UNIVERSIDADE DE SÃO PAULO

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1 UNIVERSIDADE DE SÃO PAULO Instituto de Ciências Matemáticas e de Computação Singular Milnor fibrations Maico Felipe Silva Ribeiro Doctoral Dissertation of the Graduate Program in Mathematics (PPG-Mat)

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3 SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP Data de Depósito: Assinatura: Maico Felipe Silva Ribeiro Singular Milnor fibrations Doctoral dissertation submitted to the Instituto de Ciências Matemáticas e de Computação ICMC-USP, in partial fulfillment of the requirements for the degree of the Doctorate Program in Mathematics. FINAL VERSION Concentration Area: Mathematics Advisor: Prof. Dr. Raimundo Nonato Araújo dos Santos Co-advisor: Prof. Dr. Mihai Marius Tibăr USP São Carlos April 2018

4 Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassi e Seção Técnica de Informática, ICMC/USP, com os dados inseridos pelo(a) autor(a) R484s Ribeiro, Maico Felipe Silva Singular Milnor fibrations / Maico Felipe Silva Ribeiro; orientador Raimundo Nonato Araújo dos Santos; coorientador Mihai Marius Tibăr. -- São Carlos, p. Tese (Doutorado - Programa de Pós-Graduação em Matemática) -- Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Stratifications Theory. 2. Real and Complex Singularities. 3. Milnor Sphere fibration. 4. Milnor Tube fibration. 5. Geometry and Topology of Singularities. I. Araújo dos Santos, Raimundo Nonato, orient. II. Tibăr, Mihai Marius, coorient. III. Título. Bibliotecários responsáveis pela estrutura de catalogação da publicação de acordo com a AACR2: Gláucia Maria Saia Cristianini - CRB - 8/4938 Juliana de Souza Moraes - CRB - 8/6176

5 Maico Felipe Silva Ribeiro Fibrações de Milnor singulares Tese apresentada ao Instituto de Ciências Matemáticas e de Computação ICMC-USP, como parte dos requisitos para obtenção do título de Doutor em Ciências Matemática. VERSÃO REVISADA Área de Concentração: Matemática Orientador: Prof. Dr. Raimundo Nonato Araújo dos Santos Coorientador: Prof. Dr. Mihai Marius Tibăr USP São Carlos Abril de 2018

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7 To my girls, Gabriela and Lis.

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9 ACKNOWLEDGEMENTS Agradeço a Deus por confiar a mim o dom da vida. Agradeço a minha amada esposa Gabriela, pelo amor, amizade, companheirismo, dedicação e acima de tudo, pelo o que será até o fim de minha vida a minha melhor lembrança: o nascimento da nossa flor, Lis. Agradeço aos meus pais Evandro e Penha pelo amor incondicional, pelas orações, pelas oportunidades que me concederam ao longo de toda a vida e pelos bons exemplos. Agradeço a Irani, Braz e Bruna por terem me recebido de braços abertos em sua família e em sua casa, pelas orações, pelas palavras de apoio e por todo o suporte. Agradeço especialmente ao Professor Raimundo, meu orientador, pelo incentivo, por toda dedicação, paciência, amizade e pelo privilégio de poder trabalhar ao seu lado e desenvolver esta tese sob sua orientação. Seus ensinamentos foram muito além do mundo matemático e certamente passaram a fazer parte da minha vida. Obrigado. I am very grateful to Professor Mihai Tibăr, my co-advisor, for the opportunity to work together, for the mathematical teachings, for the patience and kindness you received me in Lille on the occasion of my visit. Agradeço a Professora Maria Aparecida Soares Ruas e aos Professores João Carlos Ferreira Costa e Bruno Cesar Azevedo Scárdua por terem aceitado avaliar esse trabalho e pelas excelentes sugestões. Agradeço ao Professor Leonardo Câmara pelo auxílio na escolha do ICMC como local de doutoramento. Gostaria de agradecer aos meus grandes amigos Fernando, Karlo, Thiago e Leandro, pelo incentivo, apoio e momentos de descontração. É um honra ter a amizade de vocês. I am very grateful to Professor Daniel Dreibelbis (University of North Florida)

10 for the careful grammar correction and great suggestions to improve the text of this work. Agradeço a todos os colegas, professores e funcionários do ICMC-USP que contribuíram, de alguma forma, na realização desse trabalho.

11 An equation means nothing to me unless it expresses a thought of God. (Srinivasa Ramanujan)

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13 ABSTRACT RIBEIRO, M. Singular Milnor fibrations f. Doctoral dissertation (Doctorate Candidate Program in Mathematics) Instituto de Ciências Matemáticas e de Computação (ICMC/USP), São Carlos SP. In this work we present the most recent developments in the direction of local fibrations structures of analytic singularities. Using techniques and tools from stratification theory we prove structural theorems in the stratified sense, which will be called singular Milnor tube fibration and Milnor-Hamm sphere fibration. In addition, we present algorithms with the purpose of creating a large number of examples in this new setting and compare our results obtained with the current ones found in the literature. Our results generalize all previous result in both cases: in the classical and in the stratified ones. Key-words: Stratification Theory, Real and Complex Singularities, Stratified Fibration Structures, Regularity Conditions, Milnor Sphere fibration, Milnor Tube fibration. Mixed singularities, Local Fibration Structures, Geometry and Topology of Singularities.

