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It is known that given any smooth ($C^\infty$) simple closed curve $\gamma$ in the 2-sphere, there is a smooth diffeomorphism of $\mathbb{S}^2$ taking $\gamma$ to the standard equator $x^2+y^2=1$.

Question: if $\gamma$ is instead piecewise smooth, it is there a homeomorphism of $\mathbb{S}^2$ taking $\gamma$ to the standard equator, whose restriction to the complement in $\mathbb{S}^2$ of the set of non-smooth points of $\gamma$ is a smooth diffeomorphism?

References would be much appreciated.

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  • $\begingroup$ Remove the singular points, find a diffeomorphism of what's left to a sphere with finite subset of the equator removed that maps the smooth segments to the equator, and then fill in the points. $\endgroup$ Feb 9, 2015 at 18:28
  • $\begingroup$ @Igor: how do you know there is such a diffeomorphism? $\endgroup$ Feb 10, 2015 at 8:03
  • $\begingroup$ @Greg: this takes work of course, like everything else in surface topology. This becomes clear after one understands how isotopies on surfaces are build. Note that our curve does not spiral towards the puncture so we could just enlarge the puncture and fill and reduce everything to the boundary case etc. $\endgroup$ Feb 10, 2015 at 11:20
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    $\begingroup$ 1. It is easy to find a homeomorphism which maps your curve to some smooth curve and which is diffeomorphism on the complement to the set of non-smooth points on the curve. 2. Apply the theorem you mentioned to the new smooth curve and find diffeomorphism which maps the curve to the equator. 3. The composition of these two maps is the desired homeomorphism. $\endgroup$ Feb 11, 2015 at 10:39
  • $\begingroup$ @MaximPrasolov, that's it, starting with the identity map on the complement of a finite number of small balls centered at the non-smooth points of the curve, the diffeomorphism is progressively extended nearer and nearer the non-smooth points. The limit of this construction gives the homeomorphism in 1. Thank you! $\endgroup$
    – Nautilus
    Mar 2, 2015 at 16:40

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