Perhaps this question has already been asked on Mathoverflow. I mean this question in a global sense. A friend mentioned it to me today, and I started thinking about it. I'm not sure how to prove it. I would really love to see a proof of this using classical differential geometry techniques. I did a little bit of googling, and it seems that in higher dimensions this problem is very complicated, with many startling results (i.e. there exists a smooth 4-manifold homeomorphic but not diffeomorphic to $\mathbb{R}^4$). I will state three questions:

  1. Let $\Sigma_1$ and $\Sigma_2$ be smooth, embedded hypersurfaces in $\mathbb{R}^3$, which are homeomorphic. Are they necessarily diffeomorphic? Does it matter if they are closed (compact with no boundary) or not?

  2. If the answer to question 1. is in the affirmative, what about for general manifolds of dimension 2?

  3. Do the equivalent statements hold in dimension 3? If not, what breaks down?

Perhaps it would be instructive if the ideas for several proofs could be presented. One using classical differential geometry, one using homotopy theory (as this question seems heavily tied to the subject, at least the corresponding question in higher dimensions).

Thanks in advance for any comments or answers.

Edit: Is there an intuitive reason why this breaks in higher dimensions? How does this relate to current research in manifold theory and/or homotopy theory?


closed as off-topic by Igor Rivin, Qiaochu Yuan, Anton Petrunin, Igor Belegradek, Ricardo Andrade Nov 22 '14 at 1:27

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    $\begingroup$ It's a classical result that in dimensions $\le 3$ the smooth, PL, and topological classifications of manifolds coincide; for the case of surfaces see, for example, math.cornell.edu/~hatcher/Papers/TorusTrick.pdf and for the case of 3-manifolds see en.wikipedia.org/wiki/Moise%27s_theorem. $\endgroup$ – Qiaochu Yuan Nov 21 '14 at 20:36
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    $\begingroup$ In dimension 2 the answers follows immediately from the classification of surfaces (be they closed or not, the latter classification is more complicated though). $\endgroup$ – Benoît Kloeckner Nov 21 '14 at 20:40
  • $\begingroup$ Cool! Thanks for the references! Is there a good reference for Moise's theorem? $\endgroup$ – Michael Pinkard Nov 21 '14 at 20:57
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    $\begingroup$ Moise has written a textbook: ams.math.uni-bielefeld.de/mathscinet/search/… $\endgroup$ – ThiKu Nov 22 '14 at 1:12
  • $\begingroup$ Michael, if you'd like to learn more the keywords to look up are "smoothing theory" and "surgery theory." It would probably be easier to ask a new question (after doing some research using these keywords if you still have questions) than to try to get this one reopened. $\endgroup$ – Qiaochu Yuan Nov 22 '14 at 9:26