A Refined scissors congruence group and the third homology of $\textrm{SL}_2$

Behrooz Mirzaii; Elvis Torres Pérez

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arXiv · 2307.08872

A Refined scissors congruence group and the third homology of $\textrm{SL}_2$

Behrooz Mirzaii·Elvis Torres Pérez

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Abstract

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 $H_3(\textrm{SL}_2(A), \textrm{SM}_2(A),\mathbb{Z})$.

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BibTeX RIS

Behrooz Mirzaii, Elvis Torres Pérez. 2023-07-17. A Refined scissors congruence group and the third homology of $\textrm{SL}_2$. https://arxiv.org/abs/2307.08872

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