Geometrical Conditions for the Existence of a Milnor Vector Field
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Geometrical Conditions for the Existence of a Milnor Vector Field Maico F. Ribeiro1
· Raimundo Nonato Araújo dos Santos2
Received: 19 February 2020 / Accepted: 7 September 2020 © Sociedade Brasileira de Matemática 2020
Abstract We introduce several sufficient conditions to guarantee the existence of the Milnor vector field for new classes of singularities of map germs. This special vector field is related with the equivalence problem of the Milnor fibrations for real and complex singularities, if they exit. Keywords Singularities of real analytic maps · Milnor fibrations · Mixed singularities · Stratification theory · Topology of subanalytic sets Mathematics Subject Classification 32S55 · 58K15 · 57Q45 · 32C40 · 32S60 · 32B20 · 14D06 · 58K05 · 57R45 · 14P10 · 32S20
1 Introduction For the holomorphic functions germs f : (Cn+1 , 0) → (C, 0) with dim Sing f ≥ 0, Milnor showed that for any small enough ε > 0 the restriction map f /| f | : Sε2n+1 \ K ε → S11
(1)
is a locally trivial smooth fibration, where K ε := Sε2n+1 ∩ f −1 (0). It was proved later by D˜ung Tráng (1977) that for any small enough > 0, there exists 0 < δ , such that the restriction map f | : Bε2n+2 ∩ f −1 (Dδ \ {0}) → Dδ \{0}
B
Maico F. Ribeiro [email protected] Raimundo Nonato Araújo dos Santos [email protected]; [email protected]
1
Departamento de Matemática, Universidade Federal do Espirito Santo, Vitória 29.075-910, Brazil
2
Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação - Universidade de São Paulo, Caixa Postal 668, São Carlos 13.560-970, Brazil
123
M. F. Ribeiro, R. N. Araújo dos Santos
is the projection of a locally trivial smooth fibration, where Bε2n+2 and Dδ are the open balls in the respective spaces. It is well known that the previous fibration induces the smooth fibration (2) f /| f | : Bε2n+2 ∩ f −1 Sη1 → S11 . Moreover, it follows from the Milnor (1968) and from Lê works (D˜ung Tráng 1977) that the fibrations (1) and (2) are equivalent. In the case of real analytic map germ G : (Rm , 0) → (R p , 0), m > p ≥ 2 with Sing G ⊂ G −1 (0) as a set germ, the authors of Massey (2010) and Parameswaran and Tib˘ar (2018) gave several conditions that ensure the existence of the empty tube fibration p−1 G/G : Bεm ∩ G −1 Sηp−1 → S1 . Under special conditions, in the papers (dos Santos et al. 2013; dos Santos and Tib˘ar 2008, 2010; Cisneros-Molina et al. 2010; Pichon 2005; Pichon and Seade 2003, 2008) it was proved the existence of the sphere fibration p−1
G/G : Sεm−1 \ K ε → S1
.
More recently, in dos Santos et al. (2019, 2020), the authors extended all previous results for the case when the singular set Sing G has positive dimension and is not necessarily included in the central fibre G −1 (0), i.e., when Disc G has positive dimension. However, even in the case where Sing G = 0 it is not known whether or not these two fibrations are equivalent. This equivalence problem has been approached by several authors in the last years, first in the case Disc G = {0}, e.g., dos Santos
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