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Let $f:M\to N$ be a smooth locally trivial fibration between smooth manifolds and $L \subseteq M$ a closed submanifold of codimension $\geq 1$ such that $f|_L:L \to N$ is submersive.

Then, is $f|_{M\setminus L}:M \setminus L \to N$ a (locally trivial) fibration as well?

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Let f:R^2 → R be the projection on the first factor and let L be the closed submanifold given by {(x,y)| x>0,xy=1 } union {(x,y)| y=-1 } . I think this is a counterexample.

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  • $\begingroup$ Yes, it is. Thanks! What if $L$ is connected? Probably there is still a similar counterexample but I do not see it right away. $\endgroup$
    – KoopaTroopa
    Commented Nov 13 at 16:45

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