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.
What is known about the discrete group cohomology $H^2(\mathrm{SL}_2(\mathbb C), \mathbb C^\times)$?
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asked Mar 24, 2021 at 14:51
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2$\begingroup$ As a discrete group $H_2(\mathrm{SL}_2(\mathbf{C}))$ has continuum cardinal. Probably this is still true modulo torsion. If so, this $H^2$ is huge (maybe even of cardinal power of the continuum). A full description possibly exists in terms of Kähler differentials. Related: mathoverflow.net/a/63599/14094 $\endgroup$– YCorCommented Mar 24, 2021 at 16:00
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2$\begingroup$ I think with Matsumoto's theorem, you can identify $H_2(SL_2(\mathbb{C}),\mathbb{Z})$ with $K^M_2(\mathbb{C})$, and then you could use universal coefficient formulas to get statements about second cohomology with $\mathbb{C}^\times$-coefficients. Not sure if this is specific enough, but the torsion should be controllable, the uniquely divisible part less so. $\endgroup$– Matthias WendtCommented Mar 24, 2021 at 19:11
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1$\begingroup$ As a follow-up on the link to Milnor K-theory, elements of $H^2({\rm SL}_2(\mathbb{C}),\mathbb{C}^\times)$ should be given by $\mathbb{C}^\times$-valued Steinberg coycles. Those are maps $\mathbb{C}^\times\times\mathbb{C}^\times\to \mathbb{C}^\times:(u,v)\mapsto \{u,v\}$ which are bimultiplicative and satisfy $\{a,1-a\}=1$. That point of view may help with computations, the K-book has more information in this direction in the $K_2$-section. $\endgroup$– Matthias WendtCommented Mar 25, 2021 at 10:46
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$\begingroup$ Thanks! I know very little about K-theory, so your suggested point of view is helpful. $\endgroup$– Calvin McPhail-SnyderCommented Mar 25, 2021 at 14:10
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1 Answer
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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.
answered Mar 24, 2021 at 17:52
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1$\begingroup$ Could you please add some information on the state of art of the Milner conjecture (which I presume is made in this paper) to your answer?! $\endgroup$ Commented Mar 25, 2021 at 13:04
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1$\begingroup$ I don't know the current status. I am not an expert in this area, I merely reported something that I remembered reading a long time ago. $\endgroup$ Commented Mar 25, 2021 at 15:02
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$\begingroup$ MathSciNet will give you a list of all the papers that cite the one that I mentioned, and that will presumably include all papers that make progress on the conjecture. $\endgroup$ Commented Mar 25, 2021 at 15:11