Explicit sharbly cycles at the virtual cohomological dimension for SL_n(Z)
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913449850699776 |
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| author | Ash, Avner Gunnells, Paul E. McConnell, Mark |
| author_facet | Ash, Avner Gunnells, Paul E. McConnell, Mark |
| contents | 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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_14153 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Explicit sharbly cycles at the virtual cohomological dimension for SL_n(Z) Ash, Avner Gunnells, Paul E. McConnell, Mark Geometric Topology Number Theory Primary 11F75, Secondary 11F67, 20J06, 20E42 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). |
| title | Explicit sharbly cycles at the virtual cohomological dimension for SL_n(Z) |
| topic | Geometric Topology Number Theory Primary 11F75, Secondary 11F67, 20J06, 20E42 |
| url | https://arxiv.org/abs/2402.14153 |