“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.  

“The” Milnor Fibration, and Why-You-Should-Care.

I’m back!  After a long period of laziness, I’m back.  Mainly, because the past week, I’ve been kicking myself in the ass for losing basically all my notes over the past few months, and I have to present at the math department’s new seminar in singularity theory.  Aren’t I smart?

The topic?  A really useful technique/object used to study the topology of (complex) analytic spaces, called the Milnor fibration.  The reason for the quotes in the title is simple: there are several objects and manifold manifestations of this so-called “Milnor fibration.”  Hence, I’ll do my best to introduce “the” main idea, and hopefully walk both myself and whatever readers are out there through its different forms.

Let’s start simple, with an innocent question:  “Given a complex analytic hypersurface Y in a complex manifold M, and a point p \in Y, how does Y ‘sit inside’ M at p?”

Since we only care about the local behavior (here) of Y at p in M, without any loss of generality, we can rephrase this as follows:  M = \mathcal{U} is a connected open neighborhood of p = 0 \in \mathbb{C}^{n+1}, Y = V(f) := f^{-1}(0) for some (non-constant) complex analytic function germ f : (\mathcal{U},0) \to (\mathbb{C},0).  In this form, our innocent question becomes less hand-wavy:

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

Wtf do we mean by “sit inside”?  We want to know about the “topological type” (i.e., the homeomorphism class of the pair (V(f),\{0\})), but more importantly, we want to know how V(f) is embedded as a subset of \mathcal{U}.  For this, we mean the local, ambient, topological type of V(f) in \mathcal{U} at 0, given by the datum of a triple (\mathcal{W},\mathcal{W} \cap V(f),\{0\}), where \mathcal{W} is an open neighborhood of 0 in \mathcal{W}.  In our case, it suffices to consider triples of the form (\overset{\circ}{B}_\varepsilon, \overset{\circ}{B}_\varepsilon \cap V(f), \{0\}), where \overset{\circ}{B}_\varepsilon denotes the open ball centered at the origin in \mathbb{C}^{n+1} \cong \mathbb{R}^{2n+2}.  For the sake of horribly cramped notation, I’ll suppress the dimension, and hope you can keep up.

From general topological stuff, if we can determine the structure of a triple (B_\varepsilon, B_\varepsilon \cap V(f),\{0\}), we can understand (\overset{\circ}{B}_\varepsilon, \overset{\circ}{B}_\varepsilon \cap V(f), \{0\}).

Now, the question becomes:

“What is the topological type of the triple (B_\varepsilon, B_\varepsilon \cap V(f),\{0\})?”

But wait, it gets better (due to a result of Milnor):

Theorem (Milnor):

For \varepsilon > 0 sufficiently small, there is a homeomorphism of triples (B_\varepsilon, B_\varepsilon \cap V(f),\{0\}) \cong (Cone(S_\varepsilon),Cone(S_\varepsilon \cap V(f)), \{0\}).  Moreover, the topological type of this triple is independent of \varepsilon (for \varepsilon sufficiently small).

Where “Cone” is the topological cone. That is, if X is a topological space, Cone(X) := X \times [0,1] / X \times \{0\}.  This is just a fancy way of saying “all the interesting topological behavior occurs on the boundary of the ball.”  So…if we can figure out how this space K := S_\varepsilon \cap V(f) (called the Real Link of V(f) at 0) is embedded inside S_\varepsilon, then we’ve (more or less) answered our question.  It’s important to note that the real link of V(f) at 0 is actually a well-defined object, since the topological type of the above triple is independent of epsilon, when epsilon is chosen “small enough”.

This is basically what Milnor strove to answer with his Fibration.  But, you’ll have to wait for the sequel to see that in action.

Before I leave,  I wanted to talk about some nice cases/descriptions of the real link of V(f), K:


Let X be a real analytic subset of \mathbb{R}^{n+1}  containing the origin, such that X - \{0\} is a real analytic submanifold of \mathbb{R}^{n+1}.  Then, for \varepsilon > 0 sufficiently small, X transversely intersects the n-sphere S_\varepsilon.


Suppose, for the sake of contradiction, that no such \varepsilon exists.  Then, for all \varepsilon > 0, there is some p \in S_\varepsilon \cap X such that

T_p S_\varepsilon + T_p X \neq T_p \mathbb{R}^{n+1}

Let r(x) = \|x\|^2 be the (real analytic) “norm-squared” function on \mathbb{R}^{n+1}, so that r^{-1}(\varepsilon^2) = S_\varepsilon.  In addition, since the origin is the only critical point of r, T_p S_\varepsilon = T_p r^{-1}(r(p)).  Then, the above implies p \in Z := \Sigma(r|_X) (where \Sigma(r|_X) is the critical locus of r, restricted to X, which is a real analytic subset of \mathbb{R}^{n+1}).  Since we get such a p for all \varepsilon > 0, it follows that 0 \in \overline{Z}, so we can apply the Curve Selection Lemma: there exists a real analytic curve \gamma : [0,\eta) \to \mathbb{R}^{n+1} such that \gamma(0) = 0, and for t \in (0,\eta), \gamma(t) \in Z.  What then?

When in doubt, take the derivative of something: for t \neq 0,

r(\gamma(t))' = d_{\gamma(t)} r(\gamma(t)') = 0

since \gamma(t)' \in \text{Ker}(d_{\gamma(t)}r).  That is, r(\gamma(t)) = \text{ constant}.  BUT, r(\gamma(0)) = 0, and r \circ \gamma is a real analytic function, so we actually get r(\gamma(t)) = 0 for ALL t \in [0,\eta).  BUT, r(\gamma(t)) =0 iff \gamma(t) = 0, CONTRADICTION!

And the lemma follows.


So…here, if 0 is an isolated critical point of f, this tells us that (for small enough \varepsilon > 0), that the real link K is a smooth submanifold of S_\varepsilon, of real dimension 2n-1.

More later.