I don't know if this is quite the answer you're looking for, but elements of $H_3(PSL_2(\mathbb{C}),\mathbb{Z})$ 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, https://arxiv.org/pdf/math/0307092.pdf
[2] The volume and Chern-Simons invariant of a representation, Christian K. Zickert, https://arxiv.org/abs/0710.2049