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

TL;DR

This paper explores the relationship between the third homology of SL_2 over a ring and the refined Bloch group, establishing exact sequences and isomorphisms under certain algebraic conditions.

Contribution

It proves an exact sequence linking third homology groups and the refined Bloch group for universal GE_2-domains, and shows the refined scissors congruence group is isomorphic to a relative homology group.

Findings

Established an exact sequence involving H_3(SL_2(A)) and the refined Bloch group.

Proved the refined scissors congruence group is isomorphic to a relative homology group.

Identified conditions under which these algebraic structures are connected.

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