LAUSR

We compare the topology of the link $$L_0$$L0 of non-isolated singularities defined by real analytic map-germs $$({\mathbb {R}}^m,0) \buildrel {h} \over {\rightarrow } ({\mathbb {R}}^n,0)$$(Rm,0)→h(Rn,0), $$m > n$$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 $$0 \in {\mathbb {R}}^n$$0∈Rn 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 $$L_0 \cap \Sigma $$L0∩Σ, where $$\Sigma $$Σ denotes the critical set of h, such that away from it the topology of $$L_0$$L0 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 $$L_0 \cap \Sigma $$L0∩Σ. A particular class of real singularities we envisage in this paper are those of the type $$f\bar{g}: ({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$$fg¯:(Cn,0)→(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 $$a_f$$af-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.