Goals for the (immediate) future

The next few weeks are going to be very (very) busy, what with qualifying exams, more talks, and extremely hard homeworks, so I’m not sure how much I’ll be posting (but, I won’t bail again, last year (I hope, at least…)).

So, modulo the existence of free time, I’ve been really wanting to do a post that surveys the Milnor fibration in “all” of its various forms, from its conception to modern usage.  I’ve already talked about the “classical” case, i.e., Milnor’s original fibration on the sphere (minus the real link), as well as the fibration “in the ball”, from last time.  In addition to this, I want to talk about

  • The “general” Milnor fibration, associated to a complex analytic function on an arbitrary complex analytic space (for this, I’ll need to talk about the notion of a stratification, and various equisingularity conditions we can impose thereon).
  • A “cohomological” version of the Milnor fibration, which basically says that, for a bounded, constructible complex of sheaves, \mathcal{F}^\bullet, and a complex analytic function f: (X,0) \to (\mathbb{C},0), there is a fundamental system of compact neighborhoods of 0, \{K_n \}_n, such that all the higher direct image sheaves R^i f_*(\mathcal{F}^\bullet|_{K_n}) are locally constant (along with a “stability” result).  For this, I’ll probably need to mention some basic results/theory concerning the derived category, constructible sets, etc.
  • The “vanishing and nearby cycles” of Deligne, associated to a complex analytic function f: (X,0) \to (\mathbb{C},0).  A purely category-theoretic formulation, these produce certain complexes of sheaves whose stalk cohomology at a point is (naturally) isomorphic to the cohomology of the Milnor fiber of f at that point.  Really fucking cool, but I’ll need to do quite a bit of explaining to get to these beasts.

In addition to the last point, I want to (later on, after the above stuff) talk about

  • “Basic” Morse theory
  • Stratified Morse theory
  • Microlocal Morse theory

but that might be quite a lofty goal, at least for the time being.  Who knows, I might wake up smarter some day.

Until next time.

“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.