Topological invariants for singularities of real vector fields in dimension three
type: Item
instance of: scholarly article
title: Topological invariants for singularities of real vector fields in dimension three
zbMATH Open document ID: 1170.37013
published in: Discrete and Continuous Dynamical Systems
publication date: 2008-09-29T00:00:00Z
review text: This paper is devoted to the determination of topological invariants of absolutely isolated singularities of three-dimensional real vector fields. Following the notation of the paper, if \(\xi_0\) is the germ of an analytic vector field over \((\mathbb{R}^3, 0)\) with an isolated singularity at the origin, the oriented foliation \(\mathcal{L}_0\) defined by \(\xi_0\) is given by any germ of vector field of the form \(f\xi_0\), where \(f\) is a germ of a positive analytic function. This foliation is said to have an \textit{absolutely isolated singularity} at the origin if and only if the transform of \(\mathcal{L}_0\) under any finite sequence of blowing-ups has only isolated singularities and the exceptional divisor is invariant. As it has been shown in [\textit{C. Camacho, F. Cano} and \textit{P. Sad}, Invent. Math. 98, No. 2, 351--369 (1989; Zbl 0689.32008)], the oriented foliations with an absolutely isolated singularity at the origin, have a canonical reduction of singularities \(\pi: (M,D) \to (\mathbb{R}^3,0)\) formed by the composition of a finite sequence of blowing-ups and such that the singularities of the corresponding vector field at \(M\) have a non-nilpotent linear part. Note that the exceptional divisor is given by \(D\), that \(\partial M=D\) and that \(D = \bigcup_{i=1}^{n} D_i\), where \(D_i\) is an irreducible component created at each blowing-up. Two germs \(\xi_0\) and \(\xi_0'\) of analytic vector fields over \((\mathbb{R}^3,0)\) are said to be \textit{\(\pi\)-topologically equivalent} if there is a homeomorphism \(h: (M,D) \to (M,D)\) that respects the natural stratification of \(M\) induced by the decomposition \(D = \bigcup_{i=1}^{n} D_i\) and that induces an oriented topological equivalence between the transforms \(\mathcal{L}\) and \(\mathcal{L}'\) of each oriented foliation defined by \(\xi_0\) and \(\xi_0'\), respectively. In this paper, a complete classification for the \(\pi\)-topological equivalence in a wide class \(\mathcal{H}_\pi\)…
reviewed by: Maite Grau
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 5347947
zbMATH Keywords: qualitative theory
zbMATH Keywords: vector fields
zbMATH Keywords: topological classification
zbMATH Keywords: singularities
zbMATH Keywords: blowing-ups
zbMATH Keywords: saddle-connections
author: Clementa Alonso-González
author: Maria Izabel Tavares Camacho
author: Felipe Cano
MaRDI profile type: Publication
full work available at URL: dcds.2008.20.823
OpenAlex ID: W1997769676
DOI: 10.3934/DCDS.2008.20.823
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description: scientific article; zbMATH DE number 5347947