Isolated critical points of mappings from $\mathbf{R}^4$ to $\mathbf{R}^2$ and a natural splitting of the Milnor number of a classical fibered link. Part I: Basic theory; examples

Details Isolated critical points of mappings from $\mathbf{R}^4$ to $\mathbf{R}^2$ and a natural splitting of the Milnor number of a classical fibered link. Part I: Basic theory; examples Isolated critical points of mappings from $\mathbf{R}^4$ to $\mathbf{R}^2$ and a natural splitting of the Milnor number of a classical fibered link. Part I: Basic theory; examples

2002-03-04

arxiv.org/abs/math/0203032v2

Commentarii Mathematici Helvetici 62 (1987) 630-645

Mathematics

Geometric Topology

HyperTeX version (correcting minor typographical errors; with addenda) prepared February 2002

Scientific paper

From a fibered link in the 3-sphere may be constructed a field of not everywhere tangent 2-planes; when the fibered link is the link of an isolated critical point of a map from 4-space to the plane, the plane field is essentially the field of kernels of the derivative of the map. Homotopically, such a plane field determines two integers. I show that the sum of these integers is the Milnor number of the fibered link. Taking the mirror image of a link exchanges the integers. Various examples are computed. It is noted (proof given elsewhere) that these integers are not determined by the algebraic monodromy (or Seifert form) of the fibered link.

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