What is the 'non-intuitive' part in sphere eversion (turning inside out)? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T21:07:43Z http://mathoverflow.net/feeds/question/105824 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105824/what-is-the-non-intuitive-part-in-sphere-eversion-turning-inside-out What is the 'non-intuitive' part in sphere eversion (turning inside out)? Uday 2012-08-29T12:25:38Z 2012-08-29T13:12:57Z <p>Hello, Everybody!</p> <p>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. </p> <p>Recently, I was reading about <a href="http://en.wikipedia.org/wiki/Smale%27s_paradox" rel="nofollow">Smale's paradox</a>, 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. </p> <p>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. </p> <p>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? </p> <p>This topic is new to me. I hope the question is not too naive. Thank you in advance. </p> http://mathoverflow.net/questions/105824/what-is-the-non-intuitive-part-in-sphere-eversion-turning-inside-out/105830#105830 Answer by Mark Grant for What is the 'non-intuitive' part in sphere eversion (turning inside out)? Mark Grant 2012-08-29T12:56:47Z 2012-08-29T13:12:57Z <p>Watch <a href="http://www.youtube.com/watch?v=wO61D9x6lNY&amp;feature=gv" rel="nofollow">Outside In</a> (something we should all do anyway, to commemorate Bill Thurston's passing).</p> <p>To understand the mathematics behind sphere eversions, you should first get a good intuition for the concepts of <a href="http://en.wikipedia.org/wiki/Immersion_%28mathematics%29" rel="nofollow">immersion</a> and <a href="http://en.wikipedia.org/wiki/Regular_homotopy" rel="nofollow">regular homotopy</a>. I recommend Guillemin and Pollack's "Differential Topology" book for starters.</p> <p>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 $S^1\to S^1$.</p> <p>You'll find a few more resources related to sphere eversions on my <a href="http://www.maths.nottingham.ac.uk/personal/pmzmg/" rel="nofollow">web page.</a></p>