Where were we?
We had just equated the “innocent” question
“How does ‘sit inside’ at ?”
with the more precise(ish)
“How is the real link of at 0 embedded in ?”
Milnor’s genius idea was to realize the complement, , as the total space of a smooth, locally trivial fibration over the unit circle:
Milnor’s Fibration Theorem:
For sufficiently small, the map
is the projection of a smooth, locally trivial fibration. In addition, “the” fiber is a smooth, -dimensional, parallelizable manifold.
In reality, there are a few other objects called “the” Milnor fibration:
- (Inside the ball) For , the restriction
is a smooth, locally trivial fibration.
- (the compact fibration) For , the restriction
is a topological locally trivial fibration.
And here’s the kicker: topological (resp., smooth) locally trivial fibrations over 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 . The characteristic diffeomorphism is induced by the action of the fundamental group , and is defined upto pseudo-isotopy (endow the total space with a Riemannian metric, horizontally lift to the total space, and parallel transport “around” the circle. ) HOWEVER, we do get a unique Monodromy automorphism
and this correspondance yields a group representation
called the local monodromy.
Some more on the fiber:
- is a complex -dimensional manifold.
- has the homotopy type of a finite CW-complex.
If 0 is an isolated critical point of , then
- is homotopy equivalent to a finite bouquet of -spheres.
The number of -spheres, denoted , is called the Milnor number for at 0, and is the th Betti number of . Actually, may be calculated quite easily, via an alternative description:
where is the Jacobian Ideal of : the ideal generated by the partial derivatives of inside the local ring of germs of holomorphic functions at 0 (which is isomorphic to the ring of convergent power series in 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 are two reduced complex analytic function germs such that is homeomorphic to , then there exists a homotopy-equivalence such that commutes with the respective monodromy automorphisms.
Note that, when 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.