Timeline for Decomposition of manifolds with toroidal boundary
Current License: CC BY-SA 4.0
20 events
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| Feb 22, 2022 at 16:48 | comment | added | Moishe Kohan | @G.Blaickner: yes, it is a contradiction, proving that $M=\hat{T}$. | |
| Feb 22, 2022 at 16:39 | comment | added | G. Blaickner | Thank you very much for answering again and adding so many details! But there is still one point which I can't get my head around. If you assume that $\mathcal{M}$ is prime and the conclusion (in the compressible case) is that $\mathcal{M}=\mathcal{N}\#\hat{T}$ for some $3$-manifold different from the $3$-sphere, isn't that a contradiction? I mean, prime means that every splitting of the form $\mathcal{M}=N_{1}\# N_{2}$ implies that either $N_{1}$ or $N_{2}$ has to be a $3$-sphere, right?.... | |
| Feb 22, 2022 at 16:32 | history | edited | Moishe Kohan | CC BY-SA 4.0 |
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| Feb 22, 2022 at 16:10 | comment | added | Moishe Kohan | @G.Blaickner: No, I continue to assume that $M$ is prime (as in the previous paragraph). I will edit my answer to make this even more clear. | |
| Feb 22, 2022 at 9:51 | comment | added | G. Blaickner | After some time, I cam back to this problem and I still do not fully understand. In the paragraph starting with "Such a manifol" you assume that $\mathcal{M}$ is non prime, right? (or as explained above, equivalently, non boundary-prime). But why is the compressing disk then "necessarily non-seperating"? | |
| Jan 8, 2022 at 18:25 | comment | added | Moishe Kohan | @G.Blaickner Yes for the first sentence. I did not quite understand the second sentence. | |
| Jan 8, 2022 at 9:50 | comment | added | G. Blaickner | @MoisheKohan Do understand the refined result correctly, is it correct that (2) can be divided into manifolds of the form $\hat{T}\#\mathcal{M}$ (compressible boundary and non-prime) and into non-prime manifolds with incompressible boundary? Furthermore, the solid torus is prime and has compressible boundary, hence the list basically just contains the four possible combinations of prime vs. non-prime an compressible vs. incompressible. | |
| Jan 8, 2022 at 9:47 | comment | added | G. Blaickner | Thanks @B.Hueber for pointing that out! | |
| Jan 7, 2022 at 21:54 | history | edited | Moishe Kohan | CC BY-SA 4.0 |
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| Jan 7, 2022 at 21:48 | comment | added | B.Hueber | Thank you very much for your time, that clears it up for me :-) | |
| Jan 7, 2022 at 21:48 | comment | added | Moishe Kohan | @B.Hueber: I clarified in the edit. | |
| Jan 7, 2022 at 21:47 | history | edited | Moishe Kohan | CC BY-SA 4.0 |
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| Jan 7, 2022 at 21:45 | comment | added | B.Hueber | Thank you very much for clarification. But I still do not 100% understand. Is your proposed trichotonmy then only true for those $\mathcal{M}$ with $\partial\mathcal{M}\cong T^{2}$ which are $\partial$-prime, or also in general. (Srry if this is a stupid question, but since I am not so familiar with (de)compressibility, I have some troubles to follow the derivation) | |
| Jan 7, 2022 at 21:45 | history | edited | Moishe Kohan | CC BY-SA 4.0 |
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| Jan 7, 2022 at 21:39 | comment | added | Moishe Kohan | @B.Hueber: Item (2) should be read as "$M$ is not prime." (One of the factors need not be a solid torus.) Thank you for noticing. | |
| Jan 7, 2022 at 21:38 | comment | added | B.Hueber | Okay I see. But this does not change your proposed trichotomy, right? | |
| Jan 7, 2022 at 21:35 | comment | added | Moishe Kohan | @B.Hueber: You are right, I was sloppy and assumed implicitly that $M$ is prime to begin with. Then it will have to be boundary-prime as well. | |
| Jan 7, 2022 at 21:26 | comment | added | B.Hueber | Sorry, I am late to this question, but I was looking for a similar problem. Just a short question: Why is a $3$-manifolds with $\partial\mathcal{M}\cong T^{2}$ necessarily a $\partial$-prime? Because, it could maybe still be possible to write $\mathcal{M}$ as $\mathcal{M}=Q_{1}\#_{\partial} Q_{2}$ where $Q_{1}$ is some other $3$-manifolds with torus boundary and where $Q_{2}$ is some $3$-manifolds with spherical boundary, which is different from the closed $3$-ball. | |
| Dec 10, 2021 at 9:00 | vote | accept | G. Blaickner | ||
| Dec 9, 2021 at 20:33 | history | answered | Moishe Kohan | CC BY-SA 4.0 |