Eversion of the 6-sphere in 7-space

Question

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 $\mathbb{R}^7$?

Answer 1

(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 $C^k$.

John 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 $S^6$?

Answer 2

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...