Singularities and Topology of Hypersurfaces

Frontmatter

Chapter 1. Whitney Stratifications

Abstract

A (smooth) manifold (i.e., a C^∞-manifold without boundary, of constant dimension) enjoys the next well-known homogeneity property, see, for instance, [M4], p. 22.

Alexandru Dimca

Chapter 2. Links of Curve and Surface Singularities

Abstract

We start with a brief account of knot theory for the following two reasons. First, the links of (plane) curve singularities—which are usually regarded as the simplest class of singularities to investigate—form a special class of knots, the so-called algebraic links. Second, many of the fundamental concepts related to the local topology of a higher dimensional IHS (e.g., Seifert matrix, intersection form, Milnor fibration, Alexander polynomial) have been considered first in relation to knot theory.

Alexandru Dimca

Chapter 3. The Milnor Fibration and the Milnor Lattice

Abstract

In this section we introduce various Milnor fibrations, in particular, the global Milnor fibration associated with a weighted homogeneous polynomial. Then we discuss the basic properties of the corresponding monodromy operators. Let O_n+1 = ℂ{x₀,…,x_n{ be the ℂ-algebra of analytic function germs at the origin 0 of ℂⁿ⁺¹ and let (X, 0) be a hypersurface singularity defined by an equation f = 0, for some f ∈ O_n+1 with f(0) = 0. Here n ≥ 0 is a positive integer. There are two equivalent fibrations which, in the literature, are called the Milnor fibration of the function germ f (or of the hypersurface singularity (X, 0)).

Alexandru Dimca

Chapter 4. Fundamental Groups of Hypersurface Complements

Abstract

We have seen in Chapter 2 that a basic idea in studying a link L ⊂ S³ is to investigate the topology of its complement S³\L. In particular, the fundamental group π₁(S³\L) of this space played a crucial role. Note that since most of the spaces of interest to us are path-connected, we usually pay no attention to base points.

Alexandru Dimca

Chapter 5. Projective Complete Intersections

Abstract

Although the topology of the complex projective space ℙⁿ is well known, we recall in this section some basic facts on it. The reason for doing this is:

(i)

to fix some notation useful in the sequel; and

 

(ii)

the topology of the projective complete intersections shares a lot of properties with the topology of ℙn.

 

Alexandru Dimca

Chapter 6. de Rham Cohomology of Hypersurface Complements

Abstract

In this chapter we work with regular differential forms in the sense of Algebraic Geometry or, depending on the context, in the sense of Analytic Geometry. Fix a positive integer n ≥ 0 and consider the affine space ℂⁿ⁺¹.

Alexandru Dimca

Backmatter