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Suppose we have a smooth map between smooth manifolds $f:M \mapsto N$ where $f$ is transverse to a submanifold $X \subset N$, i.e. any point in $X$ is a regular value of $f$. We know from the implicut function theorem, $f^{-1}(x)$ is a submanifold of $M$.

Do we also have $f^{-1}(X)$ is a submanifold of $M$? Is the proof just a generalization of the previous result? If so, what is its dimension?

  • See https://math.stackexchange.com/questions/1824631/understanding-milnors-proof-of-the-fact-that-the-preimage-of-a-regular-value-is – Eric Towers Apr 05 '21 at 13:09
  • @EricTowers I don't think they are the same questions...My point is if we have a submanifold which consists of regular values, is the inverse image still a manifold. –  Apr 05 '21 at 21:00
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    https://math.stackexchange.com/questions/2803308/transverse-manifolds-intersection-is-a-manifold-again – Eric Towers Apr 06 '21 at 18:39
  • @EricTowers Thank you so much! –  Apr 06 '21 at 21:35

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