A Refined scissors congruence group and the third homology of $\textrm{SL}_2$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911757738442752 |
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| author | Mirzaii, Behrooz Pérez, Elvis Torres |
| author_facet | Mirzaii, Behrooz Pérez, Elvis Torres |
| contents | There is a natural connection between the third homology of $\textrm{SL}_2(A)$ and the refined Bloch group $\mathcal{RB}(A)$ of a commutative ring $A$. In this article we investigate this connection and as the main result we show that if $A$ is a universal $\textrm{GE}_2$-domain such that $-1 \in A^{\times 2}$, then we have the exact sequence $H_3(\textrm{SM}_2(A),\mathbb{Z}) \to H_3(\textrm{SL}_2(A),\mathbb{Z}) \to \mathcal{RB}(A) \to 0$, where $\textrm{SM}_2(A)$ is the group of monomial matrices in $\textrm{SL}_2(A)$. Moreover we show that $\mathcal{RP}_1(A)$, the refined scissors congruence group of $A$, naturally is isomorph with the relative homology group $H_3(\textrm{SL}_2(A), \textrm{SM}_2(A),\mathbb{Z})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_08872 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Refined scissors congruence group and the third homology of $\textrm{SL}_2$ Mirzaii, Behrooz Pérez, Elvis Torres K-Theory and Homology There is a natural connection between the third homology of $\textrm{SL}_2(A)$ and the refined Bloch group $\mathcal{RB}(A)$ of a commutative ring $A$. In this article we investigate this connection and as the main result we show that if $A$ is a universal $\textrm{GE}_2$-domain such that $-1 \in A^{\times 2}$, then we have the exact sequence $H_3(\textrm{SM}_2(A),\mathbb{Z}) \to H_3(\textrm{SL}_2(A),\mathbb{Z}) \to \mathcal{RB}(A) \to 0$, where $\textrm{SM}_2(A)$ is the group of monomial matrices in $\textrm{SL}_2(A)$. Moreover we show that $\mathcal{RP}_1(A)$, the refined scissors congruence group of $A$, naturally is isomorph with the relative homology group $H_3(\textrm{SL}_2(A), \textrm{SM}_2(A),\mathbb{Z})$. |
| title | A Refined scissors congruence group and the third homology of $\textrm{SL}_2$ |
| topic | K-Theory and Homology |
| url | https://arxiv.org/abs/2307.08872 |