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, H^n (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.