Timeline for Is Cohen immersion conjecture (theorem) known for vector bundles?
Current License: CC BY-SA 4.0
20 events
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| May 22, 2019 at 19:56 | comment | added | Mark Grant | @LuisA.Florit: I guess I was thinking more about the h-principle as it manifests itself in the Smale-Hirsch theory of immersions, which says roughly that there exist an immersion of M into N iff there exists a vector bundle monomorphism from TM into TN. There are some conditions, such as dim(M)<dim(N). See Hirsch's "Immersions of manifolds". | |
| May 22, 2019 at 18:04 | comment | added | Luis A. Florit | @MarkGrant: How would you use the h-principle for this? Are you suggesting we could try to prove that any vector bundle over, say, a compact Lie Group, which itself admits an immersion as a Euclidean hypersurface, also admits an immersion as a Euclidean hypersurface? BTW, I am interested in rank 8 vector bundles, although I don't think this helps. | |
| May 22, 2019 at 18:02 | comment | added | Luis A. Florit | @MikeMiller: yes, I understood this, and it is the answer I was looking for (you may write it as an answer if you wish). What I didn't understand was Mark's comment about the h-principle (that I thought it was yours, sorry). | |
| May 21, 2019 at 20:44 | comment | added | mme | @LuisA.Florit I think one of us misunderstands the other: I am claiming that any vector bundle $E \to B$, where $B$ is a closed manifold, is a smooth submanifold of the closed manifold $S(E \oplus \Bbb R)$. Therefore, the existence of a smooth immersion $S(E \oplus \Bbb R) \to \Bbb R^{2\dim E - a(\dim E)}$ implies the existence of an immersion $E \hookrightarrow \Bbb R^{2\dim E - a(\dim E)}$, by restriction. (Here $\dim E$ means dimension of the total space --- not rank). As far as I can tell, this is what you are asking for: the answers below are about improving this bound. | |
| May 21, 2019 at 20:20 | comment | added | Luis A. Florit | @MarkGrant: I didn't know that Cohen worked for manifolds with boundary. But, as Mike said, the double immerses, so we are fine. | |
| May 21, 2019 at 20:18 | comment | added | Luis A. Florit | @MikeMiller (2) Yes, the normal bundle immerses in the same Euclidean space, but why any vector bundle would? | |
| May 21, 2019 at 20:16 | comment | added | Luis A. Florit | @MikeMiller (1): Aha! That's the trick I was expecting: fibrewise compactification. I did not know that thing was a smooth manifold. | |
| May 21, 2019 at 16:47 | comment | added | mme | @MarkGrant of course, that also works --- it was only obvious to me in retrospect that Cohen's theorem was also true for compact manifolds with boundary; but after all, I was just taking the double of what you said to get rid of the boundary, and whence a proof for the boundary case follows from the proof in the closed case. | |
| May 21, 2019 at 13:33 | answer | added | Qfwfq | timeline score: 1 | |
| May 21, 2019 at 11:41 | answer | added | Ryan Budney | timeline score: 4 | |
| May 21, 2019 at 8:59 | comment | added | Mark Grant | I suspect that you can do better using the h-principle. As vague evidence for this, if you have an immersion of the base $M$ in some Euclidean space, then its normal bundle $E$ immerses in the same Euclidean space (as an immersed tubular neighbourhood). | |
| May 21, 2019 at 8:57 | comment | added | Mark Grant | @MikeMiller: Why not just immerse the unit disk bundle for some Riemannian metric? | |
| May 21, 2019 at 6:11 | history | edited | ThiKu | CC BY-SA 4.0 |
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| May 21, 2019 at 1:00 | comment | added | mme | The total space of a vector bundle is a smooth submanifold of the corresponding fiberwise one-point compactification, aka $S(E \oplus \Bbb R)$, which is a closed manifold of the same dimension $n$ as $E$ so long as the base is closed. Since the larger space immerses in $\Bbb R^{2n-a(n)}$, so does the smaller space. | |
| May 20, 2019 at 21:49 | history | edited | Luis A. Florit | CC BY-SA 4.0 |
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| May 20, 2019 at 21:08 | comment | added | Luis A. Florit | @Misha: yes, sorry. | |
| May 20, 2019 at 20:51 | comment | added | Misha | What do you mean by immersion of a vector bundle? Do you mean an immersion of the total space? | |
| May 20, 2019 at 20:12 | history | edited | András Bátkai | CC BY-SA 4.0 |
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| May 20, 2019 at 20:05 | review | First posts | |||
| May 20, 2019 at 20:12 | |||||
| May 20, 2019 at 20:04 | history | asked | Luis A. Florit | CC BY-SA 4.0 |