Artículos de revistas
On the topology of real analytic maps
Fecha
2014-06Registro en:
International Journal of Mathematics, Singapore, v.25, n.7, p.1450069-1-1450069-30, 2014
0129-167X
10.1142/S0129167X14500694
Autor
Cisneros-Molina, José Luis
Seade, José
Grulha Junior, Nivaldo de Góes
Institución
Resumen
We study the topology of the fibers of real analytic maps 'R POT.N'→'R POT.P', n>p, in a neighborhood of a critical point. We first prove that every real analytic map-germ f : 'R POT.N' → 'R POT.P', p ≥ 1, with arbitrary critical set, has a Milnor–Lê type fibration away from the discriminant. Now assume also that f has the Thom 'A IND.F'-property, and its zero-locus has positive dimension. Also consider another real analytic map-germ g : 'R POT.N' → 'R POT.K' with an isolated critical point at the origin. We have Milnor–Lê type fibrations for f and for (f, g) : 'R POT.N' → 'R POT.P+K', and we prove for these the analogous of the classical Lê–Greuel formula, expressing the difference of the Euler characteristics of the fibers 'F IND.F' and 'F IND.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 ˜g on 'F IND.F' associated to g; as the number of critical points of ˜g on 'F IND.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 'F IND.F' and 'F IND.F,G', 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 Hopf and Kervaire, expressing the Euler characteristic of a manifold (with some conditions) as the degree of a certain map.