Timeline for Non-abelian representations and their induced abelian representations
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Aug 1, 2020 at 18:06 | history | edited | YCor | CC BY-SA 4.0 |
fixed typo
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| Aug 1, 2020 at 18:05 | comment | added | Mikhail Borovoi | You can find a down-to-earth treatment of nonabelian $H^2$ in my paper :M. Borovoi, Abelianization of the second nonabelian Galois cohomology. Duke Math. J. 72 (1993), 217-239, and also in the references therein (especially in Springer's paper) and in papers referring to my paper and to Springer. | |
| Aug 1, 2020 at 18:01 | comment | added | Mikhail Borovoi | There may be more than one neutral class in nonabelian $H^2$. | |
| Aug 1, 2020 at 18:00 | comment | added | Mikhail Borovoi | You can lift $\rho$ if and only if the cohomology class is neutral. | |
| Aug 1, 2020 at 17:57 | comment | added | Mikhail Borovoi | If yes, then you get a cohomology class in $$H^2(G,{\rm ker}[{\rm Out}(H)\to {\rm Aut}(H^{\rm ab})]).$$ | |
| Aug 1, 2020 at 17:53 | comment | added | Mikhail Borovoi | First, you should require that $\rho$ lands in the image of ${\rm Out}(H)$. | |
| Aug 1, 2020 at 17:27 | history | edited | curious math guy | CC BY-SA 4.0 |
added 21 characters in body
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| Aug 1, 2020 at 17:25 | comment | added | curious math guy | You are both absolutely right! We do have a surjection if $H$ is a free group however, which is what I had in mind. I'll change the question accordingly. | |
| Aug 1, 2020 at 17:22 | comment | added | Will Sawin | @MarkWildon I don't think your example is the right one, because $H^{\mathrm{ab}} =C_2$. However, $C_5 \rtimes C_4$ works, because its abelianization $C_4$ has an automorphism, which does not lift to $C_5 \rtimes C_4$. | |
| Aug 1, 2020 at 17:20 | comment | added | Mark Wildon | The map $\mathrm{Out}(H) \rightarrow \mathrm{Aut}(H^{\mathrm{ab}})$ is not in general surjective. I had an incorrect example and I see Will Sawin has now posted a correct one. | |
| Aug 1, 2020 at 17:20 | comment | added | Will Sawin | I am pretty sure this is not a surjection in general. Why should it be? | |
| Aug 1, 2020 at 17:19 | history | edited | LSpice | CC BY-SA 4.0 |
DeclareMathOperator's
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| Aug 1, 2020 at 17:15 | history | asked | curious math guy | CC BY-SA 4.0 |