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Given a homeomorphism $f$ on a compact manifold $M$, can you find a $C^1$ diffeomorphism within an arbitrarily small neighborhood of $f$? In other words, can you make an arbitrarily small perturbation to smooth out $f$?

It seems like a natural question, and I would guess the affirmative, but I could not find an answer.

I am working on $S^1$, but would be happy with a more general answer.

P. May
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  • Ah, I thought you said $C^0$-well if there is no response here, then it will probably be directed to overflow. I think this question is a bit involved. In any case, upon no response, I've given you two people references. – Faraad Armwood Mar 25 '18 at 16:25
  • I appreciate it, Faraad. I am considering attending NDSU next year for my PhD, so it is possible we will meet each other soon and I can ask your references in person. Good luck with your studies. – P. May Mar 25 '18 at 16:46
  • Well, looking forward to hearing from you. We'd be happy to have you. – Faraad Armwood Mar 25 '18 at 17:15
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    Answered at https://math.stackexchange.com/questions/1068805/is-every-self-homeomorphism-homotopic-to-a-diffeomorphism (see the statement about approximation vs isotopy in Moishe's answer). The dimension 1 case is very easy e.g. take a PL approximation then smooth the corners. – Dap Mar 26 '18 at 11:52

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