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.
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!