On the topology of real analytic maps

Autor(es)

Cisneros-Molina, Jose Luis ^{[1, 2]} ; Seade, Jose ^[1] ; Grulha, Jr., Nivaldo G. ^[3]

Afiliação do(s) autor(es)

^[1] Univ Nacl Autonoma Mexico, Unidad Cuernavaca, Inst Math, Cuernavaca 62210, Morelos - Mexico

^[2] Abdus Salam Int Ctr Theoret Phys, I-34014 Trieste - Italy

^[3] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Dept Matemat, BR-13566590 Sao Carlos, SP - Brazil

• Tipo de documento: Artigo Científico
• Fonte: INTERNATIONAL JOURNAL OF MATHEMATICS; v. 25, n. 7 JUN 2014.
• Citações Web of Science: 4

Resumo

We study the topology of the fibers of real analytic maps R-n -> R-p, n > p, in a neighborhood of a critical point. We first prove that every real analytic map-germ f : R-n -> R-p, p >= 1, with arbitrary critical set, has a Manor Le type fibration away from the discriminant. Now assume also that f has the Thom ay-property, and its zero-locus has positive dimension. Also consider another real anal tic map-germ g : R-n -> R-k with an isolated critical point at the origin. We have Milnor Le type fibrations for f and for (f, g) : R-n -> Rp+k, and we prove for these the analogous of the classical Le-Greuel formula, expressing the difference of the Euler characteristics of the fibers F-f and F-f,F-g in terms of an invariant associated to these maps. This invariant can be expressed in various ways: as the index of the gradient vector field of a map fj on Er associated to g; as the number of critical points of (g) over tilde on F-f; or in terms of polar multiplicities. When p = 1 and k = 1, this invariant can also be expressed algebraically, as the signature of a certain bilinear form. When the germs of f and (f, g) are both isolated complete intersection singularities, we exhibit an even deeper relation between the topology of the fibers Ff and F2,9, and construct in this setting, an integer-valued invariant, that we call the curvatura integra that picks up the Euler characteristic of the fibers. This invariant, and its name, spring from Gauss' theorem, and its generalizations by Hord and Kervaire, expressing the Euler characteristic of a manifold (with some conditions) as the degree of a certain map. (AU)

• Processo FAPESP: 13/11258-9 - Estudo topológico sobre variedades singulares e sobre aplicações entre variedades suaves
• Beneficiário: Nivaldo de Góes Grulha Júnior
• Modalidade de apoio: Auxílio à Pesquisa - Regular
• Processo FAPESP: 11/03185-6 - Classes características de variedades singulares
• Beneficiário: Nivaldo de Góes Grulha Júnior
• Modalidade de apoio: Auxílio à Pesquisa - Pesquisador Visitante - Internacional
• Processo FAPESP: 09/08774-0 - Obstrução de Euler de aplicações analíticas
• Beneficiário: Nivaldo de Góes Grulha Júnior
• Modalidade de apoio: Auxílio à Pesquisa - Regular