“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 nth 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.  

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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