Timeline for Homology of Lie groups
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Aug 11, 2014 at 19:15 | comment | added | Matthias Wendt | @user, I would like to clarify: at the moment, the conjecture is not known for $SL(2,\mathbb{C})$. It is known for homology in degrees $\leq 3$ by the work of Sah, and from Morel's CDM paper, we have the equivalences of the formulations of etale and usual topological cohomology with weak homotopy invariance. However, weak homotopy invariance is at the moment only announced by Morel, and no details of proof are available yet. So I would rather say that the conjecture is not known at the moment, we will have to wait until a proof is available. | |
| Aug 11, 2014 at 15:55 | comment | added | user | @MatthiasWendt, Thank you for the clarification!! So if I understand, the only known case where the conjecture is known to be true is for $SL(2, \mathbb{C})$ and in this case the formulation with Etale cohomology and ordinary cohomology are the same (over complex numbers with finite coefficient). I will check the book. Once again thank you very much! | |
| Aug 11, 2014 at 15:33 | comment | added | Matthias Wendt | Actually, the question was answered in the case of characteristic 0. What is known about the Friedlander-Milnor conjecture can all be found in Chapter 5 of Knudson's book "Homology of linear groups". Morel's current announcement of weak homotopy invariance for $SL_2$ would imply Milnor's conjecture on the homology of $SL_2(\mathbb{C})$ but not $SL_2(\mathbb{R})$. I have reformulated the answer linked in Eric Wofsey's comment to point out more clearly that it applies to the complex Lie groups as well. | |
| S Aug 8, 2014 at 9:12 | history | suggested | ThiKu | CC BY-SA 3.0 |
Typos corrected
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| Aug 8, 2014 at 9:10 | review | Suggested edits | |||
| S Aug 8, 2014 at 9:12 | |||||
| Aug 7, 2014 at 7:46 | comment | added | user | Thanks Eric, If I'm not wrong the question in the link was not answered in the case of (real/ complex) Lie groups i.e. over a field of characteristic 0. And that is exactly what I'm asking here. | |
| Aug 6, 2014 at 22:51 | comment | added | Eric Wofsey | See mathoverflow.net/questions/163250/… for the status for algebraic groups as of a few months ago. | |
| Aug 6, 2014 at 18:44 | history | asked | user | CC BY-SA 3.0 |