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The highest upvoted answer to this old question Nice proof of the Jordan curve theorem? is a proof by Ryuji Maehara. I personally really liked/appreciated that Maehara's proof is

  • A) a fairly elementary proof (e.g. no explicit algebraic topology) and
  • B) doesn't involve any sort of approximation/limiting argument from simpler discrete things (in particular unlike e.g. Tverberg's 1980 proof, or Jordan's original proof which uses an limiting argument from polygonal curves)

I'm interested in whether there is a "similar-style" proof for the Jordan-Schoenflies theorem (where I guess "similar-style" for now means just "satisfies (A) and (B)"). Two non-examples:

  • Carathéodory's extension to the Riemann mapping theorem tells us that Jordan-Schoenflies would follow if one could prove that the interior $U$ of the Jordan curve $J$ is simply connected. This would follow from the fact that the JCT tells us that we know the complement of $U$ is $J \cup E$ (where $E$ is the exterior of the Jordan curve $J$) hence in particular connected in the Riemann sphere $\hat {\mathbb C}$, which is equivalent to $U$ being simply-connected (see e.g. Thm. 5.7 in Marshall's Complex Analysis).

    However, I'm not quite satisfied with this, because the RMT (and Carathéodory's extension of it) is definitely not "fairly elementary", and the proof of the equivalent characterization of simply connectedness in Marshall uses winding numbers, at which point we're basically doing algebraic topology in all but name.

  • The other MO thread linked Carsten Thomassen's proof using the non-planarity of $K_{3,3}$. An absolutely wonderful proof and paper, but not really fitting my "style" guideline, since it builds approximations to the desired homeomorphism using graphs, and then takes a limit.

If people have any reasons to think that there can be no such proof, I would also like to hear those thoughts as well.

Undoubtedly there are those who think such proofs are "not desirable" (perhaps even moreso for the Jordan-Schoenflies theorem instead of just the JCT), but the other MO thread had some real gems, so I think this question and potential answers to it are not entirely devoid of insight.

D.R.
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    But Maehara's proof uses Brower's fixed point theorem, that's where all complicated stuff is hidden – erz May 30 '23 at 08:55
  • @erz yes, I suppose the word "explicit" is doing some heavy lifting here. However there is an "elementary" proof of BFPT using the combinatorial Sperner's lemma, whereas things like winding number or homologous paths seem pretty firmly in algebraic topology land. (Yes the distinction is fuzzy and arbitrary, and some may say meaningless, but still, I think at least it may be less "fuzzy" to say that the Sperner lemma -> BFPT -> Maehara proof of JCT has a different "flavor" than other proofs I discussed in my post, and I want to know if there's a proof of Jordan-Schoenflies with this "flavor".) – D.R. May 30 '23 at 22:23

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