Smale's Paradox - Proof

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 in 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 mathematical visualization:

• 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.