The Milnor fibration

Abstract

We begin this chapter by explaining what a fibration is, and giving a method of establishing that certain maps are fibrations. Then we show that two maps defined explicitly in terms of an isolated curve singularity give equivalent fibrations: each of these is termed ‘the Milnor fibration’. All the more delicate topology of C is encoded in the Milnor fibration, and studying its geometry gives a very close insight into the topology and geometry associated to the singularity. In this chapter, we give some elementary properties, leading to various calculations of the Betti numbers of the fibre. A detailed study will be made in Chapter 10. Fibrations A fibration is a sort of twisted product. More precisely, a map π : E → B is (the projection of) a fibration with fibre F if each point b ∈ B has a neighbourhood U such that there is a homeomorphism φ of π -1 ( U ) onto F × U whose second component is the restriction of π. Thus to construct the homeomorphism one needs only the first component, a map onto F , which may well be defined via a map onto F b = π -1 ( b ). Properties of fibrations are derived in full in textbooks of algebraic topology, e.g. [169], and we content ourselves here with citing those we shall require. If U is any contractible subset of B , then a homeomorphism φ as above may be constructed over U . In particular, if B = S 1 , we may take U to be either the upper or the lower semicircle: U + , U - say.