Real map germs and higher open book structures

Abstract

The paper focusses on the existence of higher open book structures defined by real map germs ψ :A general existence criterion is proved, with view to weighted-homogeneous maps.Keywords Real singularities · Milnor fibration · Links · Open book decompositions · Stratifications Mathematics Subject Classification (2000) 32S55 · 57Q45 · 32C40 · 32S60 IntroductionLet ψ : (R m , 0) → (R p , 0) be a non-trivial real analytic map germ. Let V := {ψ = 0} denote the germ at 0 of the zero locus of ψ, let B m ε ⊂ R m be the open ball of radius ε at the origin, let S m−1 ε := ∂ B ε and K ε := V ∩ S m−1 ε . Milnor studied in [11, §11] the setting Sing ψ ⊂ {0}, which means that the singular locus of the map germ ψ consists of at most the origin 0 ∈ R m . He shows in [11, p. 97-99] that for any small enough ε > 0, there is a smooth fiber bundle:

Singular open book structures from real mappings

“…Several recent papers considered this situation with respect to the existence of the Milnor fibration (1), see e.g. [2,4,18]. Let (b) This fibration has a regularity property in the neighbourhood of the link, namely, it is induced by a more refined fibration with codimension fibres.…”

“…If we consider a holomorphic mapping C → C with > 1 then there might not exist fibrations like (2) 1 or the Thom regularity may fail 2 along V . In the purely real analytic setting, the mapping ψ may have neither property (a) nor property (b).…”

“…For a more general setting, Sing ψ ∩ V ⊂ {0} and any ≥ ≥ 2, an existence criterion was given in [1] and shown to be more general than the previous ones from the literature.…”

“…Thom has called such ρ the fonction tapissante. As in [2], in this paper we shall use as a function ρ the Euclidean distance function and its open balls and spheres of radius ε, denoted by B ε and S ε respectively. The open set U from Definition 1.2 will be the small ball B ε , unless otherwise stated.…”

“…After [3] and [16], we call polar weighted-homogeneous if there are non-zero integers 1 and such that gcd ( 1 ) = 1 and…”