Truncated exact sequence of homotopy groups 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. 
 A: The obvious generalization, which I'm sure you realize, is that if the fibers are $n$-connected, the projection map is an isomorphism on the first $n$ homotopy groups.  What kind of you proof you prefer is a matter of taste, but I'll note that in many cases, you can arrange a cellulation of the total space that is a twisted Cartesian product of a cellulation of the fiber and a cellulation of the base.  If the fiber does not have any low-dimension cells, the $n$-skeleton of the base and the total space are the same, and the extra cells in the $(n+1)$-skeleton do not affect the answer.
In order to look for a standard name, I searched for the phrase "simply connected fibers" in Google Scholar with quotes.  I got 87 hits in the search with quotes, including a number of good papers by well-known people.  No other phrase leapt out with this search.  So I think it's conclusive that it is the "lemma on simply connected fibers", or the "lemma on connected and simply connected fibers".
A: 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.
