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. 



Truncated exact sequence of homotopy groups
Check out the paper "A Vietoris Mapping Theorem for Homotopy," by S. Smale, Proc. Amer. Math. Soc. 8 (1957), 604-610, available at <http://www.jstor.org/stable/2033527> . 


Paraphrase of the main theorem: If $f:X\to Y$ is a proper, onto map of 0-connected, locally compact, separable metric spaces, X is $LC^n$, and each point inverse is $LC^{n-1}$ and $(n-1$-connected, then the induced homomorphism $\pi\_r(X)\to \pi\_r(Y)$ is an isomorphism for $r\le n-1$ and surjective for $r=n$. 


$LC^n$ is a local connectedness condition surely satisfied by CW complexes, which are locally contractible.


