Mike Freedman on SPC4

Mike Freedman gave a talk last week at Berkeley titled “(Still) thinking about the smooth 4-dimensional Poincare conjecture”, and I’d like to try and relate the main idea.

Back in 2009 Mike, Bob Gompf, Kevin Walker and I wrote a paper “Man and machine thinking about the smooth 4-dimensional Poincare conjecture”, in which we discussed various equivalent and stronger conjectures with more of a “three manifold flavour”, as well as proposing a way to show that certain potential counterexamples, the Cappell-Shaneson spheres, really were counterexamples, using Khovanov homology.

As it turned out, for the Cappell-Shaneson spheres that our computers could cope with Khovanov homology didn’t provide an obstruction. Shortly thereafter, Selman Akbulut posted a short paper showing that a certain integer family of CS spheres (including the two examples we considered) were all in fact standard, and not long after that Bob Gompf killed off some more. In fact, there’s been a whole flurry of work on SPC4 recently: Nash proposing some more counterexamples, Akbulut killing these off too, Akbulut proving a very general class of 4-spheres are standard, along with [1], [2] and [3]. There’s also a paper on Property 2R by Gompf, Scharlemann and Thompson.

Even though recent progress means that there are far far fewer potential counterexamples to SPC4 available, Mike has come up with another idea for detecting counterexamples! Essentially, from each proposed counterexample to the Andrews-Curtis conjecture (below), one can produce a homotopy 4-spheres. This part of the story is old news, but I’ll go over it carefully below. Mike points out that this construction also provides a family of embedded homology 3-spheres. Now any 3-manifold embeds in S^5, but not every 3-manifold embeds in S^4 (e.g. lens spaces, but see more below). The hope now is to show that one of these homology 3-spheres can not embed in the standard S^4, and thus that the homotopy 4-sphere it sits in must be exotic. Mike gave a condition on a 3-manifold Y that exactly determines whether it embeds in S^4, and talked about his ideas towards making this condition an effective test.

So, what is the Andrews-Curtis conjecture? A balanced presentation of a group is simply a presentation with equal numbers of generators and relations. The Andrews-Curtis moves on a balanced presentation are

  • “1-handle slides”: replace a pair of generators \{x, y\} by \{x, xy\}.
  • “2-handle slides”: replace a pair of relations \{r, s\} by \{g r g^{-1} s, s\} (here g is any word in the generators).
  • “handle cancellation”: add a generator along with a new relation killing it

The Andrews-Curtis conjecture says that any balanced presentation of the trivial group can be transformed by Andrews-Curtis moves to the trivial presentation. (There’s a stronger version that says handle cancellation isn’t even needed.) Despite the name, it’s widely expected to be false.

Before continuing, I should explain the funny names I’ve given the moves. Given any presentation of a group, we can build a two-complex whose fundamental group is the group: just take some 1-handles for the generators, and attach 2-handles killing the relations. The names I’ve given the moves correspond to the appropriate geometric operations on this 2-complex.

As an example, consider the presentation xyx=yxy, x^5 = y^4. It’s easy to check that this is the trivial group: y = x^{-1}y^{-1}xyx, so y^5 = x^{-1}y^{-1}x^5yx = x^{-1}y^4x = x^5. Thus y^5 = y^4, and y=1. Even though that was easy, no one has found a sequence of Andrews-Curtis move, even after having looked extremely hard! (e.g. this paper) This is just one case of the family \{ xyx=yxy, x^{n+1} = y^n \} of proposed counterexamples that appears in the paper “A potential smooth counterexample in dimension 4 to the Poincaré conjecture, the Schoenflies conjecture, and the Andrews-Curtis conjecture”, by Selman Akbulut and Rob Kirby (Topology 24 (1985), 375–390). So far the Andrews-Curtis conjecture is just a problem in group theory, but the title of that paper should clue you in that we’re about to construct some manifolds!

Given a balanced presentation P, we can build a 5-manifold Q(P)—just like we built the 2-complex above, start with a 5-ball, attach some 5-dimensional 1-handles for the generators, and then attach some 5-dimensional 2-handles for the relations. To attach the two handles, we actually need to pick links in the boundary of the 0- and 1-handles representing in \pi_1 the relation, but since these boundaries are 4-dimensional there are actually no choices to make. If we started with a presentation of the trivial group, then \partial Q(P) is a homotopy 4-sphere. Moreover, if the presentation is a counterexample to the Andrews-Curtis conjecture, then perhaps this homotopy 4-sphere is a good candidate counterexample to SPC4: you can’t make it standard just by handle slides corresponding to some Andrews-Curtis moves! Of course, you have more freedom to simplify presentations of 4-manifolds, so just having a counterexample to AC doesn’t ensure a counterexample to SPC4.

As it turns out, Bob Gompf subsequently showed that \partial Q(xyx=yxy, x^5 = y^4) is in fact the standard 4-sphere (by introducing a cancelling pair of 2- and 3-handles!). Nevertheless all the higher values of n are still open.

We can alternatively build \partial Q(P) in a different way. Start with a 4-ball, attached some 4-dimensional 1-handles for the generators, and now pick some links L realizing the relations, and attach 4-dimensional 2-handles along these. This produces a 4-manifold W(P,L), which genuinely depends on L, but whose double (the boundary of W \times I) is just our homotopy 4-sphere \partial Q(P). This way of building the homotopy 4-sphere gives us something extra; a 3-manifold Y(P, L) = \partial W(P, L) \times \{1/2\} sitting inside \partial Q(P). In fact, this 3-manifold is a homology sphere.

The challenge now is to pick some counterexample to the AC conjecture, along with a corresponding link, and then to prove that the resulting homology sphere can not be embedded in the standard 4-sphere.

