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Mar 31, 2014 at 6:25 comment added abx Have a look at SGA I, Expose 10 (here). This is in the context of the algebraic $\pi _1$, but I guess the proof extends directly to the topological one. There are no hypotheses on the varieties; the map must be proper and separable (= reduced fibers in your case).
Mar 31, 2014 at 4:49 history edited Joaquín Moraga CC BY-SA 3.0
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Mar 31, 2014 at 4:38 history asked Joaquín Moraga CC BY-SA 3.0