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Let $X$ and $Y$ be non compact complex manifolds and $f:X\to Y$ be a holomorphic fiber bundle with fibers $F$ such that $f^*:\pi_1(X)\to\pi_1(Y)$ is injective and let for any $f_1,f_2\in F$ there exists a biholomorphic map $\psi:F\to F$ such that $\psi(f_1)=f_2$ and $\psi\in Deck(X/Y)$. My question is $\pi_1(X)$ a normal subgroup of $\pi_1(Y)$? If not can we put some extra conditions(apart from finite fibers) such that $\pi_1(X)$ is a normal subgroup of $\pi_1(Y)$?

Any comments or suggestions on how to think about this question is highly appreciated. Thanks in Advance!!

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    $\begingroup$ Just having finite fibers with an abstract biholomorphism of the fiber taking any given point to any other will surely not generally guarantee normality of the subgroup (your parenthetical remark); this biholomorphism should be the restriction of a deck transformation. $\endgroup$
    – Aleksandar Milivojević
    May 25, 2022 at 7:03
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    $\begingroup$ Provided the fibers have the homotopy-type of a discrete space, your map has many of the properties of a covering map. And this is essentially an if and only if statement, via @SamNead's comment. $\endgroup$
    – Ryan Budney
    May 25, 2022 at 7:13
  • $\begingroup$ What if $F$ is homotopic to a covering space of $X$@RyanBudney? $\endgroup$
    – tota
    May 25, 2022 at 7:30
  • $\begingroup$ I have edited now @AleksandarMilivojević $\endgroup$
    – tota
    May 25, 2022 at 7:34

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Perhaps you will be interested in the long exact sequence of homotopy groups associated to a fibration.

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