Smale's paradox

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. More precisely, let

<math>f:S^2\to R^3<math>

be the standard embedding; then there is a continuous one-parameter family of immersions

<math>f_t:S^2\to R^3<math>

such that <math>f_0=f<math> and <math>f_1=-f<math>.

This 'paradox' was discovered by Stephen Smale in 1958. It is quite hard to visualize a particular example of such a turning, but there are now some movies to help you. 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'.Indeed, the degree of the Gauss map must be preserved in such "turning" — in particular it follows that there is no such turning of <math>S^1<math> in <math>R^2<math>. But the degree of the Gauss map for the embeddings <math>f<math> and <math>-f<math> in <math>R^3<math> are both equal to 1. In fact the degree of the Gauss map of all immersions of a 2-sphere in <math>R^3<math> is 1; so there is in fact no obstacle.

See h-principle for further generalizations.