Timeline for Cohomology of $\operatorname{SL}_n(\mathbb Z)$ with coefficients
Current License: CC BY-SA 4.0
9 events
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| Aug 7, 2021 at 7:54 | comment | added | Dan Petersen | Perhaps the following reference can be useful: "The cohomology of the braid group B_3 and of SL_2(Z) with coefficients in a geometric representation", Filippo Callegaro, Fred Cohen, Mario Salvetti. | |
| Aug 7, 2021 at 7:18 | history | edited | YCor |
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| Aug 6, 2021 at 20:26 | comment | added | Mariano Suárez-Álvarez | Indeed, my coefficients are $\mathbb Z^n\otimes_{\mathbb{Z}}(-)$ applied to $K^\times$ or~$\mu$. Taking this outside of the Ext will see the torsion, but the problem is probably with $\mathbb Z^n$. | |
| Aug 6, 2021 at 20:25 | comment | added | tj_ | The Mayer-Vietoris sequence holds for all coefficients (cf. Brown VII.9) and the cohomology of cyclic groups is easy. So I would guess that you can get some information from MV. | |
| Aug 6, 2021 at 20:22 | comment | added | Tom Goodwillie | Whatever the answer is, I would not think it would have much to do with $K$. What is known about the cohomology in low degrees of $SL_n(\mathbb Z)$ with $\mathbb Z^n$ coefficients? | |
| Aug 6, 2021 at 20:00 | comment | added | Mariano Suárez-Álvarez | That does work great if the coefficients have trivial action. You find, for example, that $H^2(\operatorname{SL}_2(\mathbb Z),\mathbb Z)$ is $\mathbb Z/12\mathbb Z$ and $0$ in odd degrees, so you can compute $H^2$ with values in finite cyclic groups with trivial action. | |
| Aug 6, 2021 at 19:56 | comment | added | tj_ | Can't you exploit that SL_2(Z) is the amalgam of C_4 and C_6 along C_2 and apply the Mayer-Vietoris sequence? | |
| Aug 6, 2021 at 19:38 | history | edited | Mariano Suárez-Álvarez | CC BY-SA 4.0 |
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| Aug 6, 2021 at 19:26 | history | asked | Mariano Suárez-Álvarez | CC BY-SA 4.0 |