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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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  • $\begingroup$ Why do you want to show that? What have you tried? $\endgroup$ Apr 16, 2012 at 21:13
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    $\begingroup$ The remainder is a 3-ball, not a 2-sphere!! $\endgroup$
    – user641
    Apr 17, 2012 at 5:15
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    $\begingroup$ 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. $\endgroup$
    – user641
    Apr 17, 2012 at 5:19
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    $\begingroup$ en.wikipedia.org/wiki/Hopf_theorem $\endgroup$
    – user641
    Apr 17, 2012 at 6:57
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    $\begingroup$ 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. $\endgroup$
    – user641
    Apr 17, 2012 at 7:11

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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  • $\begingroup$ Thank you for the reference. $\endgroup$
    – user27329
    Apr 21, 2012 at 22:30
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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!

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