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Plane curve singularities via divides

Norbert A'Campo, Pablo Portilla Cuadrado

Abstract

Generic relative immersions of compact one-manifolds in the closed unit disk, i.e. divides, provide a powerful combinatorial framework, and allow a topological construction of fibered classical links, for which the monodromy diffeomorphism is explicitly given as a product of Dehn twists. Complex isolated plane curve singularities provide a classical fibered link, the Milnor fibration, with its Milnor monodromy, monodromy group, and vanishing cycles. This surveys puts together much of the work done on divides and their role in the topology of isolated plane curve singularities.

Plane curve singularities via divides

Abstract

Generic relative immersions of compact one-manifolds in the closed unit disk, i.e. divides, provide a powerful combinatorial framework, and allow a topological construction of fibered classical links, for which the monodromy diffeomorphism is explicitly given as a product of Dehn twists. Complex isolated plane curve singularities provide a classical fibered link, the Milnor fibration, with its Milnor monodromy, monodromy group, and vanishing cycles. This surveys puts together much of the work done on divides and their role in the topology of isolated plane curve singularities.

Paper Structure

This paper contains 30 sections, 19 theorems, 198 equations, 45 figures.

Table of Contents

  1. Introduction
  2. Preliminary theory on mapping class groups and plane curves
  3. Mapping class groups
  4. Isolated plane curve singularities
  5. The versal deformation space and the geometric monodromy group
  6. Definition of divide and some combinatorial properties
  7. Definition of divide
  8. Adapted Morse function $f_P$ on $D$ and its extension $\theta_{P}$ on $B^4_D$
  9. General fibration theorem
  10. A connected divide yields a fibered link
  11. Divides for plane curve singularities
  12. Totally real models of plane curve singularities
  13. Divides from an embedded resolution
  14. Divides from Chebyshev polynomials
  15. Description of the Milnor fiber
  16. ...and 15 more sections

Key Result

Theorem 2.1.4

Let $\phi: \Sigma \to \Sigma$ be an orientation preserving homeomorphism that restricts to the identity on $\partial \Sigma$. Then there exists $\phi'$ isotopic to $\phi$ and a collection $\mathcal{C}$ of non-null-homotopic disjoint simple closed curves (called reduction curves) including all bound

Figures (45)

  • Figure 2.1.1: A right-handed Dehn twist acting on a properly embedded segment. The orientation of the annulus is indicated by the curved arrows.
  • Figure 3.1.1: On the left we see a divide where there are two ordinary singularities of order $3$. On the right we see a divide that is obtain from the first one after a small perturbation.
  • Figure 3.1.2: The figure eight knot.
  • Figure 3.1.3: The movement caused by an admissible isotopy near an intersection point.
  • Figure 3.1.4: Two divides that yield the same knot but that are not equivalent through admissible isotopies.
  • ...and 40 more figures

Theorems & Definitions (64)

  • Definition 2.1.1
  • Definition 2.1.3
  • Theorem 2.1.4: See Thu and Corollary 13.3 from Farb
  • Definition 2.1.5
  • Definition 2.1.6
  • Definition 2.2.2
  • Definition 2.3.2
  • Definition 2.3.3
  • Remark 2.3.4
  • Definition 3.1.1
  • ...and 54 more