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15 RESUMO RIBEIRO, M. Fibrações de Milnor singulares f. Tese de Doutorado (Candidato ao Programa de Doutorado em Matemática) Instituto de Ciências Matemáticas e de Computação (ICMC/USP), São Carlos SP. Neste trabalho apresentamos os mais recentes desenvolvimentos na direção de estruturas de fibrações locais de singularidades analíticas. Usando técnicas e ferramentas da teoria de estratificação, provamos alguns teoremas estruturais no sentido estratificado, os quais serão chamados fibração singular de Milnor sobre o tubo e fibração de Milnor-Hamm sobre a esfera. Além disso, apresentamos algoritmos com o intuito de criar uma ampla variedade de exemplos e comparamos nossos resultados com os atuais encontrados na literatura. Nossos resultados generalizam todos os previamente existentes tanto no caso clássico, quanto no sentido estratificado. Palavras-chave: Teoria de Stratificação, Singularidades Reais e Complexas, Estruturas de Fibração estratificadas, Condições de regularidade, Fibrações de Milnor sobre esferas, Fibrações de Milnor sobre Tubo, Singularidades mistas, Estruturas de fibração local, Geometria e Topologia de singularidades.

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17 LIST OF FIGURES Figure 1 The Milnor tube for G(x,y,z) = (x,y(x 2 + y 2 + z 2 )) Figure 2 Curve of critical points of G/ G restricted to the sphere Figure 3 The Milnor tube for G(x,y,z) = (xy,xz) Figure 4 Geometric representation of condition (4.2) Figure 5 Milnor-Hamm fibration for G = (xy,z 2 ) Figure 6 Stratification of G 1 (Disc G) Figure 7 Fibers of restrictions ρ1 s, Gs 1,3 and its intersections Figure 8 Fibers of restrictions ρ1 c, Gc 1,1 and its intersections Figure 9 Fibers of restrictions ρ5 c, Gc 5,4 and its intersections Figure 10 Fibers of restrictions ρ9 c, Gc 9,3 and its intersections Figure 11 Milnor-Hamm sphere fibration for G = (xy,z 2 ) Figure 12 Milnor tube blowing to sphere Figure 13 x M(G) V G Figure 14 x M(G) \V G, a(x) > Figure 15 x M(G) \V G, a(x) < Figure 16 Diagram Figure 17 M(G) V G for G(x,y,z) = (xy,xz) Figure 18 Equivalent fibrations for G(x,y,z) = (xy,z 2 ) Figure 19 Diffeomorphism between Milnor Fibers

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19 LIST OF SYMBOLS B m ε m-dimensional open ball of radius ε centered at the origin. B m ε m-dimensional closed of radius ε centered at the origin. M boundary or frontier of the Manifold M. S m 1 ε (m 1)-dimensional sphere of radius ε centered at the origin of R m. S p 1 (p 1)-dimensional sphere of radius 1 centered at the origin of R p. D η the open disk of the C of radius η centered at origin. D η the closed disk of the C of radius η centered at origin. v norm of vector v. G real gradient of the map G. M ε,η,g Milnor tube of G. M(G) Milnor set of G. M (G) stratwise ρ-nonregular points of G or strawise Milnor set. Sing G singular set of G. Disc G discriminant set of G. V G the zero locus of the map G., the Euclidean dot product. JG(x) jacobian matrix of G in x. JG(x) t transpose of the JG(x).

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21 CONTENTS 1 PRELIMINARIES Differential Topology Fiber bundles Vector bundle Ehresmann Fibration Theorem Singularity theory Stratification Whitney regularity conditions Semianalytic Whitney stratifications Thom regularity Thom-Mather Isotopy Theorem CLASSICAL MILNOR FIBRATIONS Complex settings Real settings Isolated Singular Case: tube fibration Sphere fibration: Milnor s method Milnor s example Non-isolated singular case: tube fibration Existence of the Sphere fibration Revising the sphere fibration for holomorphic functions Fibration on sphere under Thom regularity condition Comparing the fibration structure on spheres under Thom regularity at V G and the condition (2.14) MIXED SINGULARITIES Mixed functions

22 3.2 Product of mixed functions Mixed product in separable variables Algorithm for MSL class Constructing MSL functions Mixed functions with polar action Milnor tube without Thom regularity MILNOR-HAMM FIBRATIONS Milnor-Hamm fibration The case Sing G V G = {0} Maps with full image The partial Thom regularity condition Classes of maps f ḡ with -Thom regularity Mixed f ḡ: A first extension for non-isolated critical value Milnor-Hamm without -Thom regularity SINGULAR MILNOR TUBE FIBRATIONS Stratwise Milnor set Relation with Thom regularity condition Mixed f ḡ: A second extension for non-isolated critical value SPHERE FIBRATIONS Existence of Milnor-Hamm Sphere Fibration Criteria for ρ-regularity A Matrix Criterion Other criteria to ρ-regularity Some examples in the classical case: Disc G = {0} GOOD VECTOR FIELDS Existence Topological approach Approach using differentials Some Comments on the method of [CSS2]

23 8 EQUIVALENCE OF MILNOR-HAMM FIBRATIONS Mixed functions and equivalence Maps with radial action Milnor-Hamm fibrations and equivalence Diffeomorphisms between the Milnor fibers BIBLIOGRAPHY Index

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25 23 INTRODUCTION In the famous Princeton notes of 1968 [Mi], John Milnor established the fundaments for studying fibration structures for germs of complex analytic functions f : (C n+1,0) (C,0) with dimsing f 0, and for real analytic map germs G : (R m,0) (R p,0), m > p 2 with isolated singular point at the origin, i.e., Sing G = {0} as a germ of a set. In these settings Milnor proved three fibrations structures theorems: one from sphere over the circle coming from holomorphic functions, and two fibration theorems (tube and sphere fibrations) for real analytic map germs, as we survey in Chapter 2. These results became a breakthrough on the studies of topological properties of analytic singularities and the developments of analytics invariants as can be seen in several papers published after Milnor s work. In this work we present new results on the local fibration structures for real analytic singularities. Our results will be stated for analytic maps germ but most results can be extended to any definable category where the Curve Selection Lemma holds. In our setting the image of analytic sets are subanalytic subsets of the target space and therefore the natural setting for the stratifications of the target is the subanalytic category. The results presented in Chapters 1 and 2 are classical and well known in the current literature. Our new contributions starts in Chapter 3. From this chapter on, the results previously known will be indicated by at least one reference, whereas the new results proved during this project will be stated without reference. The themes and chapters on this works are organized as follows. In Chapter 1 we group some classical definitions and results necessary for the forthcoming chapters. Other classical results used in the text will be presented at the appropriate time. We start with a short presentation of the classical tools from Differential

