# What is the 'non-intuitive' part in sphere eversion (turning inside out)?

The question does not mean sphere eversion is intuitive to me! In fact, it is just the opposite and that is the purpose of this question.

Recently, I was reading about Smale's paradox, the problem of sphere eversion (turning a sphere inside out). The wiki article is quite clear and gave me a good overview of the topic. I happened to see an animation of the eversion process as well.

The problem of sphere eversion is to construct a homotopy between the inside and outside of a sphere in a three dimensional space. During the continuous deformation self-intersections of the sphere are allowed and creating creases is not allowed.

Given that we can self-intersect the sphere while the process of eversion what could be a possible obstruction to the eversion? What exactly do we mean by self-intersection? Moreover, I find it difficult to imagine why a similar process cannot be employed in the circle case? Why can't we self intersect a circle with itself to turn it inside out? Is there an easy explanation for this phenomenon?

This topic is new to me. I hope the question is not too naive. Thank you in advance.

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The 2 dimensional analog is actually impossible. And it was proven that it was possible, before people had an explicit construction on how to do it. –  Thomas Rot Aug 29 '12 at 12:40
See "Outside In" (youtube.com/watch?v=wO61D9x6lNY) for a great explanation of both the $S^2$ and $S^1$ cases. –  Henry Cohn Aug 29 '12 at 12:54
One small point: If you look at an oriented sphere immersed in $\mathbb{R}^3$, then the Gauss map gives a map from the sphere to $S^2$. Regular homotopy preserves the homotopy type of the Gauss map, so in particular if you start with an embedded sphere, then the Gauss map is degree one. Thus, after the eversion, the Gauss map must also be degree one. This means that the isotopy class of the eversion is orientation reversing (so isotopic to the antipodal map of the sphere). –  Ian Agol Aug 29 '12 at 14:57

Watch Outside In (something we should all do anyway, to commemorate Bill Thurston's passing).

To understand the mathematics behind sphere eversions, you should first get a good intuition for the concepts of immersion and regular homotopy. I recommend Guillemin and Pollack's "Differential Topology" book for starters.

To see why it is not possible to turn a circle inside out, you should read up on the Whitney-Graustein Theorem. It basically boils down to the calculation $\pi_1(S^1)=\mathbb{Z}$, once you notice that the normalized differential of an immersion $S^1\looparrowright \mathbb{R}^2$ is a map $S^1\to S^1$.

You'll find a few more resources related to sphere eversions on my web page.

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I should add that both of these results fit into the general theory of immersions developed by Smale and Hirsch (see Hirsch, Morris W. Immersions of manifolds. Trans. Amer. Math. Soc. 93 1959 242–276). –  Mark Grant Aug 29 '12 at 19:17
@Mark Thank you. The inputs are sufficient for further exploration. –  Uday Aug 30 '12 at 7:13
@MarkGrant, How did you know that the Bill at math.columbia.edu/~woit/wordpress/?p=5059 is the Bill as claimed at mathoverflow.net/users/9062/bill-thurston ? –  Pacerier Jun 22 at 21:41
@Pacerier: I never said that I did! But I have no reason to doubt that the MO user Bill Thurston was the illustrious mathematician of the same name. –  Mark Grant Jun 24 at 6:29
@MarkGrant, Well, the reason is that usually people enjoy imitating / falsifying "famous people", or just playing pranks with them. –  Pacerier Jun 26 at 23:26