Now Mike wasn’t claiming he knew how to do this; but he did explain a relatively easy theorem giving a precise characterization of when a 3-manifold embeds in the standard 4-sphere, which has a very “3-manifold flavour”. This was

Theorem. A 3-manifold Y embeds in S^4 if and only if it has an Heegaard diagram (\Sigma, \alpha, \beta) and an embedding \iota: \Sigma \to R^3, such that \iota(\alpha) is an unlink, and \iota(\beta) is an unlink.

Mike calls this a doubly unlinked embedded Heegaard diagram. The proof of this theorem is via ambient Morse theory, but I won’t try to give the details here. In fact, every homology 3-sphere has an embedded Heegaard diagram which is doubly null, meaning that the linking matrices for \iota(\alpha) and \iota(\beta) vanish. Of course, actually using this theorem to show that Y doesn’t embed would involve a lot of work. It seems that we’d need to consider all possible embeddings of \Sigma, as well as all possible systems of curves \alpha and \beta. I think Mike has some ideas about this (if you go “far out in the mapping class group”, it should be impossible that the images of the curves are unlinked), but I don’t think I could reproduce those ideas usefully here. Perhaps some of our readers can comment on how plausible this seems, and maybe if we’re lucky Mike will turn up too and say some more.

The meaning of knot homology

What I left out of my post on knot homology was, perhaps, the elevator pitch (if you’re in an elevator with a mathematician who already has some background). I’m giving a talk tomorrow which should include this stuff, and so one possibility is to just drop the slides for that on you, and let that speak for itself. Especially recommended are slides 12 and those past 18 (the rest is more standard quantum topology and categorification stuff).

But I’m not so sure that’s a wise plan. So let me try to say something more bloggy:
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A hunka hunka burnin’ knot homology

One of the conundra of mathematics in the age of the internet is when to start talking about your results. Do you wait until a convenient chance to talk at a conference? Wait until the paper is ready to be submitted to the arXiv (not to mention the question of when things are ready for the arXiv)? Until your paper is accepted? Or just until you’re confident you’ve disposed of any major errors in your proofs?

This line is particularly hard to walk when you think the result in question is very exciting. On one hand, obviously you are excited yourself, and want to tell people your exciting results (not to mention any worries you might have about being scooped); on the other, the embarrassment of making a mistake is roughly proportional to the attention that a result will grab.

At the moment, as you may have guessed, this is not just theoretical musing on my part. Rather, I’ve been working on-and-off for the last year, but most intensely over the last couple of months, on a paper which I think will be rather exciting (of course, I could be wrong). Continue reading

Man and machine thinking about SPC4

I’ve just uploaded a paper to the arXiv, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture, joint with Michael Freedman, Robert Gompf, and Kevin Walker.

The smooth 4-dimensional Poincaré conjecture (SPC4) is the “last man standing in geometric topology”: the last open problem immediately recognizable to a topologist from the 1950s. It says, of course:

A smooth four dimensional manifold \Sigma homeomorphic to the 4-sphere S^4 is actually diffeomorphic to it, \Sigma = S^4.

We try to have it both ways in this paper, hoping to both prove and disprove the conjecture! Unsuprisingly we’re not particularly successful in either direction, but we think there are some interesting things to say regardless. When I say we “hope to prove the conjecture”, really I mean that we suggest a conjecture equivalent to SPC4, but perhaps friendlier looking to 3-manifold topologists. When I say we “hope to disprove the conjecture”, really I mean that we explain an potential computable obstruction, which might suffice to establish a counterexample. We also get to draw some amazingly complicated links:

SPC4 link

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Mike on Topological Quantum Computing, at Georgia

I’m here at the 2009 Georgia Topology Conference and Mike Freedman is about to start talking about the current proposal for building a topological quantum computer. I’ll try liveblogging his talk; there’s a copy of the slides at http://stationq.ucsb.edu/docs/Georgia-20090518.pptx (PowerPoint only, sorry!) if you want to see the real thing. I think he recently gave a version of this talk in Berkeley recently, so some of you may have already heard it. I’ll fail miserably at explaining everything he talked about, but ask questions in the comments!

Mike says that the point of the talk will be to explain how it is that there’s a “topological” approach to building a computer, and try to give an idea of the mathematics, physics and engineering problems involved.

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How to get an algebra from a knot invariant

So, a couple of months ago, I gave a talk at the Max Planck Institute on knot homology, and as motivation I tried to explain why anyone studying the HOMFLY polynomial is inexorably led to the Hecke algebra. Nathan Geer, who was in the audience, asked me afterwards if there was anywhere this construction was written down, and lacking a good answer or the ambition to write a paper about it myself, I thought I would try to explain it in a blog post. It’s just applying an old TQFTologist’s trick, but old tricks often still have some new life in them.
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TQFTs via Planar Algebras (Part 2)

In my last post I explained a strategy for using n-dimensional algebraic objects to construct (n+1)-dimensional TQFTs, and I went through the n=1 case: Showing how a semi-simple symmetric Frobenius algebra gives rise to a 2-dimensional TQFT. But then I had to disappear and go give my talk. I didn’t make it to the punchline, which is how planar algebras can give rise to 3D TQFTs!

In this post I will start explaining the 3D part of the talk. I won’t be able to finish before I run out of steam; that will have to wait for another post. But I will promise to use lots of pretty pictures!

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TQFTs via Planar Algebras

So today I am giving a talk in the Subfactor seminar here at Berkeley, and I thought it might by nice to write my pre-talk notes here on the blog, rather then on pieces of paper destined for the recycling bin.

This talk is about how you can use Planar algebras planar techniques to construct 3D topological quantum field theories (TQFTs) and is supposed to be introductory. We’ve discussed planar algebras on this blog here and here.

So the first order of buisness: What is a TQFT?

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