Timeline for Elegant proof that any closed, oriented 3-manifold is the boundary of some oriented 4-manifold?
Current License: CC BY-SA 4.0
9 events
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| Dec 10, 2021 at 19:05 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
replaced the link to the arXiv front end; see https://meta.mathoverflow.net/questions/5124/is-it-time-to-replace-links-to-the-ucdavis-arxiv-frontend
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| Oct 30, 2018 at 19:22 | history | edited | Autumn Kent | CC BY-SA 4.0 |
deleted 1 character in body
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| May 6, 2011 at 2:24 | comment | added | Dylan Thurston | @Agol: We get bounds, but probably not good ones: we construct links with $O(k^2)$ crossings and so $O(k^2)$ components. The crossing bound is as good as possible by a result of Lackenby (arXiv:0810.5252). I wouldn't be surprised if you can do better in terms of number of components. | |
| May 5, 2011 at 21:56 | comment | added | Ian Agol | Can you use your argument to bound the number of components of a link needed in a Kirby diagram? This gives some estimate on the minimal rank of H_2 of a simply-connected 4-manifold bounding the 3-manifold. | |
| May 2, 2011 at 3:39 | comment | added | Dylan Thurston | @Greg: I only understood Thom's proof well enough to see that it would be at least quite difficult to make it explicit, at least difficult enough that an explicit version might well be called a different proof. | |
| May 1, 2011 at 17:12 | comment | added | Greg Kuperberg | I confess that I never understood the ideas of Thom's proof, certainly not well enough to see that it is fundamentally non-explicit. | |
| Apr 30, 2011 at 9:43 | comment | added | Dylan Thurston | @Bruno: Yes, you can. You end up showing any 4-manifold is cobordant to a connect sum of $\mathbb{CP}^2$'s and $\overline{\mathbb{CP}}^2$'s. | |
| Apr 30, 2011 at 9:28 | comment | added | Bruno Martelli | Can you apply the argument 4 to a four-manifold $M^4$? Namely, map $M^4$ to $\mathbb R^3$ and apply the same method to construct a 5-manifold bounded by $M$. Of course, at some point you should need to do some topological blow-ups on $M^4$ to get zero-signature. | |
| Apr 29, 2011 at 23:35 | history | answered | Dylan Thurston | CC BY-SA 3.0 |