On the cohomology of SLn(Z)

Abstract

Denote the virtual cohomological dimension of ${\mathrm{SL}}_n\left(Z\right)$ by ${\nu }_n=n(n-1)/2$. Let St denote the Steinberg module of ${\mathrm{SL}}_n\left(Q\right)$ tensored with $Q$. Let $S{h}_{•}\to St$ denote the sharbly resolution of the Steinberg module. By Borel-Serre duality, $H^{{\nu }_n-i}\left({\mathrm{SL}}_n\right(Z),Q)$ is isomorphic to $H_i\left({\mathrm{SL}}_n\right(Z),St)$. The latter is isomorphic to the sharbly homology $H_i\left({\left(S{h}_{•}\right)}_{{\mathrm{SL}}_n\left(Z\right)}\right)$. We produce nonzero classes in $H_i\left({\mathrm{SL}}_n\right(Z),St)$, for certain small i, in terms of sharbly cycles and cosharbly cocycles.