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

Abstract

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} .

A Note On Non-Isolated Real Singularities and Links

“…For real analytic maps, and more particularly singular such maps, we can for the purpose of this paper mention the works of H. Hamm (e.g [4]), Z. Szafraniec ( [7]) and more recently the paper of Séade, Cisneros-Molina and Snoussi ( [1]) and the preprint ( [3]) of N. Dutertre. The result presented in following paper is in fact a mere consequence of the work of the latter authors, and should ideally be called a corollary of [6,Theorem 13.5] and [7, Lemma 1].…”

Results on Milnor Fibrations for Mixed Polynomials with Non-isolated Singularities

“…This gave rise to the theory of mixed and polar weighted singularities, developed by M. Oka, Cisneros-Molina and others (see [24] for an account on the subject). Mixed singularities are defined by complex valued real analytic functions on the variables (z 1 , · · · , z n ,z 1 , · · · ,z n ) and M. Oka proved in [16] important results about Milnor Fibrations for these singularities proving they are "strongly non-degenerate", a condition that springs from looking at the associated Newton polygon, as in the holomorphic setting.…”

“…The literature on this topic is vast and we refer for instance to [24] for an updated survey paper by Seade.…”

Fibration theorems à la Milnor for differentiable maps with non-isolated singularities

“…The map f is said to be d-regular at 0 if there exists ǫ 0 > 0 such that every X ℓ \V is transverse to every sphere centred at 0 and contained in B ǫ 0 , whenever the intersection is not empty. Several works have been published in the last two decades studying Milnor fibrations for real analytic maps with either an isolated critical point or value, see for instance [28,24,27,4,25,5,10,22,1,3,13,2] or the survey articles [6,11,12,26].…”