Edited: Phil Tosteson suggested Thom's first isotopy lemma, but it does not seem to be in the direction that I'm trying to generalize. Let me reformulate my question again.

Let $N\subset M$ be a pair of the smooth manifolds and $\pi:M\to B$ a proper submersion to a smooth base. Assume moreover that the restriction $\pi_{|N}:N\to B$ is also proper and submersive, then both $\pi$ and $\pi_{|N}$ are locally trivial by applying Ehresmann's theorem separately, but I'm wondering if they can be simultaneously locally trivialized.

More precisely, I'm looking for a proof of the following argument:

Claim: For each $b\in B$, there is a neighborhood $U$ of $b$, and a fiber preserving diffeomorphism $$\psi:\pi^{-1}(U)\xrightarrow{\cong} M_b\times U,$$ satisfying $$\psi(N\cap \pi^{-1}(U))=N_b\times U$$ where $M_b=\pi^{-1}(b)$ and $N_b=\pi_{|N}^{-1}(b)$.

The example that I have in mind is a smooth family of pairs (cubic surface, cubic curve). Explicitly take a general pencil $\mathbb P^1$ of hyperplane sections of a smooth cubic threefold $X$ over $\mathbb C$ with base locus a cubic curve $E$ (as the intersection of any two hyperplane section in the pencil). Project the incidence variety $I=\{(y,t)|y\in X\cap H_t\}\subset X\times \mathbb P^1$ to the second factor produces a family of cubic surfaces. Now $M\to B$ will be restriction of $I\to \mathbb P^1$ to the locus where cubic surfaces are smooth and $N:=E\times B$.

(For a smooth family of compact complex manifolds, one can even ask for transversely holomorphic trivialization, see page 2 in this notes, I would also like to know if it works for a pair.)

  • 3
    $\begingroup$ There is a general theorem that goes under the name of "Thom's first isotopy lemma." You can read about it in Mather's "Notes on topological stability." $\endgroup$ Apr 25 '20 at 21:28
  • 1
    $\begingroup$ @PhilTosteson Thanks for your comment, I checked Thom's first isotopy lemma in Mather's notes. However, I'm not sure how this lemma is related to my question since I already assumed both $\pi$ and $\pi_{|N}$ are proper and submersive, so Ehresmann's theorem applied to $M$ and $N$ separately tell us they are both locally trivial over $B$, but the argument that I am looking for is the existence of a (local) trivialization $\psi: \pi^{-1}(U)\xrightarrow{\cong} M_b\times U$ which satisfies $\psi (N\cap \pi^{-1}(U))=N_b\times U$. $\endgroup$
    – AG learner
    Apr 26 '20 at 4:14
  • 2
    $\begingroup$ Sorry, you need the "second isotopy lemma" to see your desired outcome as a direct consequence. However, I think that the proof of the first isotopy lemma involves proving a version of your statement. $\endgroup$ Apr 28 '20 at 18:37
  • $\begingroup$ @PhilTosteson Thank you! It makes more sense to me now. I hope I can dig an elementary proof out of it, since I am working on smooth submanifold case compared to the general stratified subsets considered in the first/second isotopy lemma. $\endgroup$
    – AG learner
    May 1 '20 at 2:06

The answer is positive. There are several proofs of Eheresmann's genuine lemma; I think that each of them can be straightforwardly generalized and gives your relative version. But you can also, alternatively, deduce the relative version from the absolute one together with a classical theorem of Cerf, as follows. After restricting $B$ to a small compact ball centered at a point $b$, by a first application of the genuine Ehresmann lemma to $(M,\pi)$, you can assume without loss of generality that $M=M_b\times B$ and that $\pi$ is the second projection. Then, after restricting $B$ to a smaller compact ball centered at $b$, by a second application of the genuine Ehresmann lemma to $(N,\pi\vert N)$, there is a local trivialization of $\pi\vert N$ over $B$, in other words a diffeomorphism $$\psi_N:N\to N_b\times B$$ such that $\pi\circ\psi_N=\pi\vert N$. You can also see $\psi_N$ as a parametric family of smooth embeddings $$f_y:N_b\hookrightarrow M_b$$ parametrized by $y\in B$, namely $$(f_y(x),y)=\psi_N^{-1}(x,y)$$ It is classical (Cerf) that a parametric family of embeddings of $N_b$ in $M_b$ extends to a parametric family of self-diffeomorphisms (isotopies) of the ambiant manifold $M_b$: there is a smooth family $(F_y)$ of self-diffeomorphisms of $M_b$ such that $f_y=F_y\vert N_b$; and $F_b$ is the identity. Finally, the trivialization $\psi$ that you are looking for is just the inverse of $$(x,y)\mapsto(F_y(x),y)$$ ($x\in M_b$, $y\in B$)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.