Explicit sharbly cycles at the virtual cohomological dimension for SL_n(Z)

Abstract: Denote the virtual cohomological dimension of SL_n(Z) by t=n(n-1)/2. Let St denote the Steinberg module of SL_n(Q) tensored with Q. Let Sh_* denote the sharbly resolution of the Steinberg module St. By Borel-Serre duality, the one-dimensional Q-vector space H^0(SL_n(Z), Q) is isomorphic to H_t(SL_n(Z),St). We find an explicit generator of H_t(SL_n(Z),St) in terms of sharbly cycles and cosharbly cocycles. These methods may extend to other degrees of cohomology of SL_n(Z).