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I'm reading Voisin's proof of Ehresmann's lemma, given in the book Hodge Theory and Complex Algebraic Geometry I as theorem 9.3; the proof is essentially reproduced in this answer.

To summarize, there's a surjective proper submersion $F:M\rightarrow N$, and the theorem shows $F$ is a fiber bundle.

To do that, it looks at an arbitrary $q\in N$ and the compact embedded submanifold $S=F^{-1}(q)\subset M$. Using a tubular neighborhood $S\subset U$ with its associated retraction $r:U\rightarrow S$, a map $\Phi:U\rightarrow S\times N$ is defined by \begin{equation*} \Phi(p)=(r(p), F(p)) \end{equation*}

To finish the proof, first $U$ is shrunk down to an open neighborhood $S\subset V\subset U$ such that $\Phi:V\rightarrow S\times Y$ is an embedding. Then using properness an open neighborhood $q\in W\subset N$ is found with $F^{-1}(W)\subset V$.

I understand all of the above. It's the last step that I can't figure out: \begin{equation*} \Phi\left(F^{-1}(W)\right)=S\times W \end{equation*}

Why is this an equality? I've tried using "vertical slices": taking $y\in W$ and showing the retraction $r:F^{-1}(y)\rightarrow S$ is surjective. But I can only show it's a local diffeomorphism. I've also tried using "horizontal slices": trying to show for every $x\in S$ that $W\subset F(r^{-1}(x))$, but I've also failed to do that.

Both proofs start this step with "Clearly", so I'm wondering if I'm missing something obvious?

Hempelicious
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  • Have you tried working it out coordinatewise? A surjective submersion should locally be something like $(x_1,\ldots,x_{n+r}) \mapsto (x_1,\ldots,x_n)$. The retraction should then be expressible as a projection and you can then write down explicitly which point in $F^{-1}(W)$ should be sent to a point in $S \times W$. – Jeroen van der Meer Feb 03 '21 at 17:12
  • @JeroenvanderMeer: thank you for the suggestion. I have tried this, but the difficulty is $F$ is only locally a projection, and I haven't been able to make it work globally. If we knew $S$ was connected, that would be enough. In fact, since for every $y\in W$, $r:F^{-1}(y)\rightarrow S$ is an injective local diffeomorphism, the only thing that can really go wrong is if $F^{-1}(y)$ has fewer components than $S$. – Hempelicious Feb 03 '21 at 19:56

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