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Let $X,Y$ be smooth algebraic surfaces and $\pi:X\to Y$ be a double cover. Let $B\subseteq Y$ be the branch locus. We assume that $B$ is nef, big and smooth.

[1] says that $\pi_1(X)\cong\pi_1(Y)$ (See page 796). Why is it true?

[1]R.V.Gujar, B.P. Purnaprajna, On the Shafarevich conjecture for genus-2 fibrations, Math. Ann. (2009),343:791-800.

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In case $B$ is ample this is Corollary 2.7 of Nori's paper: "Zariski's conjecture and related problems". I am not sure whether this result is originally due to Nori, but in this paper you will surely find many helpful results in the direction you're asking for.

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