The third homology of $SL_{2}$ of real quadratically closed fields

TL;DR

This paper provides a new proof linking the third homology of special linear groups over real quadratically closed fields to algebraic K-theory, using the refined Bloch group theory.

Contribution

It offers a novel, concise proof of isomorphisms connecting third homology groups with algebraic K-theory for real quadratically closed fields.

Findings

Established isomorphisms between $H_3(SL_2(R))$ and $K_3^{ind}(R)$

Extended results to $H_3(SL_n(R))$ for $n \\geq 3$

Utilized refined Bloch group theory for proofs

Abstract

For a real closed field $\mathbf{R}$, we use the theory of the refined Bloch group to give a new short proof of the isomorphisms $H_{3}(SL_{2}(\mathbf{R}),\mathbb{Z})\cong K_{3}^{\mathrm{ind}}(\mathbf{R})$ and $H_{3}(SL_{n}(\mathbf{R}),\mathbb{Z})\cong H_{3}(SL_{2}(\mathbf{R}),\mathbb{Z})\oplus K_{3}^{M}(\mathbf{R})^{0}$ for $n\geq3$.