Vanishing zones and the topology of non-isolated singularities

Abstract

We compare the topology of the link $L_0$ of non-isolated singularities defined by real analytic map-germs $({\mathbb {R}}^m,0) \buildrel {h} \over {\rightarrow } ({\mathbb {R}}^n,0)$, $m > n$, with that of the boundary of a local non-critical level of h. We show that if the germ of h has an isolated critical value at $0 \in {\mathbb {R}}^n$ and admits a local Milnor-Lê fibration at 0, then there exists “a vanishing zone for h”. This is an appropriate neighborhood of the set $L_0 \cap \Sigma$, where $\Sigma$ denotes the critical set of h, such that away from it the topology of $L_0$ is fully determined by the boundary of the corresponding local Milnor fibre. We give conditions for the vanishing zone to be a fiber bundle over $L_0 \cap \Sigma$. A particular class of real singularities we envisage in this paper are those of the type $f\bar{g}: ({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$ with f, g holomorphic and satisfying certain conditions. We introduce for these a regularity criterium for having a local Milnor-Lê fibration, and we use this to produce an example of a real analytic singularity which does not have the Thom $a_f$-property and yet has a local Milnor-Lê fibration. Throughout this work we provide explicit examples of functions satisfying the hypothesis we need in each section.