Given two complex algebraic varieties $X$ and $Y$. If there exists a birational proper morphism $f\colon X\rightarrow Y$ then a Theorem of Grothendieck (SGA.X) say that $\pi_1^{et}(X)\cong \pi_1^{et}(Y)$. Is it possible to obtain the same isomorphism for the topological fundamental group? What conditions i have to imposse to get this?
Thanks in advance.