26 24 Introduction Topology, Singularity Theory, Stratification Theory and regularity condition as Whitney conditions and Thom regularity. We finish with the classical Ehresmann Theorem and the Thom-Mather Isotopy theorem. In Chapter 2 we present a brief survey about the local Milnor fibrations structures for real and complex singularities published in the Princeton notes [Mi]. As we will see these fibrations were the main gates for several others fibrations structures in the real and complex settings. In the first section, we recall three classic results on the existence of a locally trivial smooth fibrations for analytic complex maps given by Milnor, Lê and Hamm, see Theorem and Theorem In the second section, we present the classic result by Milnor in [Mi, Theorem 2.2.1] for real isolated singularity, and the Milnor method (see Section 2.2.2) developed in [AT1, AT2] and [ACT1] to guarantee the existence of locally trivial smooth fibration on spheres. Such a method will be adapted and used to prove more general fibration structures in Chapter 6. In the case of nonisolated singularities in Theorem 2.2.4, Theorem and Theorem , we present some tools and techniques recently developed in the literature regarding the existence of locally trivial smooth fibrations on the tube and on the sphere. We conclude this chapter by presenting Example , which shows that in the classical case (Disc G = {0}), the Theorem ] from [ACT1] is the most general result found in the literature concerning existence of Milnor s fibration on spheres. In Chapter 3 we focus on a special class of singularities called mixed singularities, which became a good fountain to look and search for new theories and examples with nice connections between the real and complex worlds. If you believe that Sometimes we have to look to the reality with complex glasses, but sometimes we should exchange the glasses and look to the complexity with real ones, you will recognize that the mixed class is worth studying on its own right. This chapter is organized in the following way: in the first section we consider some main definitions and notations about mixed functions. This will allow us to prove our main result in this section, Proposition 3.1.2, that will be used in Chapter 4 to construct classes of examples with the called Milnor-Hamm fibration. In Section 2, we extend some main results from the class of singularities of type f ḡ, f and g being holomorphic functions, to the product of mixed functions. See Corollary 3.2.1, Lemma and Proposition In the third section, our main result is Proposition 3.3.4, which is an algorithm that permits us to build a new class of functions

27 Introduction 25 called MSL. We will see that in this class the functions have isolated critical values, the Thom regularity, and also the ρ-regularity. In the last section, we prove Proposition and Lemma 3.4.3, aiming to construct a class of analytic maps germs which admits the Milnor tube fibration without the Thom regularity. The results and examples in this section extend the research started in the section 5 of [PT]. In Chapter 4 we introduce the so-called Milnor-Hamm fibration as an appropriate way to extend the classical Milnor tube fibration. This extension generalizes all previous results and allows us further generalities. For that, in the first section we introduce its definition and prove Theorem which gives a sufficient and necessary condition for its existence. Moreover, we show that several classes of maps satisfies this condition, see for instance, Proposition and Proposition concerning finitely determined map germs under the contact group structure. In the second section, the main result is Proposition 4.2.4, which gives a special class of maps where the topological type of the Milnor-Hamm fiber is uniquely defined. In the third section, we present a weaker Thom type regularity, called -Thom regularity, which also imply the existence of the Milnor-Hamm fibration. See Proposition This condition together with our Theorem allows us to construct plenty of examples. In the fourth section, in Theorem we present a first extension of the main result of the paper [PT] to case of a non-isolated critical value. This will show that the complex mixed functions of type f ḡ also provide a good class of functions to look for the Milnor-Hamm fibration structure. In analogy with Section 3.4.1, we finish this chapter with Proposition and Proposition 4.5.4, which permit us to construct a class of examples with the Milnor-Hamm fibration without -Thom regularity. In Chapter 5 we show how to extend the Milnor-Hamm fibration introducing the so-called singular Milnor tube fibration. For that, in the first section we define the stratwise Milnor set and a stratified transversality condition in order to prove its existence in Theorem We finish this first section with an example that shows that our result is not a consequence of the Thom-Mather isotopy Theorem. In the second section, we consider Corollary to compare our stratified condition with the Thom regularity condition, and in particular, we show that our result also extend the so-called Milnor-Lê fibration, see Theorem in Chapter 2. Moreover, we prove Theorem that gives a second extension of the main result of [PT] for functions f ḡ. This shows

