On the cohomology of SLₙ(Z)

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