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Say that $S^n$ "admits eversion" if the inclusion $S^n \rightarrow \mathbb{R}^{n+1}$ is regularly homotopic to the antipodal map (where a "regular" homotopy is a continuous path through immersions).

Smale proved that $S^2$ admits eversion by defining an appropriate algebraic invariant corresponding uniquely to regular homotopy classes, and noted that the group this invariant lives in is trivial. Many people didn't believe it until someone made a movie illustrating an explicit eversion.

It can be shown that $S^n$ admits eversion if and only if the tangent bundle of $S^{n+1}$ is trivial. That is, the only spheres which admit eversion are $S^0$, $S^2$, and $S^6$.

My question is: does anyone know of an explicit eversion of $S^6$ in $\mathbb{R}^7$?

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"Someone" = Thurston via the Geometry Center. – Ryan Reich Dec 1 '12 at 22:22
What do you mean by "explicit"? – Ian Agol Dec 1 '12 at 22:36
The Thurston technique is fairly general. You should be able to write out something fairly explicit for $S^6$ using his technique. – Ryan Budney Dec 1 '12 at 22:36

2 Answers 2

(That's my first post on mathoverflow. Henceforth and unfortunately I am not allowed to post comments (this needs reputation 50), so part of the present post in the answer box would better fit in the comments, sorry for this.)

Citing Sullivan's article "The Optiverse" and Other Sphere Eversions, available on the web as of today:

Models of [the Morin-Apéry] eversion were made by Charles Pugh, and Nelson Max digitized these models and interpolated between them for his famous 1977 computer graphics movie "Turning a Sphere Inside Out".

This might be the first movie. You can find it on Youtube today. The Shappiro eversion was as far as I know only published in still pictures (in Scientific American) when the movie above was realized.

The Morin-Apéry eversion was proposed after Shappiro's, which is the first realization (still according to Sullivan's article).

There is also the Geometry Center's movie outside-in, with a decisive contribution by Thurston. As already noted by Igor Rivin, this uses corrugations, in the lines of ideas of what is called the h-principle since Gromov. Note that I am not sure how deep goes the analogy between $C^1$ isometric embeddings (that cannot be taken $C^2$) and those eversions that can be chosen to be $C^k$.

Dennis Sullivan's article presents a movie that he realized in collaboration with Rob Kusner, Ken Brakke, George Francis, and Stuart Levy. It is called the optiverse. It is done by taking optimal path with respect to the Willmore energy (intergal of the square of the mean curvature). You could probably use this idea again (with considerably more computation power needed). But it is not clear how to exploit the output (a moving 6D mesh in $\mathbb{R}^7$).

The last movie I know of is called the Holiverse (see arXiv and/or Youtube).

There is of course no hope for a 7D movie, but it would still be interesting to have more information on the $S^6$ eversion. Let me introduce a question closely related to yours: Morin proved that the minimal number of generic topological transitions in an $S^2$ eversion is 14. What would the minimal number be for $S^6$?

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the link to Sullivan's article: – YCor Sep 17 '14 at 13:16
For an image of the "corrugations" mentioned here and in Igor's answer, see this earlier MO question: "C^1 isometric embedding of flat torus into ℝ^3." – Joseph O'Rourke Sep 17 '14 at 16:31
You can post comments now :) – Igor Rivin Sep 18 '14 at 16:43

The comments give a rather bogus version of history. It is true that a movie ("Outside in") was made at the geometry center, but the explicit eversion precedes the movie by three decades, and is due to Arnold Shapiro (1960), simplified by Bernard Morin in 1967. A good reference is an Intelligencer article by Morin and George Francis in 1980.

The Thurston "crinkling" technique is not due to Thurston, but rather to Nico Kuiper, who used it in the sixties to prove the amazing result that EVERY Riemannian manifolds admits a $C^1$ isometric embedding into its topological embedding dimension, and not only that, the image of the embedding can be constrained to lie in an arbitrarily small ball. This circle of ideas was later made into a science by Gromov ("the h-principle").

As for writing something explicit for $S^6,$ maybe, but where does this method get stuck for $S^4,$ e.g.? No amount of crinkling can overcome the obstruction...

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"bogus" seems kind of pointlessly charged, especially since the comments made no pretense of trying to be a full historical chronology. In that regard, you are missing major plot points, since all these ideas are variants on obstruction theory, which goes back to the beginnings of algebraic topology. – Ryan Budney Dec 2 '12 at 1:21
"bogus" is being very precise, though if you prefer "wrong", I can live with that. The OP and you credit the explicit eversion to Thurston in 1990, while the credit belongs to Shapiro and Morin a couple of decades earlier. I have no idea what you mean by "all these ideas are variants on obstruction theory". Obstruction theory tells us nothing about how to construct a homotopy/isotopy, it only tells us whether it is (im)possible in principle or not -- the Smale theorem is a good (the best?) example of the difference between existence results and constructive argument. – Igor Rivin Dec 2 '12 at 4:24
(continued) so I don't thinking I am missing anything major. All the points relevant to the plot are contained in the Francis/Morin article, which may or may not be relevant to the six-dimensional question. – Igor Rivin Dec 2 '12 at 4:26
I object to "bogus", as I objected to "someone" in my comment. The only reason I wrote what I did was to point out who was the apparently obscure mathematician referenced by the question in connection with the movie. Since I happened to know, I made the attribution of the movie slightly more precise. Since I did not happen to know, I made no pretence of correcting the historical record fully. – Ryan Reich Dec 2 '12 at 4:50
@Ryan: I see your point, but that sort of partial correction is how history gets completely distorted. In addition, the movie was a group project -- Thurston certainly supplied the idea, but, as was usual for him, he did NOT supply the details [never mind the computer implementstion). Many very smart people worked together to make that movie work. A comment along the lines of "The movie was done by a group led by Bill Thurston -- I don't know who was the first to come up with an explicit eversion" would not have triggered my bogometer. – Igor Rivin Dec 2 '12 at 5:43

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