28 26 Introduction that this class is still a good place to look for singularities with a singular Milnor tube fibration. We finish this chapter presenting another example of a map germ that has singular Milnor tube fibration and is not Thom regular. In Chapter 6, we present our main result on the existence of sphere fibrations in the more general setting, which we call the Milnor-Hamm sphere fibration. In the first section we introduce the definition of the Milnor-Hamm sphere fibration and prove our main result, Theorem 6.1.6, which gives sufficient conditions to ensure the existence of this fibration. This result is an extension of main result of [ACT1], namely, Theorem In the second section, we explore some criteria of the called ρ-regularity, which permit us to control the projection on the Milnor-Hamm sphere fibration. The main result of this section is Theorem that will be very important in Chapter 7. We finish this chapter applying our main criteria developed in the previous section for map germs with Disc G = {0}. We start Chapter 7 presenting a relationship between the equivalence problem for Milnor-Hamm fibrations and a special vector field which will be called a good vector field. Due to its importance we use this chapter to develop a study on sufficient conditions for its existence. In the first section, we prove Theorem 7.1.4, that is our first characterization for the existence of a good vector field using the coefficient a(x) introduced in Chapter 6. This is an improvement of [Han, Theorem 3.3.1, p. 26]. Moreover, our Proposition ensures the existence of a good vector field for some class of mixed complex functions. We finish the first section presenting Theorem , which is a refinement of the main results of Section 6.2 for the case Disc G = {0}. In the second section, we use tools from general topology to do a slight improvement of [Han, Lemma p. 32] in Theorem Our main result in this section is Theorem which gives a topological condition for the set M(G) \V G to ensure the existence of a good vector field. In the last section, we will address the existence problem using differentials of maps. Our main result Proposition 7.3.3, gives a second characterization for the existence of a good vector field. In the last chapter, Chapter 8, we show that for some classes of maps the Milnor- Hamm fibrations are equivalent, since they exist. For that we will use the main results developed in Chapter 7. In the first section our main results are Theorem 8.1.2, Proposition and Corollary 8.1.5, which consider the problem of equivalent fibrations in the

29 Introduction 27 class of mixed functions. In the second section, we concentrate our studies on the classes of maps with radial action. For that, we prove a first extension of [AT2, Theorem 3.1] in Theorem 8.2.5, for the case Disc G = {0}. In the sequel we consider the more general setting dimdisc G > 0 and prove our main result, Theorem 8.2.6, which is a second extension of [AT2, Theorem 3.1]. The latter result is the first approach in the literature toward the equivalence problem in this setting and it gives a partial answer to the question introduced in Chapter 7. We finish this chapter, and hence this thesis, presenting an answer to a weaker version of Conjecture

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31 29 CHAPTER 1 PRELIMINARIES In what follows and throughout this work, the word smooth will mean differentiable of class C. We will use the notation M n to mean a smooth n-dimensional manifold which is embedded into a Euclidean space R k for some k. When there is no risk of confusion, we will use the simplest notation M. The tangent space of M at p will be denoted by T p M and if G : M N is a smooth map such that G(x) = y then the differential map will be denoted by d x G : T x M T y N. 1.1 Differential Topology Next, we present the concept of transversality between spaces, which will be very important in deciding when a map is a smooth locally trivial fiber bundle. Definition Let M, N be manifolds and W N be a submanifold. Let G : M N be a smooth map. We say that G is transversal to W at x, G x W, if either G(x) / W or G(x) W and T G(x) N = T G(x) W + d x G(T x M). We say that G is transversal to W if G is transversal to W at x for any x M. Let M a smooth manifold, N M and W M submanifolds of M.

32 30 Chapter 1. Preliminaries Definition We say that N is transversal to W at x N W, N x W, if T x N + T x W = T x M. If N x W for any x N W, we say that N is transversal to W and we denote N W. Remark We remind that: (i) N W if and only if i N W, where i N : N M is the inclusion map. (ii) If N W, then N W is a submanifold of M and codimn W = codimn + codimw. (iii) If G : M N is a smooth map and W = {x} N is a point, then G W if and only if x is a regular value of G. If G : M N and H : M K are functions from a manifold M to manifolds N and K, respectively, we say that G and H meets transversally at x and we denote G x H, if both G and H are submersions at x and G 1 (G(x)) x H 1 (H(x)) Fiber bundles Definition A locally trivial fiber bundle is a quadruple (E, p,b,f), where E,B and F are topological spaces with B connected and p : E B is a continuous surjection which satisfies the local triviality condition: for any y B there exists a neighborhood V of y in B and a homeomorphism h : p 1 (V ) V F such that p = π 1 h, i.e., the below diagram is commutative p 1 (V ) p h V V F π 1 where π 1 : V F V is the natural projection. The space E is called the total space, B the base space of the bundle and F is the fiber. The map p is the map projection and the set of all (V α,h α ) is called local trivialization of the bundle.

33 1.1. Differential Topology 31 For any y B, the topological space p 1 (y) is homeomorphic to F and is called the fiber over y. If we can choose V α = B, we will say that (E, p,b,f) is a trivial bundle. In this case, one has that E is homeomorphic to B F. A classical theorem states that every locally trivial fiber bundle over a contractible base space is trivial. A smooth locally trivial fiber bundle is a locally trivial fiber bundle (E, p,b,f) such that E,B and F are smooth manifolds and all the functions above are required to be smooth maps. In this work, we will say just that the map p : E B is a locally trivial smooth fibration (or smooth projection of a locally trivial fiber bundle) if (E, p,b,f) is a smooth locally trivial fiber bundle with p a submersion. Definition Two locally trivial smooth fibrations p : E B and p : E B are said to be equivalent if there is a smooth diffeomorphism h : E E such that, p h = p, i.e., the follow diagram is commutative: E h E p B p Consequently, for all b B, h induces a map h b : p 1 (b) p 1 (b) which is a diffeomorphism Vector bundle A special class of fiber bundles, called vector bundles, are those whose fiber are vectors spaces. More precisely: the fiber bundle ξ := (E, p,b,r n ) is a real vector bundle of rank n if for every y B, p 1 (y) has a structure of a n-dimensional real vector space and for any local trivialization (V α,h α ), the restrictions h α : p 1 (y) {y} R n are isomorphisms of vector spaces.

