Given a birational proper morphism $f\colon X \rightarrow Y$
( Assume $X$ and $Y$ irreducible )
of complex algebraic varieties. 
It is always true that $f^\* \colon \pi^{et}\_1 (X)\rightarrow \pi^{et}\_1(Y)$
is an isomorphism?
I think that this follows from (SGA.1 exp X. Corollary 1.4)
But im not totally sure.


Thanks, in advance. 



Isomorphism étale fundamental group
This is true if $X$ and $Y$ are smooth, see SGA I, exp. X, Cor. 3.4. But certainly not in general: if $Y$ is a plane cubic with one node and $f$ its normalization, $X$ is simply connected while $\pi \_1^{et}(Y)\cong \hat{\mathbb{Z}}$, cf. same SGA, exp. IX, example 5.5.


