The codimension-one cohomology of SL_n Z

Abstract

We prove that H^{d-1}(SL_n Z; Q) = 0, where d = n-choose-2 is the cohomological dimension of SL_n Z, and similarly for GL_n Z. We also prove analogous vanishing theorems for cohomology with coefficients in a rational representation of the algebraic group GL_n. These theorems are derived from a presentation of the Steinberg module for SL_n Z whose generators are integral apartment classes, generalizing Manin's presentation for the Steinberg module of SL_2 Z. This presentation was originally constructed by Bykovskii. We give a new topological proof of it.

26 pages. v2: final version, to appear in Geometry and Topology

References in corpus (1)

• A stability conjecture for the unstable cohomology of SL_n Z, mapping class groups, and Aut(F_n)

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