Is there any relation between étale homotopy theory (Grothendieck-Galois theory) and the inverse Galois problem?...I mean...in classical homotopy theory, every finite group $G$ realizes as a "Galois group"...that is...one has the universal covering $p_G:EG\rightarrow BG$ for which $$Gal(p_G):=Deck(p_G)\simeq\pi_1(BG)\simeq G$$ Hence, my question would be: Is the classifying algebraic stack $\mathscr{B}G$ a sort of Eilenberg-Maclane space $K(G,1)$ in étale homotopy theory?...can this say something about the inverse Galois problem?


The inverse Galois problem is a statement about the étale fundamental group of a specific scheme. So constructing a space whose fundamental group is $G$ does not tell you whether $G$ is a quotient of the fundamental group of $\operatorname{Spec} \mathbb Q$.

An analogous problem in topology would be.

Q: Does every finite simple group $G$ occur as the deck transformation group of a covering of the wedge of two circles?

A: Yes, because every finite simple group can be generated by two elements (by an explicit calculation with CFSG), hence is a quotient of the free group on two generators, which is the fundamental group of the wedge of two circles.

You can see that the linchpin of the argument is explicit calculations involving the specific space, and involving the specific group, and the characterization in terms of $BG$ does not help much.


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