Timeline for Relation between trivial tangent bundle $\Leftrightarrow$ certain characteristic classes of tangent bundle vanish [closed]
Current License: CC BY-SA 4.0
20 events
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| Jan 28 at 14:31 | comment | added | Michael Albanese | @zeta: If you ask the question in your last comment on MSE (while abiding by the rules of asking questions there), I could write up an answer some time this week. | |
| Jan 3 at 15:43 | history | edited | zeta | CC BY-SA 4.0 |
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| Dec 29, 2023 at 23:25 | history | edited | zeta | CC BY-SA 4.0 |
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| Dec 18, 2023 at 12:53 | history | closed |
Ryan Budney Max Horn Daniele Tampieri abx Mikhail Katz |
Not suitable for this site | |
| Dec 15, 2023 at 6:07 | comment | added | zeta | I changed the title for those absent-minded readers who mistakenly think the question was re-appeared: "trivial tangent bundle $\Leftrightarrow$ $w_1 =0, w_2 =0$, and $p_1/2=$0" - it is a new question. I don't know the precise answer. | |
| Dec 15, 2023 at 6:06 | history | edited | zeta | CC BY-SA 4.0 |
I changed the title for those absent-minded readers who mistakenly think the question was re-appeared: "trivial tangent bundle $\Leftrightarrow$ $w_1 =0, w_2 =0$, and $p_1/2=$0" - it is a new question
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| Dec 15, 2023 at 5:30 | history | edited | zeta | CC BY-SA 4.0 |
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| Dec 15, 2023 at 2:35 | history | edited | zeta | CC BY-SA 4.0 |
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| Dec 15, 2023 at 2:11 | comment | added | zeta | @John Rognes; not quite, one is about the structure. the other is about the bordism group. I don't think the structure and the bordism group of the structure are the same thing? (please do not vote down or close simply based on this misunderstanding.) | |
| Dec 14, 2023 at 16:48 | review | Close votes | |||
| Dec 18, 2023 at 12:53 | |||||
| Dec 14, 2023 at 14:25 | comment | added | John Rognes | Repeat of mathoverflow.net/questions/458940/… ? | |
| Dec 14, 2023 at 13:24 | answer | added | Arye Deutsch | timeline score: 4 | |
| Dec 14, 2023 at 13:22 | comment | added | zeta | But we see Whitehead tower comes to trivializes $w_1$,$w_2$, $p_1/2$, $p_2/6$, so that 1d $w_1$ affects 0d bordism, 2d $w_2$ affects 1d bordism, 4d $p_1$ affects the 3d bordism, and 8d $p_2$ affects the 7d bordism. Thus dimensions: 0, 1, 3, 7 seem special in the bordism group? Please comment - Thanks! | |
| Dec 14, 2023 at 13:17 | comment | added | zeta | @Ryan Budney, would you kindly write an answer for comparing framing structure and string structure in d=3, other d<7 and d>=7? It seems intriguing. (Whitehead tower did not tell this full information.) | |
| Dec 14, 2023 at 12:35 | comment | added | zeta | could we say that Trivial vector bundles admit Framings which also include the String structure. But not the other way around, the String structure does Not imply Framing? However, in 3d, is that Framing if and only if String structure? They imply each other? How about when the dimensions d < 7 when all the Frame bordism group = String bordism group? Does that mean d < 7, Framing if and only if String structure? Could you clarify this and maybe you can write an answer? | |
| Dec 14, 2023 at 12:32 | comment | added | zeta | >> "There are various intermediate steps between an arbitrary vector bundle and a trivial vector bundle. Trivial vector bundles admit framings." Reply: yes thanks, I already know what you wrote. But could we say that Trivial vector bundles admit Framings which also include the String structure. But not the other way around, the String structure does Not imply Framing? | |
| Dec 14, 2023 at 1:25 | comment | added | Ryan Budney | I failed to emphasize it, but the point of Whitehead towers is that you kill low-dimensional homotopy groups of a space without affecting the higher homotopy groups. As far as I know these spaces don't all have names, but the common one for the orthogonal group is $\cdots \to spin_n \to SO_n \to O_n$. These maps kill $\pi_1$ and $\pi_0$ respectively. | |
| Dec 14, 2023 at 1:09 | comment | added | Ryan Budney | I think if you read Milnor's paper on spin structures you might get the gist of this general process. Perhaps also look at Whitehead towers in Hatcher's book. | |
| Dec 14, 2023 at 1:08 | comment | added | Ryan Budney | There are various intermediate steps between an arbitrary vector bundle and a trivial vector bundle. Trivial vector bundles admit framings. The process of crawling up the Whitehead tower of the orthogonal group is what you are interested in here. The first step of the Whitehead tower is to find a map to $O_n$ that is the fiber of a fibration such that the fiber is connected. That's $SO_n \to O_n$. In the language of Whitehead towers we have killed $\pi_0$. The next step would be to kill $\pi_1$. That's what spin structures are about. The next step is to kill $\pi_2$. . . | |
| Dec 14, 2023 at 0:49 | history | asked | zeta | CC BY-SA 4.0 |