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I am currently going through the proof of the Jordan-Schoenflies Theorem by S.S. Cairns (http://eretrandre.org/rb/files/Cairns1951_193.pdf).

I find that because it is a somewhat old article there are some conventions that are unclear to me. I wanted to ask about them and hopefully get unstuck.

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The article claims this can be quickly established by familiar methods. I tried doing it and I found that using the corollary (A) of the Jordan-Schoenflies:

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I was able to deduce that either the interior of the $b_1$ is in the interior of $b_2$ or else their interiors are disjoint. I assume by hypothesis that $b_1$ and $b_2$ both have the Jordan-Schoenflies property. This claim holds because if part of $b_1$ lied in the interior of $b_2$ I can only exit through the simple arc $b$, which forces all of $b_1$ to be inside the interior of $b_2$ or in $b$, else we have a self-intersecting curve. As a consequence the interior of $b_1$ is also contained in the interior of $b_2$. The argument if part of $b_1$ lied in the exterior of $b_2$ is similar.

I can then map all of $E$ to itself bringing $b_1$ to a circle, which makes things look nicer. But even in this scenario I don't see how deleting $b$ I can prove Jordan-Schoenflies for $b_1+b_2-b'$. Would someone have a suggestion?

2) enter image description here

I think crossing a polygon is not well-defined in this paper. If I pass the straight line $y=1$ through the vertex of a triangle $(1,0), (0,1), (0,0)$ it doesn't have an endpoint interior to it.

I suppose I can say look for each side of the polygon a small tubular neighborhood of it. Then I could define "a polygonal path crosses a polygon if it intersects one side of the polygon, and that in a neighborhood of such intersection the path has points on both sides of the tubular neighborhood of the side it intersects on." I was hoping there is an easier definition to work with and that makes this Lemma true.

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In the middle of proving Lemma 3.1 he establishes (A) and (B), which is fine:

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but then he proceeds to prove (C):

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My question is about part (2). Here $\alpha$ is defined earlier by:

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He claims proving (C) presents no difficulty, but I was not able to do it yet. It's not clear to me how $\alpha$ as defined have to come from either of $\alpha_1$ or $\alpha_2$. I would appreciate any nudge towards showing this point (2) of (C), Lemma 3.1.

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  • $\begingroup$ Theorem 2.1. Let $C$ be a closed disk with $p\subset C$. Fix a point $x\not\in C.$ For $x\ne y\not\in p$ let $[x,y]$ be the line-segment from $x$ to $y$. Now let $S(y)$ be the set of those $z\in p\cap [x,y]$ such that if $e$ is any edge of $p$ and if $z\in e \cap [x,y]$ then $e\cap [x,y]=\{z\}.$ Now for any $w\not\in p$ let $w\in E$ iff $(w=x$ or $S(w)$ has an even number of members) and let $w\in F$ iff $w\not\in E.$ It is not hard to show that $E,F$ are open & connected, with $F$ bounded $\endgroup$ Jul 25 at 0:48
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I did not read his paper but "crossing" means "meeting transversally". Here two polygonal arcs $a, b$ in the plane are said to meet transversally (or "cross") at a point $p$ if there is a local PL homeomorphism from a neighborhood of $p$ to $R^2$ which sends $a$ to an interval in the x-axis and $b$ to an interval of the y-axis (and sends $p$ to the origin).

Instead of (or in addition to) following Cairns' paper, I would suggest the more recent

Thomassen, Carsten, The Jordan-Schönflies theorem and the classification of surfaces, Am. Math. Mon. 99, No. 2, 116-130 (1992). ZBL0773.57001.

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  • $\begingroup$ I actually started my attempt at understanding the proof of the Jordan-Schoenflies with Thomassen's article, but I found it harder to tackle. For example, in Lemma 2.3 it assumes that for a Jordan curve the regions of the complement of C all have C as boundary. I got stuck then, because regions of R^2\C a priori just mean its connected components. It is probably a consequence of the curve being simple but I could not prove that and he doesn't provide a proof either. $\endgroup$ Jul 24 at 17:41
  • $\begingroup$ @viniciuscantocosta I suggest asking a separate question regarding this point. $\endgroup$ Jul 24 at 18:56
  • $\begingroup$ You're right, thanks for the suggestion. My main point with the comment was to point out I had tried looking into your suggestion before posting my question. $\endgroup$ Jul 24 at 19:07

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