Question About Exotic Smooth Structures

July 30, 2009

Question About Exotic Smooth Structures

Posted by John Baez

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Here’s a question that came up as Aaron Lauda and I have been writing A prehistory of $n$ -categorical physics. It’s about whether you can prove the existence of an exotic $\mathbb{R}^4$ with the help of Khovanov homology.

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I don’t understand this stuff at all, but here is the story I’ve heard. I’d like to know if this story contains errors. Even if it doesn’t, I’m sure it can be improved! I don’t really know who did what…

Apparently work of Freedman combined with Kronheimer and Mrowka’s work on gauge theory (Donaldson theory) proved there are knots that are ‘topologically but not smoothly slice’:

Michael H. Freedman, A surgery sequence in dimension four; the relations with knot concordance, Invent. Math. 68 (1982), 195–226.
Peter B. Kronheimer and T. S. Mrowka, Gauge theory for embedded surfaces I, Topology 32 (1993), 773–826

Think of $\mathbb{R}^3$ as a subspace of $\mathbb{R}^4$ in the obvious way, and give both these guys their ordinary smooth structure. A knot — a plain old smooth knot in $\mathbb{R}^3$ — is said to be smoothly slice if it bounds a smoothly embedded disc in $\mathbb{R}^4$ . It’s said to be topologically slice if it bounds a flat topologically embedded disc in $\mathbb{R}^4$ .

‘Flat’? Yes, this fine print is important: a topologically embedded disk in $\mathbb{R}^4$ is said to be flat if its embedding extends to an embedding of a thickened version of that disk: that is, the product of that disc and another disc. If we drop this fine print, all heck breaks loose: every knot bounds a topologically embedded disk in $\mathbb{R}^4$ .

Apparently Gompf or somebody (?) used the existence of topologically but not smoothly slice knots to give a proof of the existence of exotic smooth structures on $\mathbb{R}^4$ .

More recently Jacob Rasmussen used Khovanov homology to construct an invariant that can prove certain knots aren’t smoothly slice:

Jacob Rasmussen, Khovanov homology and the slice genus, to appear in Invent. Math., available as arXiv:math/0402131.

And in fact, we can apply this idea to a specific knot that’s known to be topologically slice, and prove it’s not smoothly slice, and thereby conclude the existence of an exotic $\mathbb{R}^4$ .

Is this right?

If so, this would be cool because it would set up the beginnings of a connection between categorified quantum groups (which underlie Khovanov homology) and exotic smooth structures in 4 dimensions. This was an old dream of Crane and Frenkel.

Posted at July 30, 2009 1:11 PM UTC

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Re: Question About Exotic Smooth Structures

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I just got some more information on this, from Aaron himself! He pointed out page 13 of Rasmussen’s paper Knot polynomials and knot homologies, which says, in part:

While we were at McMaster, Bob Gompf kindly pointed out another such application. Namely, [the invariant] $s$ can also be used to give a gauge-theory free proof of the existence of an exotic $\mathbb{R}^4$ . Indeed, Gompf has shown that to construct such a manifold, it suffices to exhibit a knot which is smoothly but not topologically slice. (See [6] p. 622 for a proof.) By a theorem of Freedman, any knot with Alexander polynomial 1 is topologically slice [5], so we need only find a knot $K$ with $\Delta_K(t) = 1$ and $s(K) \ne 0$ . It is not difficult to produce such a knot — for example, the $(-3,5,7)$ pretzel knot will do. The Khovanov homology of this knot can be calculated, either by KhoHo or using the skein exact sequence, and from there it can easily be determined that $s = -1$ .

Posted by: John Baez on July 30, 2009 2:43 PM | Permalink | Reply to this

Re: Question About Exotic Smooth Structures

Rasmussen’s paper Knot polynomials and knot homologies

A bad link (or exotic knot?)

Posted by: Toby Bartels on July 30, 2009 8:09 PM | Permalink | Reply to this

Re: Question About Exotic Smooth Structures

Fixed! Thanks.

Posted by: John Baez on July 30, 2009 9:55 PM | Permalink | Reply to this

Re: Question About Exotic Smooth Structures

Hi John, I am not sure what you are looking for. According to one of the books you told me to study last year, Wild World of Four Manifolds, Scorpan, this seems indeed to be the case for the Akbulut Cork, not the fist proof of the book, directly starting from the failure of the h-cobordism in 4 dimensions. But one mentioned in the end of that book, and sketched, using Khovanov homology by doing the procedure you mentioned. I didn’t read that, to tell you the truth, nor I have the book nearby right now to confirm, so you should take a look. I guess you can find it at ch. 13, or close to that.

