Smale's paradox

A Morin surface seen from "above"

In differential topology, Smale's paradox states that it is possible to turn a sphere inside out in a three-dimensional space with possible self-intersections but without creating any crease, a process more commonly and historically called sphere eversion (eversion means "turning inside out"). This is surprising, to those who understand regular homotopy, and is hence deemed a veridical paradox. (The name 'Smale's Paradox' generally does not appear in the traditional mathematical literature.) 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 ƒ0 = ƒ and ƒ1 = −ƒ.

History

This 'paradox' was discovered by Stephen Smale (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 Arnold S. Shapiro and Bernard Morin who was blind. On the other hand, it is much easier to prove that such a "turning" exists and that is what Smale did.

Smale's graduate adviser Raoul Bott at first told Smale that the result was obviously wrong (Levy 1995). His reasoning was that the degree of the Gauss map must be preserved in such "turning"—in particular it follows that there is no such turning of S1in R2. But the degree of the Gauss map for the embeddings f, −f in R3 are both equal to 1, and do not have opposite sign as one might incorrectly guess. The degree of the Gauss map of all immersions of a 2-sphere in R3 is 1; so there is no obstacle. The term "veridical paradox" applies perhaps more appropriately at this level: until Smale's work, there is no documented attempt to argue that 2-sphere eversion was, or was not, possible, and accordingly, subsequent attempts at explicit eversion, or to argue that it was impossible, are in hindsight. Accordingly, there never was an historical paradox associated with Smale's actual sphere eversion, merely an appreciation of the conceptual visualization subtleties by those confronting the idea for the first time.

See h-principle for further generalizations.

Proof

Smale's original proof was indirect: 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. In principle the proof can be unwound to produce an explicit regular homotopy, but this is not easy to do.

There are several ways of producing explicit examples and beautiful mathematical visualization:

See also

References

External links