34 32 Chapter 1. Preliminaries An important example of vector bundle is (T M,π,M,R n ) where the total space T M = T p M = ({p} T p M) p M p M and the projection π : T M M is given by π(p,v) = p. The vector bundle (T M,π,M,R n ) is called tangent bundle of a smooth n-dimensional manifold M. Definition We say that M is parallelizable if (T M,π,M,R n ) is trivial bundle. Definition Let ξ := (E, p,b,f) be a smooth locally trivial fiber bundle. A section of ξ is smooth map σ : B E such that p σ(y) = y for any y B. Definition Let M be a smooth manifold. A smooth vector field ν in M is a section of the vector bundle (T M,π,M,R n ). Remark If M R n, then we say that a vector field ν : M R n is tangent to M when ν(x) T x M for any x M Ehresmann Fibration Theorem Let M,N be smooth manifolds and G : M N a continuous map. We say that G is a proper map if for any compact subset K N, G 1 (K) M is a compact set. We know that if G : M N is a continuous map and M is compact, then G is a proper map. Next result gives a characterization of proper maps. Proposition Let G : M N be a continuous map between smooth manifolds. The following statements are equivalent: (i) G is a proper map. (ii) G is a closed map and for any y N the set G 1 (y) is compact. (iii) If (x k ) k N is a sequence in M such that the correspondent sequence (G(x k )) k N in N is convergent, then (x k ) k N has a convergent subsequence. The following result is a sufficient condition for a map between two smooth manifolds be the projection of a locally trivial fiber bundle. We will state it for manifolds with boundary, which will be the more general situation used in this work.

35 1.2. Singularity theory 33 Definition Let M be a smooth manifold with boundary M and N a smooth manifold without boundary. A map G : M N is a smooth submersion if both restrictions G : M \ M N and G : M N are submersions. Theorem [BJ, Mat, S4, Ehresmann Theorem] Let M be a smooth manifold with boundary M and N a smooth connected manifold without boundary. If G : M N is a smooth submersion, surjective and proper, then G is a map projection of a locally trivial smooth fibration. 1.2 Singularity theory In what follows and throughout this work we denote by G : (R m,0) (R p,0), m > p 2 a germ of non-constant analytic map and by f : (C n+1,0) (C,0) a germ of non-constant complex analytic function, unless otherwise stated. We also consider: (i) V G = G 1 (0), the zero locus of G; (ii) Sing G := {x R m rankjg(x) < p}, the singular locus of G; (iii) Disc G := G(Sing G), the discriminant of G; (iv) Im G := G(R m ), the image of G. We recall that all these objects are considered as germs of analytic set at the origin. Moreover, when Sing G V G or equivalently, Disc G = {0}, we will say that G has isolated critical value, and when Sing G = {0} we will say that G has isolated singular point Stratification Definition [GLPW] Let M be a smooth manifold and V a closed subset of M. A stratification of V is a collection W = {W α } α A of pairwise disjoint connected smooth submanifolds of M which verifies the following: (i) V = α AW α.

36 34 Chapter 1. Preliminaries (ii) W is locally finite, i.e., for any point x V, there exists a neighborhood U of x in V such that U W α /0 just to a finite number of indexes α A. (iii) W satisfies the frontier condition, i.e., if W α W β /0, where W β is the closure of W β on M, then W α W β. Each W α is called a stratum. The condition (iii) states that the frontier of a stratum is a union of strata. A stratification W of V is a refinement of a stratification W of V if and only if the strata in W are unions of the strata in W. Let W 1,...,W j be stratifications of subsets V 1,...,V j of smooth manifolds. We obtain a stratification of V 1 V j by taking its strata to be sets of the form W 1 W j with W i W i for any 1 i j. We call this the product stratification, and denote W 1 W j. Let W be a stratification of a subset V of N, and let G : M N be a smooth map transverse to W (i.e., transverse to any strata of W ). We obtain a stratification W of G 1 (V ) by taking the strata to be set of form G 1 (W α ) with W α W. We call W the induced stratification on G 1 (V ). In particular, let W be a stratification of a subset V of a smooth manifold M and U M be open. The inclusion i : U M is automatically transverse to W, so there is an induced stratification on U V which is called restriction of W to U Whitney regularity conditions Next we define the conditions of Whitney (a) and (b) for a stratification. These conditions help us to understand the behavior of a stratum when it approaches another stratum and therefore how they fit together. of V. Let M be a smooth manifold, V a subset of M and W = {W α } α A a stratification Definition Let (W i,w j ) be a pair of strata such that W i W j. Let x W i. We say that the pair (W i,w j ) satisfies the Whitney (a) condition along W i at the point x, if for any sequence (x n ) of points in W j which converges to x such the sequence of tangent spaces

37 1.2. Singularity theory 35 T xn W j converge to some space T in the appropriate Grassmannian bundle, then one has T x W i T. We say that the pair (W i,w j ) satisfies the Whitney (a) condition along W i if it satisfies the Whitney (a) condition along W i at any point x W i. Definition Let (W i,w j ) be a pair of strata such that W i W j. Let x W i. We say that the pair (W i,w j ) satisfies the Whitney (b) condition along W i at the point x, if for any sequence (x n ) of points in W j and any sequence (q n ) of points in W i which converges to x such the sequence of tangent spaces T xn W j converge to some space T in appropriate Grassmannian bundle and the sequence of lines x n q n converge to some line l then one has l T. We say that the pair (W i,w j ) satisfies the Whitney (b) condition along W i if it satisfies the Whitney (b) condition along any point x W i. Definition Let M be a smooth manifold and V a subset of M. A stratification W = {W α } α A of V is called Whitney (a) stratification (respectively, Whitney (b) stratification) if for any pair (W i,w j ) of strata of W such that W i W j, it satisfies Whitney (a) condition along W i (respectively, Whitney (b) condition along W i ). It is well known that all Whitney (b) stratification is a Whitney (a) stratification, but the converse is not true in general. In the case where the stratification is a Whitney (b) stratification we just say a Whitney stratification. The following are some interesting properties about Whitney stratifications, which will be important for the development of this work. For more details, see for instance [GLPW] and references. Theorem [GLPW, p.12] Let W 1,...,W j be Whitney stratifications of subsets V 1,...,V j of smooth manifolds. The product stratification W 1 W j is a Whitney stratification of V 1 V j. Theorem [GLPW, p.14] Let G : M N be the smooth map transverse to a stratification W of a subset V of N, and let W the induced stratification on G 1 (V ). If W is a Whitney stratification then so is W. In particular, if W is a Whitney stratification of a subset V of a smooth manifold M and U M is a open set, then the restriction of W to U is a Whitney stratification.

