Understanding the anomaly

I’ve recently been looking at the following paper in which -TQFT anomalies are treated carefully and various old constructions of Turaev and Walker are elucidated:

Gilmer, P.M. and Masbaum, G., Maslov Index, Mapping Class Groups, and TQFT, Forum Math. 25 (2013), 1067-1106.

It makes me think a lot about just what the anomaly `actually means’…

A celebration of diagrammatic algebra

Relaxing from my forays into information and computation, I’ve recently been glancing through my mathematical sibling Kenta Okazaki’s thesis, published as:

K. Okazaki, The state sum invariant of 3–manifolds constructed from the linear skein.
Algebraic & Geometric Topology 13 (2013) 3469–3536.

It’s a wonderful piece of diagrammatic algebra, and I’d like to tell you a bit about it!

SnapPy 2.1: Now with extra precision!

Marc Culler and I released SnapPy 2.1 today. The main new feature is the ManifoldHP variant of Manifold which does all floating-point calculations in quad-double precision, which has four times as many significant digits as the ordinary double precision numbers used by Manifold. More precisely, numbers used in ManifoldHP have 212 bits for the mantissa/significand (roughly 63 decimal digits) versus 53 bits with Manifold.

Banker finds a duplication in a 3-manifold table

Daniel Moskovich recently wrote about the discovery by a lawyer of a duplication in the knot tables called the “Perko pair”.

Now a banker has found another duplicate in yet another table of 3-manifolds. This time it was Ben Burton, and the duplicate appears in the Hildebrand-Weeks cusped hyperbolic census.

What’s Next? A conference in question form

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.

Regina 4.94

It’s the season for it!  For those of you who work with normal surfaces, Regina 4.94 also came out last week.  It adds triangulated vertex links, edge drilling, and a lot more speed and grunt.

Take the new linear/integer programming machinery for a spin with the pre-rolled triangulation of the Weber Seifert dodecahedral space.  Regina can now prove 0-efficiency in just 10 seconds, or enumerate all 1751 vertex surfaces in ~10 minutes, or (with a little extra code to coordinate the slicing and searching for compressing discs) prove the entire space to be non-Haken in ~2 hours.

Read more of what’s new, or download and tinker at regina.sourceforge.net.

SnapPy 2.0 released

Marc Culler and I pleased to announce version 2.0 of SnapPy, a program for studying the topology and geometry of 3-manifolds. Many of the new features are graphical in nature, so we made a new tutorial video to show them off. Highlights include

Smooth proof of Reidemeister-Singer

Every construction I know of 3-manifold invariants from Heegaard splittings factors through the Reidemeister-Singer Theorem:

Reidemeister-Singer Theorem: For any two Heegaard splittings and of a 3-manifold , there exists a third Heegaard splitting which is a stabilization of both.

This theorem is definitely part of the big story in 3-manifold topology, and is usually proven in the PL category, as for example in Nikolai Saveliev’s Lectures on the Topology of 3-manifolds. There is another nice PL proof due to Craggs, Proc. Amer. Math. Soc. 57, n 1 (1976), 143-147.

I think of a Heegaard splitting as being intrinsically a smooth topology construction (a level set of a Morse function), and so I would really like the proof of Reidemeister-Singer to live in the smooth category. I think that there should be consistent smooth and PL stories of 3-manifold topology living side by side. In the 1970’s, Bonahon wrote a smooth proof of Reidemeister-Singer, which uses Cerf Theory (naturally, because we’re investigating paths between Morse functions). Unfortunately, Bonahon’s proof was never published, and it is lost.

A year ago (but I only saw it this morning), François Laudenbach posted a smooth proof of Reidemeister-Singer to arXiv: http://arxiv.org/abs/1202.1130. I think that this is wonderful! There are too few papers like this- there is insufficient incentive to streamline the storylines of foundations. I am very happy to have found this proof, and I want such a proof to be a part of my smooth 3-manifold topology foundations.

Edit: Thanks to George Mossessian and to Ryan Budney, who point out in the comments that Jesse Johnson proved Reidemeister-Singer using Rubinstein and Scharlemann’s sweep-outs, which involves singularity theory which is much less sophisticated that Cerf Theory: http://front.math.ucdavis.edu/0705.3712
Perhaps that should be the “smooth proof from The Book” (or the “proof from The Smooth Book”)!

Lots and lots of Heegaard splittings

The main problem that I’ve been thinking about since graduate school (so around a decade now) is the following: How does the topology of a three-dimensional manifold determine its isotopy classes of Heegaard splittings? Up until about a year ago, I would have predicted that most three-manifolds probably don’t have many distinct Heegaard splittings, maybe even just a single minimal genus Heegaard splitting and then all of its stabilizations. Sure, plenty of examples have been constructed of three-manifolds with multiple distinct (unstabilized) splittings, but these all seemed a bit contrived, like they should be the exceptions rather than the rule. I even wrote a blog post a couple years back stating what I called the generalized Scharlamenn-Tomova conjecture, which would imply that a “generic” three-manifold has only one unstabilized splitting. However, since writing this post, my view has changed. Partially, this was the result of discovering a class of examples that disprove this conjecture. (I’m hoping to post a preprint about this on the arXiv in the near future.) But it turns out there is an even simpler class of examples in which there appear to be lots and lots of distinct Heegaard splitting. I can’t quite prove that they’re distinct, so in this post I’m going to replace my generalized Scharlemann-Tomova conjecture with a conjecture in quite the opposite direction, which I will describe below.

The algorithm to recognise the 3-sphere

The purpose of this post is to convince you the 3-sphere recognition algorithm is simple.  Not the proof!  Just the statement of the algorithm itself.  I find in conversations with topologists, it’s fairly rare that people know the broad outline of the algorithm.  That’s a shame, because anything this simple should be understood by everyone.