Let $f:X\to Y$ be a surjective proper morphism between connected (but might be reducible) complex quasi-projective varieties. Assume that each fiber of $f$ is connected. Can we conclude that ${\rm Im}(\pi_1(X)\to \pi_1(Y))$ is a finite index subgroup of $\pi_1(Y)$? This question has a positive answer if $X$ and $Y$ are all irreducible and $Y$ is normal.
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5$\begingroup$ A map of connected CW complexes induces a surjection on $\pi_1$ if and only if the pullback of any connected covering space is connected. Then use that if $f : X \to Y$ is a closed map of topological spaces with $Y$ connected and every fiber connected then $X$ is connected. $\endgroup$– Ben CCommented Jul 29, 2023 at 2:57
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$\begingroup$ @ Ben C. Thanks a lot! Could you please give a reference of your theorem? $\endgroup$– Higgs-BosonCommented Jul 29, 2023 at 5:45
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2$\begingroup$ A reference is Allen Hatcher's book, free online. Go to the section on covering spaces $\endgroup$– David WhiteCommented Jul 29, 2023 at 6:08
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1$\begingroup$ @ David White: thanks! I am familiar with Hatcher's book. But I did not find the theorem "A map of connected CW complexes induces a surjection on $\pi_1$ if and only if the pullback of any connected covering space is connected. " Do you know how to prove it? $\endgroup$– Higgs-BosonCommented Jul 29, 2023 at 9:49
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2$\begingroup$ @higgs-boson Fix a connected CW complex $X$. Taking the fiber over a fixed base-point induces an equivalence of categories between covering spaces of $X$ and sets with a $\pi_1$-action where the connected ones correspond to sets with a transitive action (and hence to subgroups of $\pi_1$). This will be spelled out in any algebraic topology book (sometimes going under the name "galois correspondence"). Then use the elementary group theory fact: $G \to H$ is surjective if and only if every transitive $H$-set is also transitive when viewed as a $G$-set. $\endgroup$– Ben CCommented Jul 29, 2023 at 20:29
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