“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.

Published by brianhepler

I'm a third-year math postdoc at the University of Wisconsin-Madison, where I work as a member of the geometry and topology research group. Generally speaking, I think math is pretty neat; and, if you give me the chance, I'll talk your ear off. 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 to show you a world of pure imagination. View more posts

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