If $X$ is a normal algebraic variety over $\mathbb{C}$ and $U$ an open set of $X$ then we have surjectivity on the fundamental groups $ \pi_1(U)\rightarrow \pi_1(X)$.
If $f\colon X \rightarrow Y$ is a morphism (not necessarily a fibration) of smooth algebraic varieties over $\mathbb{C}$, under some conditions on the general fiber $F$ of this morphism we can write an exact sequence:
$$ \pi_1(F)\rightarrow \pi_1(X)\rightarrow \pi_1(Y)\rightarrow 1 $$
- There exists a generalization of this facts on higher homotopy?
I'm interested in the case when $f$ is a quotient for the action of an algebraic group on $X$.
- Is there any conditions such that $f$ is a fibration?
I've read somewhere that proper and separable should be a sufficient condition. is this true?
Thanks in advance,