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Let $f:X\to Y$ be a morphism of affine varieties such that it is a fibre bundle with fibre $F$. Let $\pi_1(Y)=\Gamma$ be a free group (non abelian) of finite rank and $\pi_1(F)$ is a finite group $G$ say and $\pi_2(Y)=0$. Let $g:Z\to X$ be a topological covering projection corresponding to $\Gamma$ and assume that fibres $(f\circ g)^{-1}(c)$ are universal covers of $f^{-1}(c)$ for all $c\in Y$. Using Serre's long exact sequence of homotopy we see that $\Gamma$ and $G$ are both subgroups of $\pi_1(X)$. My question is: is $\Gamma$ a normal subgroup of $\pi_1(X)$ i.e. is $g:Z\to X$ a regular covering?

Any comments or reference how to tackle the question will be very helpful.

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    $\begingroup$ There is something not quite right here. It follows from the assumptions on $X$ and $Y$ that $F$ is a $K(G,1)$, so it cannot be finite dimensional (unless $G=\{e\}$). $\endgroup$
    – algori
    Commented May 17, 2022 at 1:18
  • $\begingroup$ Thanks @algori. I have edited the question now. $\endgroup$
    – tota
    Commented May 17, 2022 at 3:38
  • $\begingroup$ Typically $\Gamma$ would act nontrivially on $G$, so $\pi_1(X)$ would be a semidirect product, and the subgroup would not be normal. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented May 17, 2022 at 5:39
  • $\begingroup$ Can you give an example? $\endgroup$
    – tota
    Commented May 17, 2022 at 5:45

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