On the homology of special unitary groups over polynomial rings
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908307397017600 |
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| author | Bravo, Claudio |
| author_facet | Bravo, Claudio |
| contents | In this work, we answer the homotopy invariance question for the ''smallest'' non-isotrivial group-scheme over $\mathbb{P}^1$, obtaining a result, which is not contained in previous works due to Knudson and Wendt. More explicitly, let $\mathcal{G}=\mathrm{SU}_{3,\mathbb{P}^1}$ be the (non-isotrivial) non-split group-scheme over $\mathbb{P}^1$ defined from the standard (isotropic) hermitian form in three variables. In this article, we prove that there exists a natural homomorphism $\mathrm{PGL}_2(F) \to \mathcal{G}(F[t])$ that induces isomorphisms $H_*(\mathrm{PGL}_2(F), \mathbb{Z}) \to H_*(\mathcal{G}(F[t]), \mathbb{Z})$. Then we study the rational homology of $\mathcal{G}(F[t,t^{-1}])$, by previously describing suitable fundamental domains for certain arithmetic subgroups of $\mathcal{G}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06233 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the homology of special unitary groups over polynomial rings Bravo, Claudio K-Theory and Homology Group Theory Number Theory primary 20G10, 20G30, 20E08, secondary 11E57, 20F65 In this work, we answer the homotopy invariance question for the ''smallest'' non-isotrivial group-scheme over $\mathbb{P}^1$, obtaining a result, which is not contained in previous works due to Knudson and Wendt. More explicitly, let $\mathcal{G}=\mathrm{SU}_{3,\mathbb{P}^1}$ be the (non-isotrivial) non-split group-scheme over $\mathbb{P}^1$ defined from the standard (isotropic) hermitian form in three variables. In this article, we prove that there exists a natural homomorphism $\mathrm{PGL}_2(F) \to \mathcal{G}(F[t])$ that induces isomorphisms $H_*(\mathrm{PGL}_2(F), \mathbb{Z}) \to H_*(\mathcal{G}(F[t]), \mathbb{Z})$. Then we study the rational homology of $\mathcal{G}(F[t,t^{-1}])$, by previously describing suitable fundamental domains for certain arithmetic subgroups of $\mathcal{G}$. |
| title | On the homology of special unitary groups over polynomial rings |
| topic | K-Theory and Homology Group Theory Number Theory primary 20G10, 20G30, 20E08, secondary 11E57, 20F65 |
| url | https://arxiv.org/abs/2504.06233 |