# 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.

• 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. Dec 15, 2009 at 16:04
• There us a bounty for this question now, so please don't hesitate to answer, if you know the answer :) Feb 23, 2010 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.

• 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! Feb 23, 2010 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".

• Greg, thanks a lot! So I guess you indicate, that this lemma has no name :)? Or no official name? Dec 15, 2009 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. Dec 15, 2009 at 16:35
• Greg, thanks again! The paper of McDuff turned out to have also one other lemma that is very usefull! Dec 15, 2009 at 17:27
• Unfortunatelly I had to disaccept the answer since I wanted to start a bounty, I really need a reference. Feb 23, 2010 at 18:20