Explicit sharbly cycles at the virtual cohomological dimension for SL_n(Z)

Avner Ash; Paul E. Gunnells; Mark McConnell

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arXiv · 2402.14153

Explicit sharbly cycles at the virtual cohomological dimension for SL_n(Z)

Avner Ash·Paul E. Gunnells·Mark McConnell

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Abstract

Denote the virtual cohomological dimension of SL_n(Z) by t=n(n-1)/2. Let St denote the Steinberg module of SL_n(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 H^0(SL_n(Z), Q) is isomorphic to H_t(SL_n(Z),St). We find an explicit generator of H_t(SL_n(Z),St) in terms of sharbly cycles and cosharbly cocycles. These methods may extend to other degrees of cohomology of SL_n(Z).

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BibTeX RIS

Avner Ash, Paul E. Gunnells, Mark McConnell. 2024-02-21. Explicit sharbly cycles at the virtual cohomological dimension for SL_n(Z). https://arxiv.org/abs/2402.14153

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