Explicit sharbly cycles at the virtual cohomological dimension for \(\textrm{SL}_n(\mathbb {Z})\)

Abstract

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