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Does every homeomorphism of the unit sphere S^n, n=2, has diffeomorphic extension to the unit ball. I am indeed interesten about the reference of the following problem: I need a given homeomorphism $h$ of the unit sphere to approximate uniformly by a sequence of diffeomorphisms $h_i$.

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Presumably, you are looking for a homeomorphism of the unit ball whose restriction to the interior is a diffeomorphism, and whose restriction to the boundary is your given homeomorhpism. –  André Henriques Jul 12 '11 at 11:31
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I'm guessing that the answer is "yes" for $n=2$. The question is also interesting for other values of $n$... where the answer will probably be negative (but I don't know for sure). –  André Henriques Jul 12 '11 at 11:36

2 Answers 2

See

http://en.wikipedia.org/wiki/Diffeomorphism#Extensions_of_diffeomorphisms

and

http://www.math.niu.edu/~rusin/known-math/93_back/smooth_sn

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David Kalaj posted the following comments in a (now-deleted) answer: (1) Thanks. Yes this was the point of my question. I need a reference for this. (2) It seems that this, the answer to the question: For a given homeomorphism h of the unit sphere find sequence of diffeomorphisms h_i of the sphere that converges uniformly to h is YES, but i need a reference –  Anton Geraschenko Jul 12 '11 at 16:31
    
These comments do not seem to make sense. Am I missing something? –  Igor Rivin Jul 12 '11 at 18:33

Dear Igor Rivin, indeed I possed two questions:

a) Does every homeomorphism of the unit sphere S^n, has diffeomorphic extension to the unit ball which is a homeomorphism in the closed unit ball.

b) For a given homeomorphism h of the unit sphere S^2 onto itself, does there exists sequence of diffeomorphisms h_i of the unit sphere onto itself that converges uniformly to h.

My first impression was that a) implies b), but I am not sure in that.

I think both answers are YES but I need a reference (in particular I am interested in question b)

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