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Let $G$ and $H$ be compact Lie groups, Consider $Ext_{Lie}(G,H)$ the set of isomorphism of extensions of Lie groups:

$$ 1\rightarrow G\rightarrow M\rightarrow H\rightarrow 1 $$

There exists a forgetful functor $g:Ext_{Lie}(G,H)\rightarrow Ext_{Gpr}(G,H)$ obtained by using the forgetful functor $Lie\rightarrow Gpr$ from the category of Lie groups to the category of groups.

Recently Extensions of compact Lie groups , the following question has been asked: is $g$ injective ?

Let $H$ be a Lie group and $H^{\delta}$ the underlying group of $H$ endowed with the discrete topology, the canonical embedding $i:H^{\delta}\rightarrow H$ induces a morphism $f:BH^{\delta}\rightarrow BH$. In his paper entitled the homology of Lie groups make discrete, Milnor has conjectured that $f$ induces an isomorphism between the homology and cohomology with finite coefficients of $BH^{\delta}$ and $BH$.

This enables to give a partial answer to the previous question if $G$ is finite and commuative.

Question: Is the conjecture of Milnor has been already proved ? Or in what cases it is known to be true ?

Milnor, J. On the homology of Lie groups made discrete.

Comment. Math. Helvetici 58 (1983) 72-85. (eudml, authors' website, DOI: 10.1007/BF02564625).

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  • $\begingroup$ Beware that the usual, closely related Ext in homological algebra is denoted in the reverse convention (the OP's $Ext(G,H)$, as defined in the linked question, would rather usually be denoted $Ext(H,G)$). $\endgroup$
    – YCor
    Commented Mar 9, 2019 at 13:04
  • $\begingroup$ I'm pretty sure that Milnor's conjecture is still open. However, I'm unable to point to any survey or account of progress made at any point on this conjecture since it was asserted. $\endgroup$
    – YCor
    Commented Mar 9, 2019 at 13:08
  • $\begingroup$ See this question : mathoverflow.net/a/166245 $\endgroup$
    – D.-C. Cisinski
    Commented Mar 9, 2019 at 16:54
  • $\begingroup$ @Denis-CharlesCisinski Thank you for your link. The FM Milnor conjectures is the MIlnor conjecture in the Etale topos. It is true that their formulation are similar. I would look at the reference pointed in the link to see if there exist a relation between the techniques used by Milnor in his paper and the techniques used in algebraic geometry. $\endgroup$
    – Tsemo Aristide
    Commented Mar 9, 2019 at 17:23
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    $\begingroup$ If we restrict to algebraic groups over the field of complex numbers, the two conjectures are equivalent because étale cohomology and singular cohomology agree with finite coefficients for complex algebraic varieties. $\endgroup$
    – D.-C. Cisinski
    Commented Mar 9, 2019 at 17:31

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