Normal bundle of a submanifold defined by a submersion

Question

If we have a submersion between differential manifolds then the normal bundle of the preimage of a regular value is trivializable? I know it is true if the codomain has dimension one but I do not see how to generalize or if it is true.

Answer 1

Here's a good exercise for you: If $f\colon X\to Y$ is transverse to $Z$, then $W=f^{-1}(Z)$ is a submanifold whose normal bundle $N(W,X)$ is the pullback of the normal bundle $N(Z,Y)$. Of course, when $Z$ is a point, $N(Z,Y)$ is a trivial bundle.