  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.
