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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
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What do you mean by "explicit"? –  Ian Agol Dec 1 '12 at 22:36
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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

1 Answer 1

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
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"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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