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Let $f\colon X\rightarrow Y$ be a morphism of complex algebraic varieties. Let $y\in Y$ be a general point, then we have a sequence of homomorphisms of fundamental groups induced by the inclusion of the general fiber $f^{-1}(y)$ and $f$ $$ \pi_1 ( f^{-1}(y) ) \rightarrow \pi_1 ( X) \rightarrow \pi_1(Y).$$ If $X$ and $Y$ are smooth, there exists conditions such that this sequence is exact. (e.g. Generalized Zariski-van Kampen theorem and its application to Grassmannian dual varieties - Ishiro Shimada).

There exists conditions such that this sequence is exact in $\pi_1(X)$ for $X$ and $Y$ being normal?

Im interested in the particular case when $f$ is the good quotient for the action of an reductive algebraic group on $X$.

Any comment will be highly appreciated.

Thanks in advance.

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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). – abx Mar 31 '14 at 6:25

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