Importance of third homology of $\operatorname{SL}_{2}$ over a field

Question

$\DeclareMathOperator\SL{SL}$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.

Answer 1

I don't know if this is quite the answer you're looking for, but elements of $H_3(\operatorname{PSL}_2(\mathbb{C}),\mathbb{Z})$ can be identified as fundamental classes of finite-volume hyperbolic $3$-manifolds. Neumann [1] showed how to construct the Cheeger–Chern–Simons class as a map $\mathfrak{c} : H_3(\operatorname{PSL}_2(\mathbb{C}),\mathbb{Z}) \to \mathbb{C}/4\pi^2\mathbb{Z}$. Evaluating $i\mathfrak{c}([M])$ on a manifold computes the complex volume, whose real part is the volume of the hyperbolic metric and whose imaginary part is the Chern–Simons invariant.

To effectively apply this formula (you need some extra combinatorial data called a flattening) it is helpful to use a particular coordinate system on the space of $\operatorname{PSL}_2(\mathbb{C})$-bundles over the manifold, the Ptolemy coordinates [2]. I think that these were originally discovered when thinking about things from the perspective of group homology: see [2, Sections 2 and 3].

[1] Extended Bloch group and the Cheeger–Chern–Simons class, Walter Neumann.

[2] The volume and Chern–Simons invariant of a representation, Christian K. Zickert.