$\DeclareMathOperator\SL{SL}$I am reading some papers about the third homology of linear groups. In particular for the $\SL_{2}$ over a field. Why is it important to study these homologies?

I have been trying to read some papers and books (for example, Knudson - *Homology of linear groups*, Hutchinson - *A Bloch–Wigner complex for $\operatorname{SL}_2$*, Hutchinson - *A refined Bloch group and the third homology of $\operatorname{SL}_2$ of a field*, Mirzaii - *Third homology of general linear groups*) and they only mention homology stabilization theorems and the Theorem of Hurewicz but they did not go deep on this.

I only found on the paper *Scissor Congruences II* by Johan L. Dupont and Chih Han Sah that it explored the properties of the Bloch group due to its connection to their study of the Third Hilbert problem in Hyperbolic 3-space and for the case $F=\mathbb{C}$ they proved the *Bloch–Wigner Theorem* that states that there is a short exact sequence

$$0\rightarrow\mu_{\mathbb{C}}\rightarrow H_{3}(\SL_{2}(\mathbb{C}),\mathbb{Z})\rightarrow \mathcal{B}(\mathbb{C}) \rightarrow 0,$$

where $\mathcal{B}(F)$ is the Bloch group of a field $F$.

Also when $F=\mathbb{C}$ (and more generally $F^{\times}=(F^{\times})^{2}$) it is proved that $K_{3}^{\operatorname{Ind}}(\mathbb{C})=H_{3}(\SL_{2}(\mathbb{C}),\mathbb{Z})$.

Andrei Suslin proved for any infinite field $F$ in his paper *$K_{3}$ of a field and the Bloch group* that there is a natural exact sequence

$$0\rightarrow \widetilde{\operatorname{Tor}_{1}^{\mathbb{Z}}(\mu_{F},\mu_{F})}\rightarrow K_{3}^{\operatorname{Ind}}(F)\rightarrow \mathcal{B}(F)\rightarrow0$$

where $\widetilde{\operatorname{Tor}_{1}^{\mathbb{Z}}(\mu_{F},\mu_{F})}$ is the unique nontrivial extension of $\operatorname{Tor}_{1}^{\mathbb{Z}}(\mu_{F},\mu_{F})$ by $\mathbb{Z}/2$ when $\operatorname{Char}(F)\ne2$ (and $\widetilde{\operatorname{Tor}_{1}^{\mathbb{Z}}(\mu_{F},\mu_{F})}=\operatorname{Tor}_{1}^{\mathbb{Z}}(\mu_{F},\mu_{F})$ if $\operatorname{Char}(F)=2$).

So we have a relationship through $K$-theory and the Bloch group via the homology of linear groups. But I think there is more things for the studies of the homology of the linear groups.

I would appreciate any hint or indication of where I can read about it.

Geometry of configurations, polylogarithms and motivic cohomology, Section 2. $\endgroup$6more comments