Are two homeomorphic hypersurfaces of the same smooth manifold also diffeomorphic?

Question

Let $M$ be a smooth (connected, without boundary) manifold and $N_1$, $N_2$ be two smooth (connected, without boundary) hypersurfaces of $M$. Suppose $N_1$ and $N_2$ are homeomorphic. Can $N_1$ and $N_2$ be non-diffeomorphic?

I am currently working on a problem where I have shown that two smooth, compact hypersurfaces $N_1$ and $N_2$ of the same manifold $M$ are both homeomorphic (even $C^{\alpha}$-homeomorphic for some $\alpha \in (0,1)$) to the same manifold $N$. However, I would like to use some differential properties of both $N_1$ and $N_2$ and it would be suitable that they have the same differential structure.

I know there exists many examples of non-diffeomorphic manifolds which are homeomorphic, such as exotic spheres. However, I don't know if one can realize two different differentiable spheres as hypersurfaces of the same smooth manifold.

Answer 1

Here is (what I believe is) a counterexample which works for any applicable $N_1,N_2$, which additionally is compact whenever $N_1$ and $N_2$ are:

Choose two homeomorphic, but not diffeomorphic manifolds $N_1,N_2$. Let $M=(N_1\times S^1)\#(N_2\times S^1)$, where $\#$ denotes a smooth connected sum (chosen with arbitrary orientation if applicable). We can always construct this sum by modifying a sufficiently small neighborhood from each $N_i\times S^1$ factor that $M$ retains an embedded hypersurface diffeomorphic to $N_i$.

Answer 2

In fact, this can fail spectacularly.

There is a submersion $\pi:\mathbb{R}^5\rightarrow \mathbb{R}$ with the property that for any $t\in \mathbb{R}$, $\pi^{-1}(t)$ is homeomorphic to $\mathbb{R}^4$, but for any $t\neq s\in\mathbb{R}$, the inverses images are not diffeomorphic.

I learned about this fact from Ryan Budney's answer to an MO question.