Milnor Fibrations and the Thom Property for maps fḡ

Anne Pichon and José Seade

Note that erratum exists for this article.

Journal of Singularities
volume 3 (2011), 144-150

Received: 14 March 2011. Received in revised form: 25 October 2011.

DOI: 10.5427/jsing.2011.3i

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Abstract:

We prove that every map-germ fḡ: (C^n, 0) →(C, 0) with an isolated critical value at 0 has the Thom a_{fḡ}-property. This extends Hironaka's theorem for holomorphic mappings to the case of map-germs fḡ and it implies that every such map-germ has a Milnor-Lê fibration defined on a Milnor tube. One thus has a locally trivial fibration φ: S_ε – K → S^1 for every sufficiently small sphere around 0, where K is the link of fḡ and in a neighbourhood of K the projection map φ is given by fḡ/ | fḡ|.

Keywords:

Whitney stratifications, Thom a_f property, real singularities, Milnor fibrations

Mathematical Subject Classification:

Primary: 32S55, 32C05, 57Q45.

Author(s) information:

Anne Pichon	José Seade
Aix-Marseille Université	Instituto de Matemáticas, Unidad Cuernavaca,
Institut de Mathématiques de Luminy UMR 6206 CNRS	Universidad Nacional Autónoma de México
Campus de Luminy - Case 907	A. P. 273-3, Cuernavaca, Morelos, México.
13288 Marseille Cedex 9, France
email: pichon@iml.univ-mrs.fr	email: jseade@matcuer.unam.mx