Fortunantely, Scott Morrison is involved with related questions right now. Maybe you could invite him to answer such questions.

Concerning the quantization, I will tell you people once again about this article . This is the most recent, and I guess relevant article to this discussion. This group of study started with Brans, the guy from Brans-Dicke gravity, in the early 90’s, to relate exotic structures to quantization.

Posted by: Daniel de França MTd2 on July 30, 2009 7:06 PM | Permalink | Reply to this

Re: Question About Exotic Smooth Structures

The Preface of the Gompf book is online which provides a few more details. I did a Google search on Robert Gompf and Andras Stipsicz and the third returned result linked to 4-manifolds and Kirby calculus - Google Books Result

Posted by: Stephen Harris on July 30, 2009 11:42 PM | Permalink | Reply to this

Re: Question About Exotic Smooth Structures

Indeed, you can hope to use the s-invariant to show the existence of exotic 4-spheres, and thus disprove the smooth 4-d Poincaré conjecture This was one of the goals of my paper with Freedman, Gompf and Walker, although Akbulut quickly pointed out the particular examples we were working on couldn’t be exotic.

To summarise the current state of play, suppose you have a homotopy 4-sphere, and a handle presentation which has no 3-handles. Pull off the 4-handle, to produce a homotopy 4-ball, and consider the handle presentation we have of the boundary. This boundary is just the
standard 3-sphere, so there’s a sequence of Kirby moves reducing the handle presentation of the boundary to the trivial presentation. Now, look at the link consisting of the meridian curves around the original 2-handles. These are certainly slice in the homotopy ball: they bound discs transverse to the cores of the 2-handles. On the other hand, let’s carry along this link while we do our Kirby calculus simplification of the boundary. We end up with a complicated link in S^3, and there’s no particular reason why it needs to be slice in the standard 4-ball. Our hope was to calculate the s-invariant of the resulting link (the s-invariant is usually just for knots, but you can either take a band connect sum, or use a slightly fancier invariant) for a few examples, and, hoping to be outrageously lucky, get something nonzero.

Sadly, the only potential counterexamples to the smooth 4-d Poincaré conjecture that have known handle presentations with no 3-handles appear to be the integer family of Cappell-Shaneson spheres. We were trying the m=1 case of this family. And just recently, in response to our paper, Akbulut showed that the whole family was standard. We’re thus left in the unfortunate situation of having an unlikely method, without even any available cases to try!

Posted by: Scott Morrison on July 31, 2009 10:29 PM | Permalink | Reply to this

Re: Question About Exotic Smooth Structures

I’ve been trying to post a reply, but when I click “Preview”, it successfully typesets my text but shows a blank text box below, and when I try to “Post” I get an error message, saying that comment text is required. :-(

Posted by: Scott Morrison on July 31, 2009 10:32 PM | Permalink | Reply to this

Re: Question About Exotic Smooth Structures

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I’m sorry! That’s an odd failure mode — I’ve never heard of that happening. I assume the problem is gone now, since you’re successfully posting?

I’d been gonna email you about this question, but I decided to blog about it.

We’re thus left in the unfortunate situation of having an unlikely method, without even any available cases to try!

Ah well. It seems like lots of serious low-dimensional topologists have cracked their heads on these hard problems… and there are fewer left these days.

While you’re here, what do you think about this passage in A Prehistory of n-Categorical Physics:

Khovanov found a way to assign a graded chain complex to any link in such a way that its graded Euler characteristic is the Jones polynomial of that link. This new invariant is a strictly stronger link invariant [BN], but more importantly it is ‘functorial’. Khovanov homology associates to each tangle diagram a graded chain complex, and to each 2-tangle going between tangle diagrams one gets a chain map between the respective complexes [Jac1,Kh2]. The original construction of Khovanov only satisfies 2-tangle relations up to scalar multiples, so we say that it gives a ‘projective’ invariant of 2-tangles. Later Clark, Morrison, and Walker fixed this problem, so that Khovanov homology becomes a functor [CMW].

