Timeline for Is $TS^n$ diffeomorphic to an open subset of $\mathbb{R}^{2n}$
Current License: CC BY-SA 3.0
14 events
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| Jun 16, 2017 at 23:13 | comment | added | John Klein | @MikeMiller I don't have complete argument, but here's an idea: the total space is of the form $(S^4 \vee S^5) \cup_f D^9$. The attaching map $f: S^8 \to S^4 \vee S^5$ has 3 components: $\pi_8(S^4) + \pi_8(S^5) + \pi_8(S^8)$ where the second component is trivial because the bundle has a section. It's enough to show that the first component is non-trivial after suspending once into $\pi_9(S^5) \cong \Bbb Z/2$. I suspect that the unstable $J$-homomorphism $\pi_4(SO(5)) \to \pi_9(S^5)$ is an isomorphism. If true, the result follows as the tangent bundle of $S^5$ is non-trivial. | |
| Jun 16, 2017 at 22:56 | answer | added | mme | timeline score: 3 | |
| Jun 16, 2017 at 22:30 | answer | added | PVAL | timeline score: 3 | |
| Jun 16, 2017 at 22:01 | history | edited | YCor |
edited tags
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| Jun 16, 2017 at 21:57 | comment | added | mme | @JohnKlein Do you know a proof that $T^1 S^5$ is not homotopy equivalent to $S^4 \times S^5$? I'd like to try to use whatever invariant distinguishes these as an obstruction somehow. | |
| Jun 16, 2017 at 21:56 | comment | added | mme | @YCor Right, though now you probably want to phrase intersection numbers / the rest of the argument in terms of compactly supported cohomology. | |
| Jun 16, 2017 at 21:53 | comment | added | YCor | @MikeMiller I guess this argument works with "homeomorphic"? | |
| S Jun 16, 2017 at 21:36 | history | edited | Qfwfq | CC BY-SA 3.0 |
grammar and a typo
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| S Jun 16, 2017 at 21:36 | history | suggested | étale-cohomology | CC BY-SA 3.0 |
grammar and a typo
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| Jun 16, 2017 at 21:25 | review | Suggested edits | |||
| S Jun 16, 2017 at 21:36 | |||||
| Jun 16, 2017 at 21:14 | comment | added | John Klein | @MikeMiller Right! | |
| Jun 16, 2017 at 21:10 | comment | added | mme | @JohnKlein It's simpler than that; such a sphere in $\Bbb R^{2n}$ would have nontrivial self-intersection. It's not possible for any even $n$. | |
| Jun 16, 2017 at 21:05 | comment | added | John Klein | It's definitely not true in the case $n=2$. The unit tangent sphere bundle in this case has total space $SO(3) \cong \Bbb RP^3$, and it is well-known that $\Bbb RP^3$ doesn't embed in $\Bbb R^4$. | |
| Jun 16, 2017 at 20:50 | history | asked | Ali Taghavi | CC BY-SA 3.0 |