On the cohomology of SL$_n(\mathbb{Z})$

Avner Ash

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arXiv · 2402.08840

On the cohomology of SL$_n(\mathbb{Z})$

Avner Ash

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Abstract

Let St denote the Steinberg module of $SL_n(Q)$ tensored with Q. Let Sh denote the sharbly resolution of St. By Borel-Serre duality, $H^{n(n-1)/2-i}(SL_n(Z),Q)$ is isomorphic to $H_i(SL_n(Z),St)$. The latter is isomorphic to the homology of the $SL_n(Z)$-coinvariants of Sh. We produce nonzero classes in $H_i(SL_n(Z),St)$ for certain small $i$ in terms of sharbly cycles and cosharbly cocycles.

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Avner Ash. 2024-02-13. On the cohomology of SL$_n(\mathbb{Z})$. https://arxiv.org/abs/2402.08840

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