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 homeomorphic to the 4-sphere is actually diffeomorphic to it, .

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:

Our new formulation of SPC4 might be thought of as a generalization of Gabai’s Property R [euclid.jdg/1214441488, Corollary 8.3]:

If surgery on gives then is the unknot.

We point out that SPC4 is equivalent to the following (Conjecture 3.3 in the paper)

Let be a link in with a dotted p-component unlink and a framed link of p+q components. Suppose that normally generates , and that surgery on (with dotted components 0-framed) is diffeomorphic to . Then there is a sequence of moves

transforming to the empty diagram.

(you’ll have to read the paper if you want to know how “dotted” and “framed” come into the picture; essentially there are restrictions on move (3) depending on this data, which you’ll already know about if you understand link presentations for 4-manifolds) This is a generalization in the sense that Property R says that for p = 0 and q = 1, a single move suffices. The proof that this is equivalent to SPC4 is essentially trivial, and makes use of the Kirby calculus theorem. We discuss some variations of this conjecture, both weaker and stronger, that might be more approachable.

In the later half of the paper, we discuss some potential counterexamples to SPC4, the Cappell-Shaneson spheres, and explain how Rasmussen’s s-invariant (related to Khovanov homology) might give a computable obstruction preventing these homotopy spheres from being diffeomorphic to the standard sphere.

**UPDATE:** Akbulut has proved that the Cappell-Shaneson spheres are all standard; see the comments.

The basic idea is quite simple. The Cappell-Shaneson spheres have a handle presentation, due to Akbulut and Kirby (in the m=0) case, and Bob Gompf (otherwise), that **have no 3-handles**.

Pull off the top 4-handle, and think of this handle presentation as a presentation of boundary , which is certainly standard. Inside this , we can see a certain link, the meridian links of the loops along which we glue the 2-handles. This link is obviously slice in the homotopy sphere (because it’s just the meridian links, we can see the slice disks immediately!), but once we do the long sequence of Kirby calculus moves converting this presentation of into the standard (empty) presentation, this link has become tremendously complicated (see the diagram near the top of the post!) and it’s not at all obvious that it’s slice in $B^4$. Moreover, if it **isn’t**, then our homotopy ball can’t have been diffeomorphic to the standard ball!

Our problem now is to find computable obstructions to sliceness. Until recently, not many were available, and those that were (for example the invariant from knot Floer homology) were suspected to be unable to detect slice genus in homotopy balls. Rasmussen’s s-invariant, defined using a deformation of Khovanov homology, fits our requirements however! Unfortunately, computing the s-invariant, although in principal combinatorial, suffers from terrible scaling as the complexity of the link (especially its girth) increases. It seems that of the many examples coming from different Cappell-Shaneson spheres, very few cases are computable with modern hardware and the current best algorithms for computing s, and even then we need to cheat a little, and look only at various knots obtained by band connect summing the link components together. We’ve done two cases (the and Cappell-Shaneson spheres; the is known to be standard), and got both times — after more than a week of computing time on a huge machine!

If anyone out there thinks up a new way to compute the s-invariant, and can prove (perhaps via a gauge theoretic interpretation of Khovanov homology?) that the slice genus bound from the s-invariant can’t tell the difference between homotopy spheres, please let us know!

“If anyone out there thinks up a new way to compute the s-invariant, and can prove (perhaps via a gauge theoretic interpretation of Khovanov homology?)”

Well, perhaps you can find your answer in this very recent paper:

http://arxiv.org/abs/0904.1276

They argue

In case you can find any relation between the quantization of SU(2) Wess Zumino Witten sigma models and s invariants, that could be used to (dis)prove the conjecture. In the paper, they argue, as you, that what is missing is a suitable invariant to tackle the problem. Anyway, given that WZW has interesting non trivial topological S3 obstructions (http://en.wikipedia.org/wiki/Wess-Zumino-Witten_model) , the authors use this fact to find for a quantization for such models. I guess, in this obstruction, you could find a physical inspiration to try to find the answer you want.

BTW, they argue for the failure of the conjecture, but, as I said, they cannot prove for the lack of a suitable invariant.

