On Real Singularities with a Milnor Fibration

Abstract

In this article we study the singularities defined by real analytic maps $$ \left( {\mathbb{R}^m ,0} \right) \to \left( {\mathbb{R}^2 ,0} \right) $$ with an isolated critical point at the origin, having a Milnor fibration. It is known [14] that if such a map has rank 2 on a punctured neighbourhood of the origin, then one has a fibre bundle φ : S m−1 − → S 1, where K is the link. In this case we say that f satisfies the Milnor condition at 0 ∈ ℝm. However, the map φ may not be the obvious map \( \frac{f} {{\parallel f\parallel }} \) as in the complex case [14, 9]. If f satisfies the Milnor condition at 0 ∈ ℝm and for every sufficiently small sphere around the origin the map \( \frac{f} {{\parallel f\parallel }} \) defines a fibre bundle, then we say that f satisfies the strong Milnor condition at 0 ∈ ℝ m. In this article we first use well known results of various authors to translate “the Milnor condition” into a problem of finite determinacy of map germs, and we study the stability of these singularities under perturbations by higher order terms. We then complete the classification, started in [20, 21] of certain families of singularities that satisfy the (strong) Milnor condition. The simplest of these are the singularities in ℝ2 n ≅ ℂ n of the form \(\{ \sum _{i = 1}^nz_i^{{a_i}}z_i^{ - {b_i}} = 0, {a_i} > {b_i} \geqslant 1\}\) We prove that these are topologically equivalent (but not analytically equivalent!) to Brieskorn-Pham singularities.KeywordsVector FieldIntegral ClosureReal SingularityGradient Vector FieldHolomorphic Vector FieldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.