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Vanishing zones and the topology of non-isolated singularities

  • Aurélio Menegon
  • José Seade
Original Paper
  • 9 Downloads

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 fg 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.

Keywords

Milnor-Lê fibration Milnor fiber Link Vanishing zone Lê’s polyhedron Non-isolated singularity 

Mathematics Subject Classification (2000)

Primary 14J17 14B05 32S05 32S25 32S30 32S45 Secondary 14P15 32C05 

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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Universidade Federal da ParaíbaJoão PessoaBrazil
  2. 2.Universidad Nacional Autónoma de MéxicoCuernavacaMexico

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