Topological invariants of real Milnor fibres
type: Item
instance of: scholarly article
title: Topological invariants of real Milnor fibres
zbMATH Open document ID: 1033.32018
author: Zbigniew Szafraniec
published in: Manuscripta Mathematica
publication date: 2003-06-02T00:00:00Z
review text: Let \(F\) be an analytic mapping defined on an open neighbourhood \(D\) of \(0 \in {\mathbb R}^n\), \[ F:(D,0) \rightarrow ({\mathbb R}^m,0),\quad n>m, \] such that the origin is an isolated singular point of \(F^{-1}(0)\). For any regular value \(z \in {\mathbb R}^m\) which is close enough to the origin and a sufficiently small ball \(B_r=B(0,r)\), the fibre \(W_z=F^{-1}(z) \cap B_r\) is called the Milnor fibre. In complex case the topological type of the Milnor fiber does not depend on \(z\). In the real case this is no longer true. The author considers an analytic function \(g\) defined on \(B_r\) such that its restriction to the generic Milnor fibre is Morse. He studies the properties of the global index \[ \lambda (g, W_z) = \sum (-1)^{\lambda(p)}, \] where \(\lambda(p)\) is the Morse index of the critical point \(p\) of \(g_ {| W_z}\) and the sum extends over all such critical points. The main result of the paper is the proof that under some general assumptions there exist a matrix \(\Theta\) with analytic entries and analytic functions \(\theta_i\) such that \[ \lambda (g, W_z)= \text{signature}(\Theta(z))= \sum \text{sgn} : \theta_i (z). \]
reviewed by: Piotr Jaworski
Mathematics Subject Classification ID: 32S55
Mathematics Subject Classification ID: 14P15
Mathematics Subject Classification ID: 32B10
Mathematics Subject Classification ID: 58A07
Mathematics Subject Classification ID: 32C07
zbMATH DE Number: 1922437
zbMATH Keywords: Milnor fibration
zbMATH Keywords: singularities of real-analytic mappings
zbMATH Keywords: Morse functions
MaRDI profile type: Publication
full work available at URL: s00229-002-0326-4
OpenAlex ID: W2034282124
DOI: 10.1007/S00229-002-0326-4
description: scientific article; zbMATH DE number 1922437