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On the topology of the Milnor fibrations
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| Author(s): |
Rafaella de Souza Martins
Total Authors: 1
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| Document type: | Doctoral Thesis |
| Press: | São Carlos. |
| Institution: | Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB) |
| Defense date: | 2018-02-16 |
| Examining board members: |
Nivaldo de Góes Grulha Junior;
Aurélio Menegon Neto;
José Luis Cisneros Molina;
Michelle Ferreira Zanchetta Morgado
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| Advisor: | Nivaldo de Góes Grulha Junior; José Antonio Seade Kuri |
| Abstract | |
In this thesis two types of problems related to the famous Milnor Fibration Theorem and Milnor-Lê Fibration Theorem for the real case with non-isolated critical values will be addressed. Primarily, we assure the fibrations of type Milnor-Lê for the germ F : (X, 0) → (Y, 0) subanalytic with X and Y subanalytic spaces on C \\ {0} a subanalytic connected curve in Y and over a smooth analytical subspace W ⊂ Y of dimension p, n &ge p ≥ 2, under some conditions. In particular, we show the existence of the fibrations about the discriminants of subanalytical map-germ, if this not been studied in the literature, usually the set of critical values are disregarded. Finalizing our analysis of this subanalytic category, we certify that there exist the fibrations of type Milnor-Lê to f : (X, 0) → (Rp, 0), with dimension of X greater than p ≥ 2, subanalytic and X subanalytic with non-isolated critical values, setting d -regularity. We address these problems using results of the rugose vector fields. In a second part, we present a new necessary and sufficient criterion to verify the important transversality property of a real map-germ f of class Cl, l ≥ 1. Using a recent developed tool, D-regularity, we verify conditions for the existence of the fibrations of map-germ Ψ F, X : (Cn, 0) → (C, 0) non holomorphic, given by Ψ (z, z̄) = Σnj=1 kjtjzj ajzb<sup<j, aj, bj ≥ 0 with aj = bj for at least one j and aj ≠ bj for at leeast one j = 1, ..., n. We note that Ψ F, X are mixed weighted homogeneous polynomials with R+-action. We consider ΨF, X : (R2n, 0) → (R2, 0) real analytic map-germ. We studied the topology of these fibrations, noting that the discriminant has dimension 1 and therefore has both the fibrations known. Lastly we show a homeomorphism between the fibers of the regular values and the critical values for a case special this family. (AU) | |
| FAPESP's process: | 13/22718-0 - On the Topology of the Milnor Fibrations |
| Grantee: | Rafaella de Souza Martins |
| Support Opportunities: | Scholarships in Brazil - Doctorate |