On the cohomology of SL$_n(\mathbb{Z})$

Abstract

Let St denote the Steinberg module of $SL_n(Q)$ tensored with Q. Let Sh denote the sharbly resolution of St. By Borel-Serre duality, $H^{n(n-1)/2-i}(SL_n(Z),Q)$ is isomorphic to $H_i(SL_n(Z),St)$. The latter is isomorphic to the homology of the $SL_n(Z)$-coinvariants of Sh. We produce nonzero classes in $H_i(SL_n(Z),St)$ for certain small $i$ in terms of sharbly cycles and cosharbly cocycles.