# 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.

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You surely mean birational "projective varieties" or "compact manifolds" or some other rigidity thing, otherwise I can say that $\mathbb{P}^1$ and $\mathbb{C}^\times$ are birational, but have different $\pi_1$'s. As for the lemma, I've never seen it named, just as a consequence of the long exact sequence. – Charles Siegel Dec 15 '09 at 16:04
There us a bounty for this question now, so please don't hesitate to answer, if you know the answer :) – Dmitri Feb 23 '10 at 18:18

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.

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Dear Allan, thank you very much for the answer! I will go through the article for a bit, but I think, this is exactly the reference I was looking for:) Huge thanks! – Dmitri Feb 23 '10 at 19:29

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".

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Greg, thanks a lot! So I guess you indicate, that this lemma has no name :)? Or no official name? – Dmitri Dec 15 '09 at 16:26
Or rather, what Google Scholar indicates, because without it I'm not an authority. Google Scholar can show you common phrases in papers. It is convincing enough that "simply connected fibers" is a common phrase, and that there is no particular other common phrase to go with it. The top match, by the way, is an influential paper by Dusa McDuff. – Greg Kuperberg Dec 15 '09 at 16:35
Greg, thanks again! The paper of McDuff turned out to have also one other lemma that is very usefull! – Dmitri Dec 15 '09 at 17:27
Unfortunatelly I had to disaccept the answer since I wanted to start a bounty, I really need a reference. – Dmitri Feb 23 '10 at 18:20