Immersions associated with holomorphic germs

Abstract

A holomorphic germ Φ : (C 2 , 0) → (C 3 , 0), singular only at the origin, induces at the links level an immersion of S 3 into S 5 . The regular homotopy type of immersions S 3 S 5 are determined by their Smale invariant, defined up to a sign ambiguity. In this paper we fix a sign of the Smale invariant and we show that for immersions induced by holomorphic gems the sign-refined Smale invariant Ω is the negative of the number of cross caps appearing in a generic perturbation of Φ. Using the algebraic method we calculate Ω for some families of singularities, among others the A-D-E quotient singularities. As a corollary, we obtain that the regular homotopy classes which admit holomorphic representatives are exactly those, which have non-positive sign-refined Smale invariant. This answers a question of Mumford regarding exactly this correspondence. We also determine the sign ambiguity in the topological formulae of Hughes-Melvin and Ekholm-Szűcs connecting the Smale invariant with (singular) Seifert surfaces. In the case of holomorphic realizations of Seifert surfaces, we also determine their involved invariants in terms of holomorhic geometry. A. Némethi and G. PintérAs we will see, Imm hol (S 3 , S 5 ) is not symmetric with respect to a sign change of Z, hence, in order to identify the subset Imm hol (S 3 , S 5 ) without any sign-ambiguity, we will fix a 'canonical' generator of π 3 (SO (5)). This will be done via the ismorphisms π 3 (U (3)) → π 3 (SO(6)) → π 3 (SO(5)) and by fixing a canonical generator in π 3 (U (3)) (see 4.2). Sometimes, to emphasize that we work with the Smale invariant with this fixed sign convention, we refer to it as the sign-refined Smale invariant. Our second goal is to determine the correct signs (compatibly with the above choice of generators) in the existing topological formulas, which were stated only up to a sign-ambiguity.1.2. The set Imm hol (S 3 , S 5 ). One expects that the analytic geometry of holomorphic realization imposes some rigidity restrictions, and also provides some further connections with the properties of complex analytic spaces. Mumford already in 1961 in his seminal article [16] asked for the characterization of the Smale invariant of a holomorphic (algebraic) immersion in terms of the analytic/algebraic geometry. This article provides a complete answer to his question. A more precise formulation of our guiding questions are: Question 1.2.1.(a) Which are the regular homotopy classes Imm hol (S 3 , S 5 ) and Emb hol (S 3 , S 5 ) represented by holomorphic germs?(b) How can a certain regular homotopy class be identified via complex singularity theory, that is, via algebraic or analytic invariants of the involved analytic spaces? Furthermore, if some Φ realizes some Smale invariant (e.g., if its Smale invariant is zero), then what kind of specific analytic properties Φ must have?The main results of this paper provide the following answer in the case of immersions. Theorem 1.2.2. (a) Imm hol (S 3 , S 5 ) is identified via the sign-refined Smale invariant Ω(f ) by the s...

On k-folding map-germs and hidden symmetries of surfaces in the Euclidean 3-space

“…Milnor showed that two isolated hypersurface singularities with same topological type have the same Milnor number. In the case of our invariants, a similar result can be obtained using the results in [28] and [30] (see [4] for details).…”

The boundary of the Milnor fibre of certain non-isolated singularities

“…We set S 3 = Φ −1 (S 5 ǫ ) = ∂B ǫ , diffeomorphic to S 3 , and we treat it as the usual Milnor-ball boundary 3-sphere. Recall that the immersion associated with Φ at the level of local neighbourhood boundaries is Φ| S 3 : S 3 → S 5 [21].…”

“…Write (X, 0) := (im(Φ), 0) and let f : (C 3 , 0) → (C, 0) be the reduced equation of (X, 0). Note that (X, 0) is a non-isolated hypersurface singularity, except when Φ is a regular map (see [21]). We denote by (Σ, 0) = (∂…”

“…In some sense it plays the role of the Milnor fibre, e.g. its (singular) boundary is the image of the stable immersion Φ| S 3 : S 3 S 5 associated with Φ [21], and it is homotop equivalent with a wedge of 2spheres [17,16]. For such germs Φ the disentanglement has a huge literature, it is even more studied than the Milnor fibre.…”

“…Here, the last identity follows from the algebraic definition of C(Φ), as the codimension of the Jacobian ideal of Φ (cf. [15,21,10]). In our case this ideal is genereted by t and d(s, t).…”