Selman Akbulut posted today a proof that Cappell-Shaneson spheres are diffeomorphic to S^4 for any value of m:

http://arxiv.org/abs/0907.0136

I guess you can give your computers a rest!!! :)

In the article “Man and Machine…” it is statated that “…there are several constructions which give potential counterexamples [to SPC4].”

Can you tell me some of these potential counterexamples?

Bob Gompf has checked Akbulut’s argument, and it looks good — what a pity! (It seems to make the second half of our paper a bit pointless, and similarly a fair bit of work I did last summer shepherding along computer calculations…) It’s very much in the spirit of Bob’s older argument that the m=0 Cappell-Shaneson sphere is standard.

I’m curious if Akbulut’s paper coming out yesterday was just coincidentally immediately after ours. I’m guessing not, and that our paper prompted him to put this up. The second question then is if he’s known this for a while or if it’s a recent observation. Oh well :-)

But don’t you know more potential counter examples to SPC4?

There are a few things still to try:

* the Gluck construction on 2-knots (cut out a 2-knot, and paste it

back in with the other framing) can produce homotopy spheres. Cameron

Gordon thought about this a while ago, although his effort was mostly

directed to the opposite problem; ensuring that the Gluck construction

gave something not a homotopy sphere.

* the Cappell-Shaneson spheres aren’t actually exhausted by integer

family that Akbulut just killed. In the definition of a CS sphere,

there’s a certain monodromy matrix in SL(3,Z), up to some

equivalences. For each trace, there are only finitely many

equivalence classes (and exactly one for traces -4 through +9, or

thereabouts). The integer family that Gompf found handle presentations

for realise exactly one of each trace, and these are the ones Akbulut

killed.

* take any balanced presentation of the trivial group which looks like

a counterexample to the Andrews-Curtis conjecture, and use that to

write a handle presentation of a 5-manifold. The boundary is then a

good potential counterexample.

The problem with all of these is that for our attack via the s-

invariant to work, we need a handle presentation of the 4-mfld with no

3-handles. For most of the sorts of examples above, we don’t even have

a handle presentation, let alone sone reason to expect it might not

need 3-handles.

33 years later….

by the samae person (AK)!

correction to my previous msg:

33 years later…

by the same person (Akbulut)…

Hi Scott,

There is a follow up to that Akbulut article in June, this time by Gompf.

*******

http://arxiv.org/abs/0908.1914

More Cappell-Shaneson spheres are standard

Authors: Robert E. Gompf (The University of Texas at Austin)

(Submitted on 13 Aug 2009)

Abstract: Akbulut has recently shown that an infinite family of Cappell-Shaneson homotopy 4-spheres is diffeomorphic to the standard 4-sphere. In the present paper, a strictly larger family is shown to be standard by a simpler method. This new approach uses no Kirby calculus except through the relatively simple 1979 paper of Akbulut and Kirby showing that the simplest example with untwisted framing is standard. Instead, hidden symmetries of the original Cappell-Shaneson construction are exploited. In the course of the proof, we give an example showing that Gluck twists can sometimes be undone using symmetries of fishtail neighborhoods.

******

Can you explain what Cappell-Shaneson spheres are still not know to be standard? And if just a part of class of spheres are standard, why does Gompf is already thinking that SPC4 is true?

Cheers,

Daniel

Dear Scott,

I new paper was uploaded today, where in Theorem 1.15 the generalized poicaré is restated yet in another form, but this time using homotopy equivalence and an invariant I never heard about, the “complexity” Definition 2.3. Here is the paper:

http://arxiv.org/abs/0909.0168

Cheers,

Daniel

Dear Scott,

Take a look at this paper:

http://arxiv.org/abs/1001.1538

Topologically slice knots with nontrivial Alexander polynomial

Matthew Hedden, Charles Livingston, Daniel Ruberman

(Submitted on 10 Jan 2010)

Let C_T be the subgroup of the smooth knot concordance group generated by topologically slice knots and let C_D be the subgroup generated by knots with trivial Alexander polynomial. We prove the quotient C_T/C_D is infinitely generated, and uncover similar structure in the 3-dimensional rational spin bordism group. ****Our methods also lead to the construction of links that are topologically, but not smoothly, concordant to boundary links.****

Does it mean that examples exotic sphere can now be found?

Cheers,

Daniel.