New open-book decompositions in singularity theory

Aguilar-Cabrera, Haydée

doi:10.1007/s10711-011-9622-z

Cited by 8 publications

(15 citation statements)

References 25 publications

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“…When n − p is odd, so that the link K of f is even-dimensional, this invariant equals 1 2 χ(K), the Euler characteristic of the link, and it coincides with the Euler characteristic of the Milnor fibre. When n−p is even this construction only gives an invariant in Z/2Z, so we modify as follows.…”

Section: Introductionmentioning

confidence: 97%

“…Its counter-part for real singularities R n → R p , n > p, also comes from Milnor's book, and there are several interesting articles on the topic published by various people in the 1970s and 1980s, as for instance by E. Looijenga, P. T. Church and K. Lamotke, N. A'Campo, B. Perron, L. Kauffman, W. Neumann, A. Jacquemard and others. Later, in the mid 1990s, a new wave of interest on the topic arose, and nowadays the study of Milnor fibrations for real singularities is an active field of research (see for instance [44,5,38,9,39,35,36,6,1] or the survey article [12]).…”

Section: Introductionmentioning

confidence: 99%

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On the topology of real analytic maps

2014

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We study the topology of the fibers of real analytic maps ℝn → ℝp, n > p, in a neighborhood of a critical point. We first prove that every real analytic map-germ f : ℝn → ℝp, p ≥ 1, with arbitrary critical set, has a Milnor–Lê type fibration away from the discriminant. Now assume also that f has the Thom af-property, and its zero-locus has positive dimension. Also consider another real analytic map-germ g : ℝn → ℝk with an isolated critical point at the origin. We have Milnor–Lê type fibrations for f and for (f, g) : ℝn → ℝp+k, and we prove for these the analogous of the classical Lê–Greuel formula, expressing the difference of the Euler characteristics of the fibers Ff and Ff,g in terms of an invariant associated to these maps. This invariant can be expressed in various ways: as the index of the gradient vector field of a map [Formula: see text] on Ff associated to g; as the number of critical points of [Formula: see text] on Ff; or in terms of polar multiplicities. When p = 1 and k = 1, this invariant can also be expressed algebraically, as the signature of a certain bilinear form. When the germs of f and (f, g) are both isolated complete intersection singularities, we exhibit an even deeper relation between the topology of the fibers Ff and Ff,g, and construct in this setting, an integer-valued invariant, that we call the curvatura integra that picks up the Euler characteristic of the fibers. This invariant, and its name, spring from Gauss' theorem, and its generalizations by Hopf and Kervaire, expressing the Euler characteristic of a manifold (with some conditions) as the degree of a certain map.

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Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

On the topology of real analytic maps

2014

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“…For the triple (2, 3, 5), the group is the binary icosahedral group and L f is Poincaré's homology 3-sphere. Notice that if we order p, q, r so that p ≤ q ≤ r then the condition 1/p + 1/q + 1/r > 1 is satisfied only for the triples (2, 2, r), for every r ≥ 2, (2, 3, 3), (2,3,4) and (2,3,5). In all cases the singularity we obtain is a rational double point, also called a Klein or Du Val singularity.…”

Section: Low Dimensional Manifolds Ifmentioning

confidence: 99%

“…(2) If n − p = 3, non-trivial examples exist for (5, 2) and (8,5), and perhaps for (6, 3). In particular, if p = 2, such examples exist for all n ≥ 4.…”

Section: Foundations and First Stepsmentioning

confidence: 99%

On Milnor’s fibration theorem and its offspring after 50 years

Seade

2018

Bull. Amer. Math. Soc.

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To Jack, whose profoundness and clarity of vision seep into our appreciation of the beauty and depth of mathematics.Abstract. Milnor's fibration theorem is about the geometry and topology of real and complex analytic maps near their critical points, a ubiquitous theme in mathematics. As such, after 50 years, this has become a whole area of research on its own, with a vast literature, plenty of different viewpoints, a large progeny and connections with many other branches of mathematics. In this work we revisit the classical theory in both the real and complex settings, and we glance at some areas of current research and connections with other important topics. The purpose of this article is two-fold. On the one hand, it should serve as an introduction to the topic for non-experts, and on the other hand, it gives a wide perspective of some of the work on the subject that has been and is being done. It includes a vast literature for further reading. IntroductionMilnor's fibration theorem in [189] is a milestone in singularity theory that has opened the way to a myriad of insights and new understandings. This is a beautiful piece of mathematics, where many different branches, aspects and ideas, come together. The theorem concerns the geometry and topology of analytic maps near their critical points.Consider the simplest case, a holomorphic map (C n+1 , 0) f → (C, 0) taking the origin into the origin, with an isolated critical point at 0. As an example one can have in mind the Pham-Brieskorn polynomials:(0.1) z → z a 0 0 + · · · + z an n , a i ≥ 2 for all i = 0, 1, · · · , n .Since f is analytic, there exists r > 0 sufficiently small so that 0 ∈ C is the only critical value of the restriction f | Br , where B r is the open ball of radius r and center at 0. Set V := f −1 (0) and V * := V \ {0} .

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“…It is important to notice that our approach is exhaustive, we have computed all conditions that a polar weighted homogeneous polynomial must satisfy in order to have an isolated critical point. In our classification we get some known families for example: the twisted Brieskorn-Pham polynomials [34] and the family studied by Haydée Aguilar [2].…”

Section: Vertices Of the Dual Graph Of The Minimal Resolution Of Xmentioning

confidence: 99%