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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?

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    I don't see how cobordism would help. I would try using the implicit function theorem instead.2012-02-23
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    show that $f$ is proper and look at http://en.wikipedia.org/wiki/Ehresmann%27s_fibration_theorem2012-02-23
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    You are missing some hypothesis to conclude that $f$ is proper. For example, without any additional hypothesis, we have a counterexample $M=(0,2)$, $N=S^1$ (the unit circle in the complex plane), and $f(t)=e^{2\pi i t}$.2014-04-28

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