Timeline for Non-abelian Ext functor and non-abelian $H^2$
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Aug 4, 2020 at 15:16 | comment | added | Mikhail Borovoi | Thank you, @DonuArapura, for the example. | |
| Aug 4, 2020 at 15:11 | comment | added | Donu Arapura | Thanks again for the interesting answer. Concerning an example. Let $F$ be a nonabelian free group. Choose a surjection $f:F\to G$, where $G$ is finite and nontrivial. Let $K$ be the kernel. $K$ is again nonabelian and free, so it must have trivial centre. By your answer, the nonabelian $H^2$ consists of a single element. This cannot be neutral, because the sequence can't split, as $F$ is torsion free. | |
| Aug 3, 2020 at 21:12 | comment | added | Mikhail Borovoi | @LSpice: I think that my preprint is not quite relevant to this answer, and so I did not name it on purpose. | |
| Aug 3, 2020 at 21:07 | comment | added | LSpice | Of course it's your right not to name your paper if you don't want, and I apologise for an unwanted edit; but too many old MO comments become less useful when "here"-type links fade, so I'll put the name of "this preprint" here if it's all right: Borovoi - Extending the exact sequence of nonabelian $H^1$, using nonabelian $H^2$ with coefficients in crossed modules. | |
| Aug 3, 2020 at 21:03 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
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| Aug 3, 2020 at 20:31 | history | edited | LSpice | CC BY-SA 4.0 |
Links to references and comment; name of preprint; some proofreading
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| Aug 3, 2020 at 20:25 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
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| Aug 3, 2020 at 20:19 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
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| Aug 3, 2020 at 16:55 | comment | added | Donu Arapura | Let me know if you prefer that I ask these as separate questions, rather than as comments. | |
| Aug 3, 2020 at 16:54 | comment | added | Donu Arapura | This is a very nice answer. I do have a couple of questions. (1) When you say $\eta(E)$ is neutral, does that mean there exists a lift of $b$ to $G\to Aut(K)$, such that $E$ is the semi direct product? So then it is clear, there may be more than one, or perhaps none. (2) What is the relationship to the old results of Eilenberg-Maclane "Cohomology theory in abstract groups. II." Annals (1947)? As I recall, they show that the set of extension classes is either empty (with obstructions in $H^3$) or parameterized by $H^2(G, Z)$, where $Z$ is the centre of $K$. | |
| Aug 3, 2020 at 13:06 | vote | accept | curious math guy | ||
| Aug 3, 2020 at 11:11 | comment | added | Mikhail Borovoi | If you are interested in nonabelian $H^2$ in Galois cohomology, please ask a separate question, and I will answer it when I have time | |
| Aug 3, 2020 at 10:58 | history | answered | Mikhail Borovoi | CC BY-SA 4.0 |