With the appearance of 2-tangle invariants one might expect that there is a braided monoidal 2-category lurking behind Khovanov homology. However, this is not quite the case, at least for the original formulation, since Khovanov homology does not respect the monoidal structure of tangles given by the disjoint union. However, Bar-Natan gave an interpretation of Khovanov homology where one works with complexes of formal direct sums of pictures consisting of nonintersecting arcs and circles; morphisms between these complexes are given by cobordisms between these pictures. He called this setup the ‘picture world’ [BN,BN2]. In this framework Khovanov’s invariant of 2-tangles can be described in a way where the monoidal structure is preserved. In fact, the work of Clark, Morrison, and Walker is done in this ‘picture world’ language so that it can be viewed as giving a braided monoidal 2-category. Only in turning this picture world into concrete algebra does one lose the nice monoidal properties of the picture world. However, experts in the field are quite good at computing in the picture world [BM].

It is worth mentioning that the authors in this field have chosen to study higher categories with duals in a manner that does not distinguish between ‘source’ and ‘target’. This makes sense, because duality allows one to convert input to outputs and vice versa. In 1999, Jones introduced ‘planar algebras’ [Jones2], which can thought of as a formalism for handling certain 2-categories with duals. In his work on Khovanov homology, Bar-Natan introduced a structure called a ‘canopolis’ [BN2], which is a kind of categorified planar algebra. The relation between these ideas and other approaches to $n$ -category theory deserves to be clarified and generalized to higher dimensions.

I don’t really understand this stuff very well; I’ve just been editing some stuff Aaron wrote, trying to strip it of distracting technical details to help get across the big picture to nonexperts. One thing that makes me feel funny is the stuff about whether Khovanov homology is a functor or not, and especially whether it’s monoidal or not.

The first paragraph seems to be saying that first Khovanov homology wasn’t functorial with respect to composition of 2-morphisms, and then you came along and fixed some signs and it was. That’s clear enough.

But the second paragraph tells a more mysterious story. Why do we lose the ‘nice monoidal properties’ when we leave the picture world and go to the realm of ‘concrete algebra’? Wouldn’t that just mean we’re using the wrong ‘concrete algebra’?

I’d also be glad to hear about anything else in the above passage that makes you feel funny. Well, at least I think I’d be glad — maybe I won’t be when you actually tell me.

Posted by: John Baez on August 1, 2009 9:28 AM | Permalink | Reply to this

Re: Question About Exotic Smooth Structures

John, do you mind summarizing the current status of this post? In the original post, you were unsure whether the existence of exotic smooth structures on $\mathbb{R}^4$ could be proved from Khovanov homology. Then in the comment, it seems that Aaron found a statement in Rasmussen’s paper showing that indeed this is the case. So the mystery is over? There is a gauge-theory-free proof of an exotic $\mathbb{R}^4$? Is this the only gauge-theory-free construction known? (Not that it is really gauge-theory-free, of course: all these knot polynomials ultimately derive geometrically from gauge theory, don’t they?)

Posted by: Bruce Bartlett on August 1, 2009 12:50 PM | Permalink | Reply to this

Re: Question About Exotic Smooth Structures

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Hi, Bruce! Glad you’re back! It must have been a long time — it looks like you forget how to do TeX on this blog.

So the mystery is over?

Yep — not the underlying mystery of the universe, nor even the mystery of how to get interesting 4d TQFTs from Hopf categories, but at least the mystery I was asking about.

There is a gauge-theory-free proof of an exotic $\mathbb{R}^4$ ?

Yes, I think so! As explained here, you just need to:

take a $(-3,5,7)$ pretzel knot, show it’s topologically slice by computing its Alexander polynomial (a purely combinatorial business) and applying a result of Freedman,
then show it’s not smoothly slice by computing its $s$ invariant (another purely combinatorial business, involving Khovanov homology) and applying a result of Rasmussen,
and then apply the result of Gompf, whose proof I believe is gauge-theory-free, saying the existence of a knot that’s topologically but not smoothly slice implies the existence of an exotic $\mathbb{R}^4$ !

Not that I personally could do any of this stuff, mind you…

Is this the only gauge-theory-free construction known?

Dunno. It’s the only one I’ve heard of.

(Not that it is really gauge-theory-free, of course: all these knot polynomials ultimately derive geometrically from gauge theory, don’t they?)

Well, the reason I’m talking about this stuff in the last section of that prehistory is that this is indirect evidence that purely combinatorial invariants of 4d piecewise-linear topology, computed using categorified quantum groups, are smart enough to know about Donaldson theory… which was Crane and Frenkel’s dream.

