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I've just seen the wonderfully done movie Sphere Inside Out, one about the Smale's paradox.

And the first question came in mind is that, why it has to be so ugly? Why turning an ultimately simple sphere inside-out is so complicated? Although this movie has demonstrated in so many view points and a variety of ways, I still cannot fully accept this truth in my mind. What's the critical point in this whole pack of steps, what exactly did Smale find to solve this problem? And are there other methods that is also easy to understand?

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Smale observed a version of what's now known as The Smale-Hirsch Theorem.

see: http://en.wikipedia.org/wiki/Immersion_%28mathematics%29

This states that if a manifold $M$ has dimension strictly less than the dimension of the manifold $N$, then the space of all immersions of $M$ in $N$ has the same homotopy-type of the space of fibrewise-linear bundle injections $TM \to TN$. Smale did it for the case $M$ is a sphere, and $N$ a Euclidean space, and Hirsch later dealt with the general case.

As stated in the Outside-In video you refer to, Smale's inspiration was the Whitney-Graustein Theorem. This is the Smale-Hirsch theorem but for $M=S^1$ and $N=\mathbb R^2$. Smale's dissertation generalized this to $N$ any surface. So it was only natural for him to think of the case $M$ a sphere and $N$ a Euclidean space.

There is something of a meta-reason as to why you'd expect any inversion of the sphere to be somewhat complicated. The fact that the sphere and its reverse are connectable in the space of immersions, via Smale's proof this boils-down to the fact that $\pi_2 SO_3$ is trivial. This is a rather subtle fact and it relates to several interesting phenomena, such as $\pi_1 SO_3$ having two elements. In the video these phenomena are paralleled with "the belt trick" section.

John Sullivan (of T.U. Berlin) was very interested in the question of "how simple can you make an inversion of the sphere?" What he did is perform a "relaxation" on the Boy surface inversion, to minimize the bending energy of the entire 1-parameter family of the inversion. He created this:

http://video.google.com/videoplay?docid=-761214833095493063

Off the top of my head I forget who is responsible for this result (perhaps it was Sullivan) but you can attach a notion of "complexity" to sphere inversions, depending on how complicated the multiple-point set becomes, how many transitions and such it has. If I recall correctly, Sullivan's Optiverse has minimal complexity. edit: oh, the video mentions this result is Bernard Morin's.

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Hmm, apparently I should have credited Kusner. I've heard all this information first-hand from Sullivan, perhaps it shows. :) –  Ryan Budney Nov 8 '11 at 6:52
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