Low Dimensional Topology

February 5, 2012

The 3-sphere has interesting triangulations

Filed under: 3-manifolds,Combinatorics,Triangulations — dmoskovich @ 1:06 am
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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.

I think of quantum topology as being the mathematical discipline which strives to make mathematical sense of topological results coming from quantum field theory. The reason that these results don’t make a-priori mathematical sense is that they involve path integrals. So, being a bit cynical for effect, physicists take a whole big sum of infinities, throw in some “physical intuition”, “hide” what the conveniently call “hidden strata”, and generate a number. Quantum topologists try to recover that same number using rigourous mathematical reasoning- and my quasi-theological belief is that the universe makes sense, so that the eventual mathematical explanation is going to detect “what is really going on”, or “what really occurs in our universe”.

The problem with QFT path integrals is that they integrate over huge infinite-dimensional spaces without measures which make sense. Two basic attempts to rigourize path integrals are to limit oneself to paths satisfying some nice properties (integrate over the moduli space of pseudoholomorphic curves, for example) and get Gromov-Witten invariants or whatever. The underlying ideology would be that reality is an almost complex manifold. The second attempt is to discretize differential geometry, and replace path integrals by convergent series over simplices. From my quantum topological perspective, it is into this second approach that the Benedetti-Ziegler paper fits. The underlying ideology would be that nothing is trully infinitesmally small, and that the universe is made of zillions of tiny cells (why must they be simplicies?).

Try to translate a path integral into a series over simplices, and one encounters combinatorial problems such as the one which the Benedetti-Ziegler paper considers. The big open problem which is the background of the paper is to determine the growth rate of triangulations of the 3-sphere by N 3-simplices, as N grows to infinity. If it’s exponential, you win. Any more than that, and the discrete approach fails (at least in this incarnation), and the series you get (the “partition function for gravity”) is just as bad as the original path integral. This paper identifies an important obstacle that must be overcome in order to prove exponential growth rate, namely that not all triangulated 3-spheres are “locally constructible”.

Determining this growth rate actually looks like a really interesting problem, with close relation to “low dimensional topologist topics” such as bridge index of knots and what have you (counterexamples to the “all triangulations of the 3-sphere are locally constructible” conjecture are identified by making use of high bridge-number knots). I wonder whether the conjecture will turn out to be true- I hope not, because I find it hard to believe that our world is actually made up of zillions of eentsy weentsy tetrahedra, and it would fit my prejudices perfectly if all approaches except for the “real world” approach were to fail, and if mathematical rigour and reality were to coincide! So I’d be tickled pink if the growth rate of triangulations of a 3-sphere turned out to be more than exponential in N, and only the growth rate of more sophisticated “realistic” cell decompositions of a 3-sphere (whatever those were to be- I like this idea) were to turn out to be exponential in N.

Of course, the problem is interesting in its own right, and not only for S3. Choose your favourite 3-manifold. What is the growth rate for the number of its triangulations?


  1. Best description I’ve read of the differences between physicists and mathematicians working on the same area.

    Comment by isomorphismes — June 25, 2012 @ 4:08 pm | Reply

  2. Isn’t theresomething about Planck length/time — how could reality go smallerthan that?

    Comment by isomorphismes — June 25, 2012 @ 4:25 pm | Reply

  3. […] blog post mentioning just one of the interesting features at the Low Dimension Topology Blog: The 3-sphere has interesting triangulations. For us, this doesn’t matter much since the action is supposed to be invariant under […]

    Pingback by Notes on Lattice Topological Field Theory in Three Dimensions, Part VI | Christopher Duston's Home — November 1, 2013 @ 2:45 pm | Reply

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