“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 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.

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