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 ∈ ℝ^m. 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 ∈ ℝ^m 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 ℝ^{2 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.