TITLE: What is known about the discrete group cohomology $H^2(\mathrm{SL}_2(\mathbb C), \mathbb C^\times)$?
QUESTION [5 upvotes]: The cohomology ring of $\mathrm{SL}_2(\mathbb C)$ as a topological group is straightforward (it's generated by a Chern class), but what is known in the discrete case? I'm particularly interested in $H^2$ with coefficients in $\mathbb C^\times = \mathbb C \setminus \{0\}$, especially if there are explicit formulas for the classes.

REPLY [8 votes]: There is a paper by Milnor called On the homology of Lie groups made discrete which contains many results on this kind of problem, as well as references to related literature.  However, I don't think that it directly answers your question.