For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense:

Conjecture FM: For a reductive algebraic group $G$ over a separably closed field $k$, the map $BG^\delta \to BG$ induces an isomorphism on $H^*_{\mathrm{et}}(-,\mathbf{Z}/\ell)$ for any $\ell$ invertible on $k$; here $G^\delta = G(k)$ with the discrete topology.

What is the status of FM today? Here is what google tells me:

1) Friedlander and Mislin proved the conjecture for $k = \overline{\mathbf{F}_p}$ in their 1984 paper "Cohomology of classifying spaces of complex Lie groups and related discrete groups."

2) There is a 2011 paper by Morel showing special cases of this result, and establishing a connection with $\mathbf{A}^1$-homotopy theory by proving the so-called "weak Friedlander-Milnor" conjecture.

In particular: is there a single example of a field $k$ of characteristic $0$ where we know FM?


Here is what I know about the status of the question (as of today):

At the moment, there is no field of characteristic 0 for which the Friedlander-Milnor conjecture is known for a semisimple algebraic group. From Matsumoto's theorem, it follows for $H_2$ of semisimple groups, from the work of Suslin it follows in the stable range and for the limit groups $GL_\infty$, $Sp_\infty$ and $O_\infty$, and from the work of Sah it follows for $H_3(SL_2)$. And of course, Friedlander and Mislin proved it for the algebraic closure of a finite field. Those results can be found in the book "Homology of linear groups" by K. Knudson.

As far as I know, and modulo checking some details, what I think Morel establishes in the paper (via a construction of transfers on $\mathbb{A}^1$-classifying spaces of split semisimple linear algebraic groups) is the equivalence of the following statements about change-of-topology maps (whenever they make sense):

  1. weak homotopy invariance, i.e., the canonical map $G(k)\to G(k[\Delta^\bullet])$ induces an isomorphism in mod $\ell$ homology, where $k[\Delta^\bullet]$ is the standard polynomial cosimplicial object.

  2. Milnor's version of the Friedlander-Milnor conjecture, i.e., the canonical change-of-topology map $G^\delta\to G$ induces an isomorphism in mod $\ell$ homology, for $G$ a Lie group.

  3. Friedlander's generalized isomorphism conjecture, i.e., the canonical map $BG(k)\to BG^{\operatorname{et}}_k$ from discrete to the étale classifying space induces an isomorphism in mod $\ell$ homology, for $G$ a linear algebraic group over an algebraically closed field $k$.

The original proof strategy then was to prove homotopy invariance, which however has turned out to be a lot more subtle than expected: Theorem 1.2 in the 2009 paper is wrong, and Theorem 7 in the 2011 paper of Morel is not known at the moment (partly my fault). In fact, a strong form of homotopy invariance does not seem to hold for arbitrary semisimple groups. I wrote a paper on counterexamples to homotopy invariance, which is published as:

M. Wendt. On homotopy invariance for homology of rank two groups. J. Pure Appl. Algebra. 216(10), 2012, pp. 2291--2301.

Even worse, weaker forms of homotopy invariance also seem to generally fail. There is another paper

K. Hutchinson and M. Wendt. On third homology of $SL_2$ and weak homotopy invariance. ArXiv:1307.3069.

In this paper, there are many examples of fields where the natural map $H_3(SL_2(k),\mathbb{Z})\to H_3(BSL_2(k[\Delta^\bullet]),\mathbb{Z})$ fails to be an isomorphism. The counterexamples known at the moment all depend on non-triviality of square classes, so they do not say anything about quadratically closed fields.

The current status is that there is a new announcement on Morel's webpage claiming a proof of weak homotopy invariance for $SL_2$ over quadratically closed fields. This together with the rigidity result in his 2011 paper would then imply the Friedlander-Milnor conjecture for $SL_2$ - both Milnor's form of the conjecture for $SL_2(\mathbb{C})$ and Friedlander's generalized isomorphism conjecture for $SL_2$ over any algebraically closed field. Milnor's form of the conjecture for $SL_2(\mathbb{R})$ would not follow from the announced result as $\mathbb{R}$ is not quadratically closed.


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