) ought to be taken very seriously, and it is my irrational belief that these will eventually turn out to be the most fundamental invariants in some precise mathematical sense.

It is fascinating to me, then, that input from physics seems to be leading towards a fundamental rethink of the basic definitions of smooth and PL manifolds. I feel like we had some sub-optimal definitions, which we worked with for sociological reasons (definitions are made by people, and people are not perfect), and maybe in the not too distant future there will be a chance to put more convenient definitions in place. Maybe the real world (physics) will force it on us. Let me tell you, then, about some of the papers I’ve been (casually) flicking through recently (the one I’m most excited about is Kirillov’s On piecewise linear cell decompositions).

New SnapPy 1.5, now with better manifolds

Marc Culler and I have released version 1.5 of SnapPy, which you can get from the usual place. The main new feature is greatly improved manifold censuses, including the ability check if a given manifold is in one; here are many examples of now to use the new censuses.

I never blogged about version 1.4, since while there were major changes they were mostly under the hood, principally moving to the current version of IPython and making snappy (mostly) compatible with Python 3.2.

The 3-sphere has interesting triangulations

There is a recent interesting question on MO regarding a paper by Benedetti and Ziegler which I found most interesting.

Upon reading the question, I right away downloaded and printed Benedetti and Ziegler’s paper, after which I sat down to glance through it. My first impression is that it constitutes a really high-class piece of mathematics; the exposition is clear enough that a non-specialist can sit down and enjoy it, and the results are deep and interesting. There’s something really inspirational about a paper like that.