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I am extremely impressed by the work that has been done constructing sphere eversions, and other similar explicit geometrical proofs. In particular, surely nobody can fail to be impressed by the famous video 'Outside In' by Bill Thurston and collaborators, and book (Amazon, free pdf) by Scott Carter.

However, we already know when such eversions can be constructed, as the embeddings of $k$-spheres up to regular homotopy in $\mathbb{R}^n$ have been characterized by Smale's theorem. So here is my question:

What is the mathematical value, aside from the aesthetic quality of the proofs, of finding explicit regular homotopies?

Let me be clear: these proofs are extraordinarily beautiful, and exhibiting beauty is (in my opinion) absolutely valid as a primary goal of mathematics. But, just for the purposes of this question, I would like to put this aside.

To put my question in a different way: suppose I exhibited in an ingenious way some new explicit regular homotopy $M \sim M'$ of immersed manifolds, such that $M$ and $M'$ were already known in a nonconstructive sense to be regular homotopic. Other than aesthetic appreciation, why would a homotopy theorist be interested in studying my result? Are there, for example, important open conjectures about the minimal length of regular homotopies?

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There a variety of answers to your question. I have a particular bias. Let me transpose your question to another context.

Say someone comes up with a "computation" of the unstable homotopy groups of spheres. It's just a list of finite abelian groups, given via their Smith Normal Form presentation. The person provides no context for how they computed those groups. So it would be quite difficult to use that computation to say much about any specific homotopy-class of map $S^m \to S^n$, except in the special case where one knows the group is trivial. But even then, the computation provides no way to explicitly construct an extension $D^{m+1} \to S^n$.

In my opinion, constructing explicit eversions of the sphere is very much like the above. One may very well know a space is connected, but there is value in showing how one connects two seemingly disconnected points in a space.

This question should likely be community wiki.

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It depends on what you view as the purpose of mathematics is. If your view is constructivist (mine is), then proving that some object (in your case, a sphere eversion) exists is only the first (some would say zeroeth) step, making which tells you that there is actually something to look for. Until you have actually found the object, you don't know anything, quite literally, so are still in the domain of religion, not mathematics.

Just one man's opinion.

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  • $\begingroup$ I suspect (though I am not certain) that Smale's proof of the existence of sphere eversions was constructive in principle, i.e. you could chase through the proof and come up with an explicit construction, though it might be extremely complicated. $\endgroup$ – Eric Wofsey Jan 16 '15 at 1:20
  • $\begingroup$ @EricWofsey You suspect incorrectly. Smale found an obstruction, which lives in some homotopy group, which turns out to be trivial. MR0104227 (21 #2984) Reviewed Smale, Stephen A classification of immersions of the two-sphere. Trans. Amer. Math. Soc. 90 1958 281–290. $\endgroup$ – Igor Rivin Jan 16 '15 at 1:40
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    $\begingroup$ I don't see how that is inherently nonconstructive. You may be able to chase through the proof that the relevant homotopy group is trivial to get an explicit homotopy, which then translates into an explicit solution of the original problem. For what it's worth, Wikipedia claims Smale's proof was essentially constructive. $\endgroup$ – Eric Wofsey Jan 16 '15 at 1:46
  • $\begingroup$ @EricWofsey From what I have read about it I recall (albeit dimly) that Smale was quite surprised when the construction was discovered. $\endgroup$ – Igor Rivin Jan 16 '15 at 1:55
  • $\begingroup$ Smale's proof, in principle, with a lot of work, could have been made into something constructive. But it would take some non-trivial work. It's not constructive in a traditional sense, but in a more "in principle, if you're talented and can work hard, you can turn this into something constructive". $\endgroup$ – Ryan Budney Jan 16 '15 at 4:19
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Inspired by Thurston's "on proof an progress in mathematics", one can say that the goal of mathematics is to improve our understanding of mathematical objects.

The value of any construction which is simple enough to be made explicit, or visualized, of a phenomenon that has been abstractly known to exist, is then obvious: it improves our understanding of the phenomenon. This applies to sphere eversions as well as $C^1$ embeddings of flat tori in $\mathbb{R}^3$, for example. It also applies to new, simpler proofs of known theorems.

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