In 1959 Stephen Smale gave a proof that the group of diffeomorphisms of the 2-sphere has the homotopy-type of the subgroup of linear diffeomorphisms, i.e. the Lie Group O_3. His proof went in two steps:

Step 1: Diff(S^2) has the homotopy-type of O_3 x Diff(D^2,S^1). The latter object here is the group of diffeomorphisms of the 2-disc which are the identity on the boundary.

Step 2: Show Diff(D^2,S^1) is contractible.

Step 1 is a general argument, that Diff(S^n) has the homotopy-type of O_{n+1} times Diff(D^n, S^{n-1}), the proof of which is very much in the spirit of the isotopy extension theorem, and the classification of tubular neighbourhood theorem, but `with parameters’.

Step 2 is a rather specific argument, which, at its core involves the meatiest theorem on our understanding of first-order ODEs in the plane: the Poincare-Bendixson theorem. His clever application of Poincare-Bendixson theorem allows him to reduce the proof to the theorem that Diff(D^1,S^0) is contractible, which has many simple and elegant proofs.

Smale’s proof has a bit of the spirit of an inductive proof. It leads one to the question, what about the homotopy-type of Diff(S^3)? Perhaps because we can’t imagine anything different, it would make sense for Diff(S^3) to have the homotopy-type of O_4. At the level of path-components this was proven by Cerf in 1968, in one of the first applications of the subject now called Cerf Theory. The full proof by Allen Hatcher was given in 1983. Around this time the problem of showing Diff(S^3) has the homotopy-type of O_4 began to be called “The Smale Conjecture”.

I think it’s fair to say that most major theorems in 3-manifold theory at present have several different proofs (classification Seifert-fibred manifolds with 3 singular fibres over S^2 might be one of the few cases where there is only one proof), or at least, several variations on one proof. But the Smale Conjecture has found no alternate proofs. People have hoped that perhaps a `geometrization with parameters’ theorem could be used on the space of all metrics on S^3, but the metric collapses along families of spheres — this is much like the difficulties Hatcher encounters in his original proof, but Hatcher was just dealing with families of manifolds, while a geometrization proof would be in a category of Riemann manifolds.

Hatcher suggested a possible alternative framework to prove the Smale Conjecture. The idea is to show that the component of the trivial knot, in the space of smooth embeddings Emb(S^1, S^3) has the homotopy-type of the subspace of great circles. Hatcher gave a few other equivalent formulations of the Smale Conjecture — the one he used makes the Smale Conjecture looks like `the Alexander Theorem with parameters’ i.e. that the space of smooth embeddings Emb(S^2, R^3) has the homotopy-type of the subspace of (parametrized) round spheres. Hatcher’s proof is essentially a souped-up version of Alexanders proof; roughly speaking it involves a rather careful cutting of families of spheres into simpler families.

The embedding space Emb(S^1, S^3) has been studied in many ways over the years. Jun O’Hara had the idea of putting a “potential function” on this space, much in the spirit of Morse theory. He used a function derived (in spirit) from electrostatics. Imagine the knot as carrying a uniform electric charge along its length and write down the integral for the potential energy of the system. Technically O’Hara allowed for less physically inspired “energies” but this is the basic idea. In the 80’s and 90’s it was proven that for O’Hara’s potential function flow in the negative gradient direction makes sense, and that there are local minimizers in the space. Recently it was proven that a C^1 embedding which is a critical point of this energy functional, is necessarily a C^\infty smooth embedding. So there has been plenty of progress.

Of course, what one would really want to prove is that the only critical points of this functional on the component of the trivial knot are the great circles themselves. That would allow for a Morse-theoretic argument that the unknot component of Emb(S^1, S^3) has the homotopy-type of the great-circle subspace, and give a new, rather appealing proof of the Smale conjecture.

References:

O’Hara. Energy of a knot, Topology, 30 (2): 241–247

Freedman, He, Wang. Möbius energy of knots and unknots, Annals of Mathematics, Second Series, 139 (1): 1–50

He. The Euler-Lagrange equation and heat flow for the Möbius energy. Communications in Pure and Applied Mathematics.

Blatt, Reiter, Schikorrra. Harmonic Analysis Meets Critical Knots. TAMS Vol 368, no 9, sept 2016, pg 6391–6438

Does Grayson’s proof that circles flowing by curvature in the plane converge to round points give another proof of Smale’s theorem? http://www.jstor.org/stable/1971486 I guess the tricky thing with showing that the round unknot is the only critical point for O’Hara’s energy among unknots is to input the fact that the loop is unknotted.

Comment by ianagol — October 2, 2016 @ 11:49 pm |

Yes, I believe it does. If you do the kind of fibre-bundle argument Hatcher does at the end of the Smale Conjecture paper you convert the problem of studying Emb(S^1,D^2) into studying the homotopy-type of the space of proper embeddings of an interval into D^2 (with fixed endpoints). You then use the fibre bundle Diff(D^2) –> Emb(I, D^2 rel endpoints), and show the fibre has the homotopy-type of a product of two copies of Diff(D^2).

It’s not clear how to use the fact that you’re on the unknot component to your advantage. Finding critical points of these energy functionals isn’t so easy. I’m probably going to give a related project to an undergrad — finding critical points of this functional, not on the unknot space, but on the space of a “big” satellite knot. There’s at least enough knowledge of the homotopy-type to get a sense for how to `cast the net’.

Comment by Ryan Budney — October 3, 2016 @ 12:04 am |

I was wondering if Dynnikov’s paper giving an algorithm for the unknot is related. Specifically Hatcher shows that the Smale conjecture is equivalent to proving that the space Emb(S^2,R^3) is contractible. Dynnikov in his paper defines a complexity for embedded spheres and uses `moves’ to (strictly) monotonically simplify spheres to spheres of 0 complexity. His 0-complexity spheres

intersect planes containing the z-axis along simple closed curves so it seems spheres of 0-complexity can flow to the standard sphere by, say, using Grayson’s theorem to flow all these simple closed curves to standard circles. Dynnikov’s spheres and moves are combinatorially defined but I don’t see why they couldn’t be done smoothly to smooth spheres. One difficulty maybe is that he does these moves to spheres in a `general position’ with respect to the `foliation’ of R^3 by planes containing the z axis.

Comment by Panos Papazoglou — April 21, 2017 @ 5:30 am |

In principle, yes, perhaps Dynnikov’s ideas could be used to inspire an explicit potential function on the space of knots. But there are several rather large technical obstructions — Dynnikov’s constructions are diagrammatic so you would need to “smooth” it to the entire space and (this might be a reflection of my hazy recollection of what Dynnikov does) secondly, Dynnikov has a rather large decision procedure so it’s unclear how to make anything geometric out of something not so simple.

But that sounds like a fun project.

Comment by Ryan Budney — April 21, 2017 @ 2:15 pm |

Just to clarify: I meant that Dynnikov’s paper seems to define a `flow’ on the space of embedded spheres (not embedded circles) in R^3 which takes any sphere to the standard sphere. The difference between knots and spheres is that his moves seem to simplify spheres ie they decrease strictly the complexity of the sphere, while they don’t do this for knots. Such a flow on the space of spheres would also imply the Smale conjecture (if I understand Hatcher correctly). Well, I suppose somebody has thought about it and there is some problem, maybe his procedure is not continuous on the set of spheres.

I wonder whether you mean that a `flow’ on the space of embedded spheres taking any sphere to the round sphere induces a flow on the space of unknotted circles taking any unknotted

circle to the round circle. I don’t see how would that work though.

Comment by Panos Papazoglou — April 25, 2017 @ 6:57 am |