A difficult question about diffeomorphism about submanifold

Let $M$ and $N$ be two smooth manifolds, and $f: M \to N$ be a submersion , ${{f}^{-1}}(y)$ is compact for all $y$ in $N$. Then prove for any $x$ in $N$ there is an open neighborhood $U$ of $x$ such that ${{f}^{-1}}(U)$ is diffeomorphic to $U\times {{f}^{-1}}(x)$. I've thought this problem for a long time, but I don't know to use which method. Cobordism can help to solve this question?

Solution 1:

As Blah said in the comments, if we can show that $f$ is proper (and in fact it is!), your statement is just Ehresmann fibration theorem.

To give a proof of Ehresmann fibration theorem, we will use theorem of tubular neighborhoods. This result says that a neighborhood $W$ of $M_y := f^{-1}(y)$ in $M$ is diffeomorphic to a neighborhood of $M_y$ in its normal bundle, and in particular that there exists a differentiable retraction $r : W \to M_y$. But then , consider the map $$(r,f):W \to M_y \times N.$$ This map has a differential (which is invertible along $M_y$). As $M_y$ is compact, there exists an open set $W' \subset W$ containing $M_y$ such that $(r,f)_{|W'}$ is an embedding. Finally, as $f$ is proper, $W'$ contains an open set $W''$ of the form $f^{-1}(U)$, where $U \subset N$ is a neighborhood of $y$. Then clearly $(r,f)(W'') = M_y \times U$ and we have shown that $(r,f)$ is a diffeomorphism from $f^{-1}(U)$ to $M_y \times U$.