38 36 Chapter 1. Preliminaries Semianalytic Whitney stratifications In this section, K denote either R or C. Let U be a open set in K m. We remind that a map G : U K p is called analytic map (or analytic function, if p = 1), if it is locally given by convergent power series in m variables over the field K. In the case K = C, these are simply the holomorphic maps. Moreover, if I is a set of analytic functions, then the zero locus of I, is defined by V (I ) := {x K n G(x) = 0, G I }. When I = {G 1,...,G j } we will write V (G 1,...,G j ) instead of V (I ). The definitions and results of this sections can be found in [ML, Section 5]. Definition An analytic set X is a closed set in K n such that for all x X, there exists an open neighborhood W of x K n and a finite collection G 1,...,G j of analytic functions such that V (G 1,...,G j ) = W X. Next we consider the definition of semianalytic set, which will be more frequent in this work. Definition A subset X R m is a semianalytic set if, for all x X, there exists an open neighborhood W of x such that W X is a finite union of subsets of the form V (G 1,...,G k ) {x W H j (x) > 0, j = 1,...,l} where G 1,...,G k,h 1,...,H l are analytic functions on W. Definition A subset X C n is a semianalytic set if, for all z X, there exists an open neighborhood W of z such that W X is a finite union of subsets of the form V (G 1,...,G k ) {x W H j (x) 0, j = 1,...,l} where G 1,...,G k,h 1,...,H l are holomorphic functions on W. In this case, the subset X has classically been called an analytically constructible subset. Remark Note that, by negating functions, the definition of a semianalytic subset can also contain H α (x) < 0 and so, after taking unions, we can also obtain H α (x) 0, H α (x) 0 and H α (x) 0. The following are some important properties of semianalytic subset of R n.

39 1.2. Singularity theory 37 Theorem [ML, Theorem 6.3 ] Let X be a semianalytic subset of K. (i) The collection of semianalytic subsets of K is closed under finite unions, finite intersections, and taking complements. (ii) Every connected component of X is semianalytic. (iii) The family of connected components of X is locally finite. (iv) X is locally connected. (v) The closure and interior of X is semianalytic. Next, we present the Curve Selection Lemma, an important result in Singularity Theory. It exists in both real and complex form. A proof can be found in [Mi, chapter 3], see also [Lo, chapter 2]. Theorem (Curve Selection Lemma). Let X be a semianalytic subset of R m and x X. Then there exists a real analytic curve γ : [0,δ) R m with γ(0) = x and γ(t) X for any t (0,δ). Let X be a semianalytic subset of C n and z X. Then there exists a complex analytic curve γ : D η C n with γ(0) = z and γ(t) X for any t D η \ {0}. An interesting application of the Curve Selection Lemma is the following proposition, which say that analytic functions have isolated critical values Corollary Let G : U K be an analytic function, where U is a open set of K m with 0 U a critical point. Then G(0) is a isolated critical value. Definition Let X be a subset of R m. A semianalytic Whitney stratification (respectively, semianalytic stratification) W = {W α } α A of X is a Whitney stratification (respectively, stratification) of X such that each stratum W i is an analytic submanifold of R m and a connected semianalytic subset of R m. Theorem [W, Theorem 19.2] Any semianalytic set has a Whitney stratification. Any stratification has a refinement which is a Whitney stratificaton.

40 38 Chapter 1. Preliminaries Remark This Follows from the proof of Theorem that any semianalytic set has a semianalytic Whitney stratification. Theorem [ML, Theorem 7.14] Let X be a semianalytic subset of R m and W a Whitney stratification of X. Then for any enough small ε > 0, the sphere Sε m 1 transversely intersects all stratas W α of W such that 0 W α. We conclude this section with the next definition which plays a important rule in the development of the Section 5.1. Let G : (R m,0) (R p,0) be an analytic map germ. By the classical stratification theory, there exist germs of locally finite semianalytic Whitney stratifications (W,S) of the source and of the target 1 of G (thus every stratum is a nonsingular manifold, open and connected, as germ at the respective origins) such that G becomes a stratified submersion, and hence Disc G is a union of strata. In particular: (i) G maps a stratum of W onto a stratum of S, (ii) The restriction G : W α S β is a submersion, where W α W, and S β S. Definition We call such pair (W,S) a regular stratification of the map germ G Thom regularity Let G : (R m,0) (R p,0) be an analytic map germ and W a Whitney stratifications of G. Let W α and W β be strata of W, for which W α W β and the restrictions G W α and G Wβ have constant ranks. Let x W α. The two following definitions can be found in [GLPW] and [Mat]. Definition We say that W β is Thom regular over W α at x relative to G or, equivalently, the pair (W β,w α ) satisfies the Thom a G -condition at x, if the condition below holds: ( ) let {x n } W β sequence, such that x n converge to x. If kerd xn (G Wβ ) converge to a plane T, in the appropriate Grassmann bundle, then kerd x (G W α ) T. 1 In fact, in our case S is a subanalytic stratification. For more detail see [BM]

