CHAPTER 2 - THE MILNOR FIBRATION

Summary

Although the objects studied in this book are mostly germs of spaces and of mappings, many of the results we are going to discuss concern representatives of germs. It is then convenient to have a distinguished class of representatives whose members are sufficiently alike (for instance, are mutually topologically equivalent). In this chapter we describe such classes: in §A we do this for an isolated analytic singula and in §B we generalize this to families of such singularities. This leads to the notion of a ‘good representative’. The next section concentrates on the geometric monodromy representation. In §D we discuss an even better (hence smaller) class of representatives: the excellent ones. This section stands somewhat apart, in that we state results whose proofs are merely sketched (and presuppose some knowledge of stratification theory). The reason is that although we don't make use of these facts, it is good to be aware of them.

The link of an isolated singularity

In this section X is an analytic set in an open ∪ ⊂ C^N and x ∈ X is such that X−{x} is nonsingular of constant dimension n. The main result will be that at x, X is homeomorphic to a cone over a C^∞-manifold and that this manifold is unique up to diffeomorphism.

(2. 1) Curve Selection Lemma. Let V be an open neighbourhood of p ∈ ℝ^m and let f₁, …, f_K, g₁, …, g_ℓ be real-analytic functions on V such that p is in the closure of Z ≔ {x ∈ V : f_κ(y) = 0 κ=1, …, k; g_λ(y) > 0 λ=1, …, ℓ}.