Posted by: John Baez on August 1, 2009 3:00 PM | Permalink | Reply to this

Re: Question About Exotic Smooth Structures

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Thanks a stack. A nifty fact to put in the bag!

Posted by: Bruce Bartlett on August 6, 2009 10:52 AM | Permalink | Reply to this

Re: Question About Exotic Smooth Structures

Happened in here from google. Just a quick note. I posted the details of step 3 over at mathoverflow.

Posted by: Kelly Davis on March 15, 2011 9:08 AM | Permalink | Reply to this

Exotic Smoothness and Quantum Gravity; Re: Question About Exotic Smooth Structures

This is a really fun paper!

Exotic Smoothness and Quantum Gravity
Authors: Torsten Asselmeyer-Maluga
Comments: 16 pages, 1 Figure, 1 Table subm. Class. Quant. Grav.
Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Geometric Topology (math.GT)

Since the first work on exotic smoothness in physics, it was folklore to assume a direct influence of exotic smoothness to quantum gravity. Thus, the negative result of Duston (arXiv:0911.4068) was a surprise. A closer look into the semi-classical approach uncovered the implicit assumption of a close connection between geometry and smoothness structure. But both structures, geometry and smoothness, are independent of each other. In this paper we calculate the “smoothness structure” part of the path integral in quantum gravity assuming that the “sum over geometries” is already given. For that purpose we use the knot surgery of Fintushel and Stern applied to the class E(n) of elliptic surfaces. We mainly focus our attention to the K3 surfaces E(2). Then we assume that every exotic smoothness structure of the K3 surface can be generated by knot or link surgery a la Fintushel and Stern. The results are applied to the calculation of expectation values. Here we discuss the two observables, volume and Wilson loop, for the construction of an exotic 4-manifold using the knot $5_{2}$ and the Whitehead link $Wh$. By using Mostow rigidity, we obtain a topological contribution to the expectation value of the volume. Furthermore we obtain a justification of area quantization.

Posted by: Jonathan Vos Post on March 30, 2010 6:38 PM | Permalink | Reply to this

Fictional treatment; Re: Exotic Smoothness and Quantum Gravity; Re: Question About Exotic Smooth Structures

All of this, explicit mention of John Baez, and pages of citations are contained in my recent Science Fiction short story, which I can email to people on request. It opens:

Exotic Smooth Manifolds and a Better World
By
Jonathan Vos Post
Draft 2.0 of 5 p.m. 2 April 2010, 24 pages, approx, 5,700 words

1.0 Introduction to Khovanov Homology and Paradise Lost

There were only two of us in the world who knew enough about exotic smooth structures for me to bring my wife back from the dead.

Let me put that another way. There were only two Mathematical Physicists in the world who knew enough about exotic smoothness and quantum gravity for me to get access to a better world, one in which my wife had never died.

I have to tell you a little about my wife, how I lost her, and a little about exotic structures, because there’s no other way to explain why I’m leaving you this final letter. You can skip the hyper-technical stuff. I can’t help it. I’m a Mathematician. Just take them as the kind of jargon that sailors would be barking in a novel about an 1802 Napoleonic naval battle. Groove on the metaphors, and skip over whatever is too obscure, so as to keep with the main story, which is all about love, sex, trans-universal travel, and death. Like all the best stories.

Posted by: Jonathan Vos Post on April 8, 2010 4:05 PM | Permalink | Reply to this

Re: Fictional treatment; Re: Exotic Smoothness and Quantum Gravity; Re: Question About Exotic Smooth Structures

I’d like an email of that short story!

Posted by: Bruce Bartlett on April 8, 2010 9:10 PM | Permalink | Reply to this

Re: Fictional treatment; Re: Exotic Smoothness and Quantum Gravity; Re: Question About Exotic Smooth Structures

Me too
I’d like to share it with another Stasheff, Christopher, for the obvious reason.

Posted by: jim stasheff on April 9, 2010 12:53 PM | Permalink | Reply to this

Re: Fictional treatment; Re: Exotic Smoothness and Quantum Gravity; Re: Question About Exotic Smooth Structures

Done. Feel free to comment to me by email or openly in this thread.

Posted by: Jonathan Vos Post on April 10, 2010 11:08 AM | Permalink | Reply to this

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July 30, 2009