Timeline for Is it possible to prove that surfaces with compact boundary are homeomorphic by glueing disks to the boundary components?
Current License: CC BY-SA 4.0
17 events
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| Mar 22, 2021 at 19:46 | comment | added | Fernando Oliveira | @Lee Mosher. I don't really know how to prove this two facts, but believe that a proof could be written using triangulations. | |
| Mar 22, 2021 at 19:46 | comment | added | Fernando Oliveira | @Lee Mosher . I wrote a detailed proof (using your ideas) that given two collections of pairwise disjoint disks, there is a homeomorphism of the surface that maps one collection to the other. Just to fill safe. I used two facts, that I believe are correct: 1) If $C$ is a simple closed curve contained in the surface, then $C$ has a neighborhood $N$ homeomorphic to an open annulus or a Mobius band. 2) If $C$ and $D$ are two simple closed curves contained in the plane and $C$ is contained in the region bounded by $D$, then the region bounded by $C$ and $D$ is homeomorphic to an annulus. | |
| Mar 1, 2021 at 0:29 | comment | added | Lee Mosher | That's correct. And you don't need any higher dimensional topology to understand the application of the annulus lemma. | |
| Feb 28, 2021 at 16:11 | comment | added | Fernando Oliveira | @LeeMosher I read the argument for one disk, and I think I understand the idea. As for details, I am not familiar with topology in higher dimensions. When you say "nicely embedded disk" I hope that in dimension two topological obstructions do not exist and all embedded disks are nicely embedded thanks to the Jordan - Schoenflies Theorem. Is it like this? | |
| Feb 26, 2021 at 14:44 | comment | added | Lee Mosher | For a single disc, the question is answered in all dimensions here (see the "For the rest..." paragraph), as an application of the Annulus Theorem. That proof should easily generalize to a finite disjoint union of discs. | |
| Feb 23, 2021 at 23:15 | comment | added | Fernando Oliveira | @MikeMiller I am trying to write a paper on the dynamics of homeomorphisms of surfaces and I would like to explain why Kerekjarto / Ian Richards' characterisation of homeomorphic surfaces without boundary in terms of genus, orientability and ends is also true for surfaces with compact boundary. I don't want bother you or make you waist time. A reference that I could understand with the existence of a homeomorphism of a surface that maps a collection of n disjoint to another would be perfect. If you could sketch a proof with little effort I could go through the details. Thanks | |
| Feb 23, 2021 at 14:47 | comment | added | mme | I am sure there are many nice references but due to my own ignorance I don't know any (other than my own unpublished course notes). I can write a proof if you want; if a reference is more important I will have to punt on that. | |
| Feb 23, 2021 at 14:43 | comment | added | Fernando Oliveira | @MikeMiller Thank you. I am not a topologist, but I thought that a result like this should already exist. Would you please give me a reference for the unique disk lemma? Thanks again | |
| Feb 22, 2021 at 11:42 | comment | added | mme | The unique disc lemma is basically a form of the Schoenflies theorem (or more accurately a form of the annulus theorem in dimension 2). | |
| Feb 22, 2021 at 11:40 | comment | added | mme | What you need ultimately is the "unique disc lemma": if $M$ is a surface without boundary, any two embeddings $i,j: D^2 \to M$ are nearly isotopic: there is a family of homomorphisms $F_t: M \to M$ so that $F_0$ is the identity and $i F_1 = j$ or $i F_1 = jr$, where r is reflection of the disc. In particular there is an ambient isotopy taking the image of one to the image of the other. You can then extend this to embeddings of k disjoint discs. Now you recover S from M by deleting the interiors of these discs, and the isotopies above justify that the result is unique up to homeomorphism. | |
| Feb 22, 2021 at 4:18 | history | edited | Fernando Oliveira | CC BY-SA 4.0 |
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| Feb 22, 2021 at 4:15 | comment | added | Fernando Oliveira | As for the isotopy idea, would you please explain more carefully how it helps to solve the problem? Thanks | |
| Feb 22, 2021 at 4:12 | comment | added | Fernando Oliveira | Thanks, of course the same number of boundary components. I forgot to write it. | |
| Feb 22, 2021 at 0:53 | comment | added | Moishe Kohan | You need equal number of boundary components for this to hold. | |
| Feb 21, 2021 at 23:39 | history | edited | Fernando Oliveira | CC BY-SA 4.0 |
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| Feb 21, 2021 at 23:39 | comment | added | Anubhav Mukherjee | One can always isotope the homemorphism such that it is identity on a disk. | |
| Feb 21, 2021 at 23:32 | history | asked | Fernando Oliveira | CC BY-SA 4.0 |