“The” Milnor Fibration: Classical Case

Where were we?

We had just equated the “innocent” question

“How does $V(f)$ ‘sit inside’ $\mathcal{U}$ at $0$ ?”

with the more precise(ish)

“How is the real link of $V(f)$ at 0 embedded in $S_\varepsilon$ ?”

Milnor’s genius idea was to realize the complement, $S_\varepsilon - K$ , as the total space of a smooth, locally trivial fibration over the unit circle:

Milnor’s Fibration Theorem:

For $\varepsilon > 0$ sufficiently small, the map

$\frac{f}{\| f\|}: S_\varepsilon - K \to S^1$

is the projection of a smooth, locally trivial fibration. In addition, “the” fiber is a smooth, $2n$ -dimensional, parallelizable manifold.

In reality, there are a few other objects called “the” Milnor fibration:

(Inside the ball) For $1 >> \varepsilon >> \delta > 0$ , the restriction

$f: \overset{\circ}{B}_\varepsilon \cap f^{-1}(\partial \mathbb{D}_\delta) \to \partial \mathbb{D}_\delta$

is a smooth, locally trivial fibration.

(the compact fibration) For $1>> \varepsilon >> \delta > 0$ , the restriction

$f: B_\varepsilon \cap f^{-1}(\partial \mathbb{D}_\delta) \to \partial \mathbb{D}_\delta$

is a topological locally trivial fibration.

And here’s the kicker: topological (resp., smooth) locally trivial fibrations over $S^1$ are completely classified by the fiber and the so-called characteristic homeomorphism (resp., diffeomorphism) of the fiber.

I’ll refer to the first two fibrations as “the” Milnor fibration (for now…), and the Milnor fiber as $F_f$ . The characteristic diffeomorphism $h : F_f \to F_f$ is induced by the action of the fundamental group $\pi_1(S^1,1)$ , and is defined upto pseudo-isotopy (endow the total space with a Riemannian metric, horizontally lift $[\gamma(t)] \in \pi_1(S^1,1)$ to the total space, and parallel transport “around” the circle. ) HOWEVER, we do get a unique Monodromy automorphism

$h_* : H_*(F_f;\mathbb{Z}) \overset{\thicksim}{\to} H_*(F_f;\mathbb{Z})$

and this correspondance yields a group representation

$\rho: \pi_1(S^1,1) \to Aut(H_*(F_f;\mathbb{Z}))$

called the local monodromy.

Some more on the fiber:

Theorem (Milnor):

$F_f$ is a complex $n$ -dimensional manifold.
$F_f$ has the homotopy type of a finite CW-complex.

If 0 is an isolated critical point of $f$ , then

$F_f$ is homotopy equivalent to a finite bouquet of $n$ -spheres.

The number of $n$ -spheres, denoted $\mu := \mu(f,0)$ , is called the Milnor number for $f$ at 0, and is the $n$ th Betti number of $F_f$ . Actually, $\mu$ may be calculated quite easily, via an alternative description:

$\mu(f,0) = \text{dim}_\mathbb{C} \left ( \frac{\mathcal{O}_{\mathcal{U},0}}{\mathcal{J}_f} \right )$

where $J_f$ is the Jacobian Ideal of $f$ : the ideal generated by the partial derivatives of $f$ inside the local ring $\mathcal{O}_{\mathcal{U},0}$ of germs of holomorphic functions at 0 (which is isomorphic to the ring of convergent power series in $n+1$ complex variables).

I don’t have it in me now to include examples, but I’ll update this post later on (after I’ve actually slept) with some good ones.

For a closing comment, some reassurance that this all is ACTUALLY useful:

Theorem(Topological Invariance) [Le-Tessier]

For a reduced hypersurface, the homotopy type of the Milnor fiber is an invariant of the local, ambient, topological type of the hypersurface. That is, if $f,g : (\mathcal{U},0) \to (\mathbb{C},0)$ are two reduced complex analytic function germs such that $(V(f),\{0\})$ is homeomorphic to $(V(g),\{0\})$ , then there exists a homotopy-equivalence $\alpha : F_f \to F_g$ such that $\alpha$ commutes with the respective monodromy automorphisms.

Note that, when $0$ is an isolated critical point, this implies that the Milnor number is ALSO an invariant of the local, ambient, topological type of the hypersurface at the origin.

Stay tuned.

“The” Milnor Fibration: Classical Case

Published by brianhepler

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