Smale's paradox

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A Morin surface seen from "above".

In differential topology, Smale's paradox states that it is possible to turn a sphere inside out in 3-space with possible self-intersections but without creating any crease, a process often called sphere eversion (eversion means "to turn inside out"). More precisely, let

$f\colon S^2\to \R^3$

be the standard embedding; then there is a regular homotopy of immersions

$f_t\colon S^2\to \R^3$

such that $f_0=f\,$ and $f_1=-f\,$ .

1 History
2 Proof
3 Visualization
4 See also
5 References
6 External links

[edit] History

This 'paradox' was discovered by Stephen Smale in 1958. It is difficult to visualize a particular example of such a turning, although some digital animations have been produced that make it somewhat easier. The first example was exhibited through the efforts of several mathematicians, including one who was blind, Bernard Morin. On the other hand, it is much easier to prove that such a "turning" exists and that is what was done by Smale.

The legend says that when Smale was trying to publish this result, the referee's report stated that although the proof is quite interesting, the statement is clearly wrong 'due to invariance of degree of the Gauss map'.^{[citation needed]} Indeed, the degree of the Gauss map must be preserved in such "turning"—in particular it follows that there is no such turning of $S^1\,$ in $\R^2$ . But the degree of the Gauss map for the embeddings $f\,$ and $-f\,$ in $\R^3$ are both equal to 1. In fact the degree of the Gauss map of all immersions of a 2-sphere in $\R^3$ is 1; so there is in fact no obstacle.

See h-principle for further generalizations.

[edit] Proof

Smale's original proof was nonconstructive: he identified (regular homotopy) classes of immersions of spheres with a homotopy group of the Stiefel manifold. Since the homotopy group that corresponds to immersions of $S^2 \,$ in $\R^3$ vanishes, the standard embedding and the inside-out one must be regular homotopic, but it does not produce a regular homotopy.

There are two classes of constructive proofs:

the method of half-way models: these consist of very special homotopies. This is the original method, first done by Shapiro and Phillips via Boy's surface, later refined by many others. A more recent and definitive refinement (1980s) is minimax eversions, which is a variational method, and consist of special homotopies (they are shortest paths with respect to Willmore energy). The original half-way model homotopies were constructed by hand, and worked topologically but weren't minimal.

Thurston's corrugations: this is a topological method and generic; it takes a homotopy and perturbs it so that it becomes a regular homotopy.

[edit] Visualization

Sphere eversion is difficult to visualize; a computer animation of eversions (minimax and corrugations) was produced in the video Outside In.

[edit] See also

[edit] References

Nelson Max, "Turning a Sphere Inside Out", International Film Bureau, Chicago, 1977 (video)
Anthony Phillips, "Turning a surface inside out, Scientific American, May 1966, pp. 112-120.
Smale, Stephen A classification of immersions of the two-sphere. Trans. Amer. Math. Soc. 90 1958 281–290.