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Do you know where I can find (preferably online) a proof for the possibility of a sphere eversion?

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    $\begingroup$ Maybe this paper, or some of the references cited in its introduction $\endgroup$ Nov 18, 2018 at 19:05
  • $\begingroup$ @AkivaWeinberger Yes, that was what I wanted. Thank you :) $\endgroup$ Nov 18, 2018 at 19:08

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It's a question of proof acceptance:
Let you impress by the famous video 'Outside In' by Bill Thurston and collaborators ...

Or you may wish to invest more time and look into Scott Carter's book.

Added in edit:
A less informal & more technical reference is Bednorz & Bednorz, as proposed in Akiva Weinberger's comment to the OP.

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  • $\begingroup$ I watched that video today, it's why I made the post :) I actually wanted a more formal proof, if possible. The book is pretty big, but it also seems to be on the more intuitive side, no? I just looked at a couple of pages of the eversion, though $\endgroup$ Nov 18, 2018 at 19:00
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    $\begingroup$ @J.Dionisio The original proof used what is called the $h$-principle, and was a particular form of what became called "Smale-Hirsch theory". It was extremely inexplicit, and proceeded by studying the space of all immersions of $S^2$ into $\Bbb R^3$, and found that there is only one path-component in this space. Therefore there must exist a path between any of your favorite two immersions! The existence of explicit formulas giving this eversion is a newer development. $\endgroup$
    – user98602
    Nov 18, 2018 at 23:13
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    $\begingroup$ I think, though, that the general concept of the $h$-principle came quite a bit later than Smale's original 1957 proof of sphere eversion. Smale's proof, however inexplicit, is pretty elegant and not too hard to digest. $\endgroup$
    – Lee Mosher
    Nov 21, 2018 at 19:24

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