As it turns out, the Milnor fibration I talked about last time, i.e., the normalized map
is diffeomorphic to a much “nicer” fibration:
where $\partial latex D_\eta$ is the boundary of disk about the origin in of radius . And, this is a (smooth) locally trivial fibration for sufficiently small . Intuitively, I think this one makes more sense. You can picture the total space as an open “tubular neighborhood” of the fiber inside the closed ball about the origin.
The proof is basically just an application of
Let be a smooth map between the smooth manifolds and . Then, if is a proper submersion, it is a smooth locally trivial fibration over .
In addition, if is a closed submanifold such that the restriction is still a submersion, then is a smooth locally trivial fibration that is compatible with .
Usually, one takes the closed submanifold to be , in the case where is a smooth manifold with boundary.
Proof(of the Milnor fibration, isolated critical point)
Not too hard. First, we need to pick epsilon. Choose small so that transversely intersects , and such that . Now, delta.
For all choices of , the restricted map
is proper, via a quick application of the Heine-Borel theorem. The rest of the proof is just Ehresmann’s theorem.
By our choice of , has no critical points on the interior . Therefore, any critical that occur must be on the “bounding sphere”, . So, we consider the critical points of the function .
BUT, by our choice of , transversely intersects , and, by “stability” of transversality, there is an open neighborhood of the origin in consisting entirely of regular values of . WLOG, we can suppose this neighborhood is of the form for some . Throw away the origin, and the claim follows by Ehresmann’s theorem.