Timeline for Existence of smooth structures on topological $3$-manifolds with boundary
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2019 at 11:26 | comment | added | Dennis | Thanks a lot! :) | |
| Apr 12, 2019 at 23:25 | comment | added | Misha | Yes, this all works for non-orientable manifolds. | |
| Apr 12, 2019 at 23:07 | comment | added | Dennis | Thank you for your answers, it already helped a lot! I have one last question: In the cited paper by Munkres it is stated at the beginnign of the first chapter that the manifold $M$ is assumed to be oriented. In remark 2.9 he then says that the results so far also hold for the non-orientable case. Does that mean that the upcoming theorem 2.10 also holds for non-orientable manifolds? | |
| Apr 12, 2019 at 18:54 | comment | added | Misha | Read projecteuclid.org/download/pdf_1/euclid.ijm/1256059559 and you will see that smoothing works even for manifolds with boundary. Munkres starts with the assumption that your manifold is triangulated. For triangulations of topological 3-manifolds (with boundary), read Moise's book. | |
| Apr 12, 2019 at 18:46 | comment | added | Misha | Your argument is insufficient since it does not guarantee that $\partial M$ will be smooth in $\tilde{M}$. However, proofs of existence of a smooth structure work for manifolds with boundary as well. | |
| Apr 12, 2019 at 18:38 | comment | added | Neal | I think your argument has to be careful of counterexamples like the Alexander Horned Sphere, although I don't have time right now to precisely check if the AHS is indeed one. | |
| Apr 12, 2019 at 18:30 | review | First posts | |||
| Apr 12, 2019 at 18:32 | |||||
| Apr 12, 2019 at 18:30 | history | asked | Dennis | CC BY-SA 4.0 |