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There's a well-known paper by Milnor, "On the homology of Lie groups made discrete," that discusses the relation between the homology of a Lie group $G$ and the underlying discrete group $G^\delta$. He showed that solvable Lie groups have the same homology with finite coefficients as their underlying discrete groups. Morel recently announced a proof of the Friedlander-Milnor conjecture that implies this is also true for complex algebraic Lie groups.

I've always been curious about this problem, and I've discussed it with a number of other mathematicians. One question that I haven't been able to sort out about it was to what extent the homology of a Lie group with coefficients in an arbitrary abelian group $A$ made discrete is determined, e.g. whether the homology groups being nonzero, or finitely generated, is independent of ZFC. This problem involves aspects of both manifolds and the projective resolution of modules, and hence it might be plausible that this is the case.

Are there any results, positive or negative, about the dependence of (co)homology of $G^\delta$ on the underlying model of set theory?

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This looks like a very interesting question! Perhaps you could include a little more background details how the aspects of manifolds and projective resolutions of modules fit into your question. (It's been a while since I thought about such things and I think there may be other set theorists that feel the same.) – François G. Dorais Jan 28 '10 at 19:39
=\ I deleted my comment right after posting it, because I wasn't really sure that I liked what I'd written, so I deleted it before I saw your response, and now I don't remember what I'd written. – Harry Gindi Jan 28 '10 at 20:47 As far as I understand (I haven't studied the proof) the claim is that the consistency of ZFC implies the consistency of both positive and negative solutions to Whitehead problem. – algori Jan 28 '10 at 21:38
Tyler, are you asking whether the homology of a given fixed Lie group can depend on the set-theoretical background in which it is computed? Or are you asking about whether there are interesting general statements about homology that are independent of ZFC? – Joel David Hamkins Jan 28 '10 at 21:52
Basically, yes; I'm asking if, given a fixed Lie group (as a subgroup of GL_n(R)) and a coefficient group A, whether the homology and cohomology might be independent of ZFC, e.g. depending on the continuum hypothesis. @Francois: I'll try and add more details later. – Tyler Lawson Jan 29 '10 at 0:06

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