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

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...