Timeline for The Schoenflies Theorem on two dimensional surfaces
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 5, 2021 at 19:34 | comment | added | mme | The correct statement is that if $C,D$ are separating curve in the interior of a compact connected surface $S$ (with or without boundary), then there is a homeomorphism $\varphi: S \to S$ with $\varphi(C) = D$ if and only if $S \setminus C$ and $S \setminus D$ are homeomorphic. This follows essentially from the classification of compact surfaces and what I have called to you before the "unique disc lemma" though some work needs to be done (take closure on each side, cap off boundaries, etc). As Ryan suggests above this is all much much easier if you deal with smooth surfaces, curves, etc. | |
| Mar 5, 2021 at 19:22 | comment | added | Fernando Oliveira | @Mike Miller I was just wandering if there were a version of Schoenflies Theorem that holds for two sided simple closed curves $C$ that are connected components of the frontier of an open connected subset $U$ of $S$. If $N$ is an annular neighbourhood of $C$ then we need that one component of $N\setminus C$ is not contained in $U$. $S$ can be any boundaryless surface. | |
| Mar 4, 2021 at 21:30 | comment | added | Fernando Oliveira | All observations were correct, thanks. $𝑆$ can not have boundary. Also, if $N$ is a neighbourhood of $C$ homeomorphic to an annulus, then one component of $N\setminus C$ is not contained in $U$. | |
| Mar 4, 2021 at 19:05 | comment | added | Josh Howie | It will also be necessary to assume that $S$ has empty boundary. If $\partial S\neq\emptyset$ and $\partial S$ is connected, then one can take $U$ as a regular open neighborhood of $\partial S$. Then the frontier $C$ is a connected 2-sided circle, but $\partial\overline{U}=C\cup\partial S$. | |
| Mar 4, 2021 at 18:57 | comment | added | mme | Yes, one needs the additional assumption that C is separating (which implies that it is 2-sided). | |
| Mar 4, 2021 at 18:52 | history | answered | Josh Howie | CC BY-SA 4.0 |