This is a question about a name of a very useful lemma, that permits one in particular to show that smooth birational complex projective varieties have isomorphic fundamental groups. If this lemma has no name, I would like at least to have a reference (if it exits). The lemma can be seen as a truncated version of the basic fact, that if we have a locally trivial fibration (say of finite dimensional CW complexes) $F\to E\to B$ then we get a long exact sequence

$\to \pi_i(F)\to \pi_i(E)\to \pi_i(B)\to \pi_{i-1}(F)\to$

**Lemma.** Let $E\to B$ be a surjective map of finite dimensional $CW$ complexes,
such that every fiber is connected, simply connected and is a
deformation retract of a small neighbourhood.
Then $\pi_1(E)=\pi_1(B)$.

**Question.**
Do you know the name of such a lemma, or of some of its generalizations? Is there a reference for this?

The result about $\pi_1$ of birationaly equivalent varieties follows since any birational transformation can be decomposed in blow-ups and blow downs along smooth submanifolds. And it is not hard to check that the conditions of lemma are satisfied for such elementary blow ups.