“The” Milnor Fibration: some proof

As it turns out, the Milnor fibration I talked about last time, i.e., the normalized map

f/|f| : S_\epsilon - V(f) \cap S_\epsilon \to S^1

is diffeomorphic to a much “nicer” fibration:

f: B_\epsilon \cap f^{-1}(\partial D_\eta) \to \partial D_\eta

where $\partial latex D_\eta$ is the boundary of  disk about the origin in \mathbb{C} of radius \eta.  And, this is a (smooth) locally trivial fibration for sufficiently small \epsilon, \eta.  Intuitively, I think this one makes more sense.  You can picture the total space as an open “tubular neighborhood” of the fiber V(f) = f^{-1}(0) inside the closed ball B_\epsilon about the origin.

The proof is basically just an application of

Ehresmann’s Theorem

Let f: M \to N be a smooth map between the smooth manifolds M and N. Then, if f is a proper submersion, it is a smooth locally trivial fibration over N.

In addition, if Q \subseteq M is a closed submanifold such that the restriction f|_Q is still a submersion, then f|_Q is a smooth locally trivial fibration that is compatible with f.

Usually, one takes the closed submanifold Q to be \partial M, in the case where M is a smooth manifold with boundary.

Proof(of the Milnor fibration, isolated critical point)

Not too hard.  First, we need to pick epsilon.  Choose \epsilon small so that V(f) transversely intersects S_\varepsilon, and such that \overset{\circ}{B}_\epsilon \cap \Sigma f \subseteq V(f).  Now, delta.

For all choices of \delta > 0, the restricted map

f: B_\epsilon \cap f^{-1}(\overset{\circ}{\mathbb{D}}_\delta^*) \to \overset{\circ}{\mathbb{D}}_\delta^*

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 \epsilon, f has no critical points on the interior B_\epsilon^\circ \cap f^{-1}(\overset{\circ}{\mathbb{D}}_\delta^*).  Therefore, any critical that occur must be on the “bounding sphere”, S_\epsilon \cap f^{-1}(\overset{\circ}{\mathbb{D}}_\delta^*).  So, we consider the critical points of the function f|_{S_\epsilon}.

BUT, by our choice of \epsilon, V(f) transversely intersects S_\epsilon, and, by “stability” of transversality, there is an open neighborhood of the origin in \mathbb{C} consisting entirely of regular values of f|_{S_\epsilon}.  WLOG, we can suppose this neighborhood is of the form \overset{\circ}{\mathbb{D}}_\delta for some \delta > 0.  Throw away the origin, and the claim follows by Ehresmann’s theorem.

 

 

Author: brianhepler

I'm a second-year math postdoc at the University of Wisconsin-Madison, and I think math is pretty neat. Especially the more abstract stuff. It's really hard to communicate that love with the general population, but I'm going to do my best.

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