41 1.2. Singularity theory 39 We will say that W β is Thom regular over W α relative to G or, equivalently, the pair (W β,w α ) satisfies the Thom a G -condition, when the condition ( ) is satisfied for any point x W α. Definition Let G : (R m,0) (R p,0) be an analytic map germ and consider a regular stratification (W,S) of the map G as in Definition We say that the pair (W,S) is a Thom stratification of G when it satisfies the Thom regularity condition: for any pair of strata W α,w β W, for which W α W β, W β is Thom regular over W α relative to G. In such a case the triple (W,S,G) is called a Thom stratified mapping and for short we will say that G is a Thom mapping. One may weaken the above definition as follows. Definition Let G : (R m,0) (R p,0) be an analytic map germ. We say that G is Thom regular at V G (or, for short, that G is Thom regular, or that G has Thom regularity) if there exists a regular stratification (W,S) of the map germ G, as in Definition such that 0 is a point stratum in S, that V G is a union of strata of W, and the Thom a G -condition is satisfied at any stratum of V G. Let G : (R m,0) (R p,0) be an analytic map germ such that Sing G V G as germ of set. Consider a stratification W := {W α } α A of V G such that in a neighborhood U of the origin, W := {U \V G } {W α U} α A is a Whitney stratification. Remark Let G : (R m,0) (R p,0) be an analytic map germ such that Sing G V G as germ of set. If there exists a Whitney stratification W as above such that G is Thom regular at V G, (i.e., the pair (U \V G,W α ) satisfies the Thom a G -condition for any stratum W α ), we will say that G is Thom regular at V G in the classical sense, if it is necessary to emphasize that G has isolated critical value. The following result, due to H. Hironaka, guarantees that holomorphic functions have Thom regularity. Theorem [Hi, Corollary 1 p.248] Let f : E C be an algebraic map from a complex algebraic set E into a nonsingular complex curve. Then f has Thom regularity.

42 40 Chapter 1. Preliminaries In the case of hypersurfaces in C n+1, Lê Dũng Tráng and H. Hamm have proved this result in [HL, Theorem p.322] using the Łojasiewicz inequality, which we shall remind in the following. Definition [LZ, Łojasiewicz inequality] Let f : U C be a holomorphic function, where U C n is a open set such that 0 U and f (0) = 0. We say that f satisfies the Łojasiewicz inequality at the origin if, there exists a neighborhood W of 0 in U such that for any z W, one has c f (z) θ f (z) for some θ (0,1) and some c > 0. Remark It is well known that all holomorphic function satisfies the Łojasiewicz inequality. Unfortunately, for real analytic map germs G : (R m,0) (R p,0), the Thom regularity at V G is not generally satisfied, however it is well known that if the map germ has a isolated singular point at the origin then it is Thom regular at V G. The next result, which is a particular case of Proposition in Section 4.3, shows the importance of an analytic map germ G with isolated critical value be Thom regular at V G. Proposition Let G : (R m,0) (R p,0) be an analytic map germ with isolated critical value which is Thom regular at V G. Then there exists ε 0 > 0 such that, for any 0 < ε < ε 0, there exists η, 0 < η ε, such that the restriction map is a smooth submersion. G : S m 1 ε G 1 (B p η \ {0}) B p η \ {0} (1.1) In other words, since Sε m 1 V G /0, the Thom regularity at V G implies that for any ε > 0 small enough, there exists a neighborhood N ε of Sε m 1 V G in Sε m 1 such that for any x N ε \V G one has Sε m 1 x G 1 (G(x)).

43 1.2. Singularity theory Thom-Mather Isotopy Theorem The following is a version of the Thom-Mather isotopy theorem, which may be understood as an extension of Ehresmann Theorem. We can find a complete proof of this result in [Mat]. Theorem [Mat, Th2, Thom-Mather first isotopy theorem] Suppose that G : X N is a smooth, proper, stratified submersion. Then G is a (topological) locally trivial fibration, and the local trivializations can be chosen to respect the strata and to be diffeomeomorphisms when restricted to strata. In other words, Theorem states that for any q Im G, there exists an open neighborhood U of q in N and a homeomorphism h : G 1 (U) U G 1 (q), such that the restriction map G G = π h, where π is the projection from U G 1 (q) onto U. 1 (U) Moreover, if S = {S α } α A is a stratification of X, then for any stratum S α the restriction of h is a diffeomorphism from S α G 1 (U) to U (S α G 1 (q)). Therefore, we can say that G 1 (U) and U G 1 (q) have the same stratified topological type, and the same smoothness structure along the strata.

44

45 43 CHAPTER 2 CLASSICAL MILNOR FIBRATIONS In this chapter we will give a brief summary of the classical important results and their further developments found in the literature concerning the studies of fibrations structures on a neighbourhood of singularities of analytic map germs. Many of the results presented here will be generalized in this work, so for some of them we will present an idea of the proof, whenever we judge its importance for the next chapters. 2.1 Complex settings In this setting, it was shown that given a representative of an holomorphic function germ f : U C n+1 C with U an open set in C n+1, f (0) = 0, there exists a small enough real number ε 0 > 0 such that for any 0 < ε ε 0, φ := f f : S2n+1 ε \ K ε S 1 (2.1) is a smooth projection of a locally trivial fiber bundle, where K ε = f 1 (0) Sε 2n+1 called the link of the singularity at the origin. In chapters 5, 6 and 7 of [Mi], Milnor gave differentiable and topological descriptions of the link and the fibers F θ = φ 1 ( e iθ ), where e iθ S 1, showing that independent of the dimension of the singular locus, the fiber is a (2n)-dimensional smooth parallelizable manifold with the homotopy type of a n-dimensional CW-complex. is

