# “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.