# Homology of Lie groups

Let $G$ be a Lie group and $G^{\delta}$ the underlying group (with discrete topology). Obviously, we have a continuous map of groups $i:G^{\delta}\rightarrow G$ which induces a map between classifying spaces $BG^{\delta}\rightarrow BG$. My question is about the induced map in homology $i_{\ast}:H_{\ast}(BG^{\delta}, \mathbb{F}_{p})\rightarrow H_{\ast}(BG, \mathbb{F}_{p}).$ If I understand well, the map $i_{\ast}$ is conjectured to be an isomorphism (for any finite field $\mathbb{F}_{p}$).

I have read a paper by Milnor about this subject and I have heard about Morel's results in this direction (unfortunately I cannot understand Morel's Talk on the subject). So I'm wondering about the current statement and results about the conjecture (if I stated it correctly).

• See mathoverflow.net/questions/163250/… for the status for algebraic groups as of a few months ago. – Eric Wofsey Aug 6 '14 at 22:51
• Thanks Eric, If I'm not wrong the question in the link was not answered in the case of (real/ complex) Lie groups i.e. over a field of characteristic 0. And that is exactly what I'm asking here. – user Aug 7 '14 at 7:46
• Actually, the question was answered in the case of characteristic 0. What is known about the Friedlander-Milnor conjecture can all be found in Chapter 5 of Knudson's book "Homology of linear groups". Morel's current announcement of weak homotopy invariance for $SL_2$ would imply Milnor's conjecture on the homology of $SL_2(\mathbb{C})$ but not $SL_2(\mathbb{R})$. I have reformulated the answer linked in Eric Wofsey's comment to point out more clearly that it applies to the complex Lie groups as well. – Matthias Wendt Aug 11 '14 at 15:33
• @MatthiasWendt, Thank you for the clarification!! So if I understand, the only known case where the conjecture is known to be true is for $SL(2, \mathbb{C})$ and in this case the formulation with Etale cohomology and ordinary cohomology are the same (over complex numbers with finite coefficient). I will check the book. Once again thank you very much! – user Aug 11 '14 at 15:55
• @user, I would like to clarify: at the moment, the conjecture is not known for $SL(2,\mathbb{C})$. It is known for homology in degrees $\leq 3$ by the work of Sah, and from Morel's CDM paper, we have the equivalences of the formulations of etale and usual topological cohomology with weak homotopy invariance. However, weak homotopy invariance is at the moment only announced by Morel, and no details of proof are available yet. So I would rather say that the conjecture is not known at the moment, we will have to wait until a proof is available. – Matthias Wendt Aug 11 '14 at 19:15