Singularities of vector fields on IR-3

Abstract

In his well known paper 'Singularities of vector fields', Takens made a topological classification of vector fields up to codimension 2 and introduced a semialgebraic stratification to distinguish the different cases; from dimensions greater than or equal to 3 he had to use the notion of 'weak-C-0-equivalence'. In this paper we show how to classify singularities of vector fields on R-3 up to codimension 4 for the notion of C-0 equivalence. To separate the different cases we use a semianalytic stratification and show that a semialgebraic one is not possible, even for the notion of weak-C-0-equivalence. Up to codimension 3 the stratification is semialgebraic. We will always suppose that the vector fields are C-infinity, although it will be clear that the results are valid for C-r, with r sufficiently big. We provide a complete, but short, survey of the different techniques to be used, referring to the existing literature for precise calculations and pictures. We put much emphasis on the new results.