Explicit sharbly cycles at the virtual cohomological dimension for SLn(Z)
Abstract
Denote the virtual cohomological dimension of SLn(Z) by t=n(n-1)/2. Let St denote the Steinberg module of SLn(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 H0(SLn(Z), Q) is isomorphic to Ht(SLn(Z),St). We find an explicit generator of Ht(SLn(Z),St) in terms of sharbly cycles and cosharbly cocycles. These methods may extend to other degrees of cohomology of SLn(Z).
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