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Does anyone know a reference for the 2-dimensional version of the Schoenflies theorem? To be precise, I'd like a reference for the fact that every continuous, 1-1 map $S^1\rightarrow \mathbb{R}^2$ extends to a homeomorphism $\mathbb{R}^2 \rightarrow \mathbb{R}^2$. The discussions of the Jordan Curve Theorem that I can remember don't prove this stronger statement.

This statement is mentioned on the Wikipedia page for the Schoenflies problem . I looked through several papers on the generalized Schoenflies problem (which requires extra hypotheses in higher dimensions to rule out things like the Alexander Horned Sphere), but no luck...

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    $\begingroup$ Check out Moise's book: ams.org/mathscinet-getitem?mr=488059 $\endgroup$
    – Ian Agol
    Commented May 24, 2012 at 23:01
  • $\begingroup$ See Berenstein-Gay "complex variables". Krantz and Bell have several papers on this as well. $\endgroup$
    – Andrés E. Caicedo
    Commented May 24, 2012 at 23:04
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    $\begingroup$ Another possible source is Kai-Uwe Bux's "Notes on Geometric Topology" which has a section on the Schoenflies theorem. This can be found on his webpage at Bielefeld. $\endgroup$
    – Allen Hatcher
    Commented May 25, 2012 at 3:12
  • $\begingroup$ I thought it was in Bing's book. $\endgroup$
    – Scott Carter
    Commented May 25, 2012 at 6:42

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Thomassen's paper on triangulating surfaces addresses this as well. See: Triangulating surfaces

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  • $\begingroup$ Thanks! This seems like a very nice and elementary proof. Hales' formal proof of the Jordan Curve theorem is based on Thomassen's ideas. I'll accept this answer since Thomassen states the theorem in exactly the form I was asking about. $\endgroup$
    – Dan Ramras
    Commented May 25, 2012 at 20:20
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In the smooth case the idea is to take a linear height function on the plane, which is generically Morse on the curve. Apply the Jordan curve theorem + basic Morse theory, this tells you the compact region bounded by the curve is a union of discs, glued together along common arcs, and the "gluing pattern" is that of a tree. An induction argument finishes it.

If you really need it for the topological category that's a fair bit more work. Larry Siebenmann has a recent article on this

L C Siebenmann 2005 Russ. Math. Surv. 60 645

His article seems to have pretty much all the historic references.

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  • $\begingroup$ Do you know a ref in smooth case? (preferably a popular book) $\endgroup$
    – Anton Petrunin
    Commented Apr 15, 2021 at 0:47
  • $\begingroup$ @AntonPetrunin: sorry, no. This is the argument in dimension 3 created by Alexander, simplified to the 2-dimensional case. I like the write-up of the 3-dimensional case in Hatcher's 3-manifolds notes. Section 7 of the Siebenmann notes outlines a few proofs sort of like this, in the 2-dimensional but PL case. $\endgroup$
    – Ryan Budney
    Commented Apr 15, 2021 at 1:03

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