The third homology of $S{L}_2$ of real quadratically closed fields

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 $H_3(S{L}_2(\mathbf{R}),\mathbb{Z})\cong K_3^{\mathrm{ind}}(\mathbf{R})$ and $H_3(S{L}_n(\mathbf{R}),\mathbb{Z})\cong H_3(S{L}_2(\mathbf{R}),\mathbb{Z})\oplus K_3^M(\mathbf{R}{)}^0$ for $n\ge 3$.

9 pages