# A relative version of Ehresmann's theorem

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.)

• 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." – Phil Tosteson Apr 25 at 21:28
• @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 $\psi (N\cap \pi^{-1}(U))=N_b\times U$. – AG learner Apr 26 at 4:14
• 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. – Phil Tosteson Apr 28 at 18:37
• @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. – AG learner May 1 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 $$(x,y)\mapsto(F_y(x),y)$$ ($$x\in M_b$$, $$y\in B$$)