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I want to show that any orientation preserving self-homeomorphism of a 2-sphere $S^2$ is isotopic to identity.

Any help or reference is appreciated.

Edit; I want to show this to prove the following. Suppose we have two solid torus and we have a homeomorphism of the boundaries. The manifold obtained by indentifying boundaries via the homeomorphism depends only on the image of the meridian. To show this, first cut out the cylinder neighborhood of a meridian and glue it to the other solid torus. The reminder is homeomorphic to $B^3$. So if I can prove the question above, I can finish this proof.

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    Why do you want to show that? What have you tried?2012-04-16
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    @Henning I edited to include the motivation of the question. I don't know exactlly where to start.2012-04-16
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    The remainder is a 3-ball, not a 2-sphere!!2012-04-17
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    And it is very easy to see there is only one way to glue a 3-ball...2012-04-17
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    For the original question: a homeomorphism is isotopic to a diffeomorphism for $S^2$; this was proved by Munkres for general surfaces. This diffeomorphism has degree 1, and hence is homotopic to the identity. But homotopy=isotopy for closed orientable surfaces.2012-04-17
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    @SteveD Thanks. I corrected it. How can I prove that degree 1 deffeomorphism is homotopic to the identity?2012-04-17
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    http://en.wikipedia.org/wiki/Hopf_theorem2012-04-17
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    Again, I will say this: since the remainder is a 3-ball, you don't need any of this: it is very easy to see you can glue the 3-ball in in a unique way. Nothing about the 2-sphere is relevant to your main question.2012-04-17
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    @SteveD Thank you for the reference. Could you show me how to prove there is one way to glue a 3 ball?2012-04-21

2 Answers 2

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This is proven for diffeomorphisms in the following paper:

Earle, C.J. and Eeels, J. "The Diffeomorphism Group of a Compact Riemann Surface". Bull. Amer. Math. Soc. 73 (1967) 557–559.

In particular, this paper proves that the space of orientation-preserving diffeomorphisms of $S^2$ that fix three points on the circle is contractible. Note that a path in the space of diffeomorphisms is precisely an isotopy.

I don't know a reference that extends this to homeomorphisms -- we would need a proof that every homeomorphism is isotopic to a diffeomorphism.

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    Thank you for the reference.2012-04-21
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ecce answer!

https://mathoverflow.net/questions/39403/connectivity-of-the-group-of-orientation-preserving-homeomorphisms-of-the-sphere

I think that the 2 dimensional case is much easier!