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Timeline for H_3 of SL(n,Z) and SL(n,F_p)

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Jun 29 at 15:15 comment added Theo Johnson-Freyd @NoahB I don't know. If you had asked about $SL(n,\mathbb{F}_q)$, then Quillen's methods tell you everything, if I recall correctly, by comparing $H_*BSL(n,q)$ to $H_*BSU(n)$. I seem to believe that there is a comparison map between $K_*$ and $H_* BSL$, but this comparison gets worse as the degree increases, something like how the Hurewicz comparison between $H_*$ and $\pi_*$ starts out very accurate, and then gets worse and worse. Maybe someone with actual knowledge will chime in.
Jun 28 at 23:27 comment added Noah B Thank you for the great answer! Do you know of any results of this type that would help compute higher dimensional homology groups of $SL(n,\mathbb{Z})$, for $n$ in the stable range?
Jan 2, 2017 at 7:38 history edited მამუკა ჯიბლაძე CC BY-SA 3.0
As suggested by the author of the answer I added some details.
Jan 2, 2017 at 2:23 vote accept Sarah
S Jan 1, 2017 at 22:40 history answered Theo Johnson-Freyd CC BY-SA 3.0
S Jan 1, 2017 at 22:40 history made wiki Post Made Community Wiki by Theo Johnson-Freyd