# Extension of a homeomorphism

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 $h_i$.

-
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
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

-
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)

-