Milnor–Hamm sphere fibrations and the equivalence problem

Geometrical Conditions for the Existence of a Milnor Vector Field

“…More recently, in dos Santos et al (2019Santos et al ( , 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.…”

Milnor–Hamm Fibration for Mixed Maps

“…, where the fibers of G intersect it transversely, we say that G is ρ-regular. 2 The following condition (3) was used in Araújo dos Santos et al (2019a) and Araújo dos Santos et al (2019b) to ensure the ρ-regularity for G. We will see below that it is a condition that provides us the existence of a locally trivial smooth fibrations,…”

“…Condition (3) ensures that the restriction (4) is the projection of a locally trivial smooth fibration. As explained in Araújo dos Santos et al (2019b) and Araújo dos Santos and Ribeiro (2018), if Disc G is radial, the fibration (4) may be composed with the canonical projection π := s/ s : C k \{0} → S 2k−1 to get a locally trivial smooth fibration…”

“…Different from what was done in Chen et al (2014), we consider the problem of the existence of local fibration structures associated to mixed maps, in the classical sense where the discriminant set Disc G = G(Sing G) is just the origin and also the more general fibration structures where Disc G have positive dimension, like those called Milnor-Hamm tube fibrations and Milnor-Hamm sphere fibration defined in the recent papers Araújo dos Santos et al (2019a) and Araújo dos Santos et al (2019b). We present in Sect.…”

“…2.4 a series of new mixed maps built from classes of mixed function defined in recent works, as Oka (2015Oka ( , 2018aOka ( , b, 2019a, Parameswaran and Tibȃr (2018), Ribeiro (2019) and Grulha and Martins (2020). Moreover, we study the equivalence between Milnor-Hamm fibrations by using the approach introduced in Araújo dos Santos et al (2019b) and the geometrical conditions presented in Araújo dos Santos and Ribeiro (2018).…”