46 44 Chapter 2. Classical Milnor Fibrations In addition, whenever Sing f = {0} Milnor associated to the singular point of f a multiplicity denoted by µ( f ), later named by several authors the Milnor number of the singularity, given by the topological degree of the map ε f f : S2n+1 ε Sε 2n+1. In this case it was shown that the fiber F θ has the same homotopy type of a bouquet of n-dimensional spheres µ( f ) i=1 Sn i, with µ( f ) spheres on the bouquet. In 1976, Lê Dũng Tráng in his article [Le] proved the existence of a general fibration structure on a complex analytic sets. We explain below an idea of such construction as described in the paper [CA]. Let X be an analytic set in an open neighbourhood U of the origin 0 C n+1. Let f : (X,0) (C,0) be a germ of holomorphic function. Theorem [Le, Milnor-Lê Fibration] For any small enough ε > 0, there exists η, with 0 < η ε, such that is a topological locally trivial fibration. f : B 2n+2 ε X f 1 (D η \ {0}) D η \ {0} (2.2) Proof. (Idea) Let W be a Whitney stratification of X and small enough ε > 0 such that the closed ball B 2n+2 ε intersects only a finite number of strata of X and such that the sphere Sε 2n+1, boundary of B 2n+2 ε, intersects all such strata transversally. According to Theorem , we can always choose this stratification in such way f is Thom regular at V f. By Proposition , this implies that for 0 < η ε the fibers of the map (2.2) intersect transversally the strata of X Sε 2n+1 and thus it is a stratified submersion. Now the result follow from Theorem An important point to notice here is that this topological fibration structure becomes a smooth fibration if X \V f is a non-singular analytic set in C n+1. See details in [Le, Ham]. As a particular case of the previous theorem one can state:

47 2.1. Complex settings 45 Corollary [Le, Existence of Milnor-Lê (tube) fibration] Let f : ( C n+1,0 ) (C,0) be a holomorphic function germ. Then, there exists small enough ε > 0, such that for any 0 < δ ε, the map f : B 2n+2 ε f 1 (D δ \ {0}) D δ \ {0} (2.3) is the projection of a locally trivial smooth fibration. In addition, for any small enough ε, there exists η, 0 < η ε, such that f : B 2n+2 ε f 1 ( S 1 η) S 1 η (2.4) is the projection of a locally trivial smooth fibration. Moreover, the fibrations (2.1) and (2.4) are equivalent. In the case of germs of holomorphic maps G : (C n+p,0) (C p,0) that is an ICIS - Isolated Complete Intersection Singularity, Hamm proved the following result. Theorem [Ham] Let G := (G 1,...,G p ) : (C n+p,0) (C p,0), p 1, be an ICIS at 0. Then, is a locally trivial smooth fibration. G : B 2(n+p) ε G 1 (B 2p η \ Disc G) B 2p η \ Disc G This fibration was also called the Milnor fibration and it generalizes the previous isolated singular case for functions. The discriminant set Disc G is a complex hypersurface of C p. Hence, it does not disconnect the complement B 2p η \ Disc G and the fiber F of G has the same diffeomorphism type. Moreover, F is a real 2n-dimensional smooth manifold with the homotopy type of a bouquet of n-dimensional spheres µ i=1 Sn i where now µ := rank H n (F,Z), the rank of the homology in the middle dimension with integer coefficient. [Lo]. One of the richest sources of information on ICIS is Looijenga s classical book

48 46 Chapter 2. Classical Milnor Fibrations 2.2 Real settings Isolated Singular Case: tube fibration Given a representative of G : (R m,0) (R p,0), m > p 2, Milnor proved that if G has isolated critical point at the origin 0 R m, then for any small enough ε > 0, there exists η, 0 < η ε, such that the restriction map G : B m ε G 1 (Sη p 1 ) Sη p 1 (2.5) is a smooth projection of a locally trivial fiber bundle. More precisely, it was proved that: Theorem [Mi, Theorem 11.2, p. 97] Let G : (R m,0) (R p,0) be a real analytic map germ such that Sing G = {0} as germs of an analytic set. Then, there exists ε 0 > 0 such that, for each ε, 0 < ε ε 0, there exists η, 0 < η ε, such that the complement of an open tubular neighborhood of (the link) K ε = V G Sε m 1 is the total space of a smooth fiber bundle over the sphere Sη p 1. Each fiber F is a smooth compact (n p)-dimensional manifold bounded by a copy of K ε. If the link K ε is not empty for any small enough ε > 0 it is a (m p 1)- dimensional closed smooth submanifold of the sphere and the fiber is (p 2)-connected. On the other hand, if the link K ε is empty then the manifold B m ε G 1 (Sη p 1 ) is diffeomorphic to the sphere Sε m 1 and the above fibration (2.5) becomes a Hopf-type fibration. It is well known that this case is only possible for the pairs of dimensions (m, p) {(4, 3),(8, 5),(16, 9)}, according to [CL, Lemma 1, p. 151]. Geometrically, a standard picture for the total space B m ε G 1 (Sη p 1 ) is as in the Figure 1 below, in the case the link K ε is not empty for any small enough ε. The boundary manifold B m ε G 1 (Sη p 1 ) looks like a tube surrounding the special fiber V G. For this reason several author called this space the Milnor tube. From now on, we will denote the Milnor tube by M ε,η,g.

49 2.2. Real settings 47 Figure 1 The Milnor tube for G(x,y,z) = (x,y(x 2 + y 2 + z 2 )) Sphere fibration: Milnor s method Concerning the sphere fibration in this real setting, Milnor presented the following remark without a proof [Mi, see remark on p.99]: with a little more effort one can prove that the entire complement Sε m 1 \ K ε also fibers on Sη p 1. In order to make it more precise, in [AT1, AT2] and [ACT1], the authors gave a complete prove of this remark. In the next, we show the main points of this remark together with an idea of the prove of Theorem We will split the proof in some steps and later, in section 6.1, we will use these steps again to extend some results of this section. Denote by G : U R p, 0 U a germ representative with Sing G = {0}. Then, if V G \ {0} is not empty it is an analytic manifold which is transversal to all small enough spheres. With this in mind one can track the following steps: Step 1: Under the condition Sing G = {0} one can shrink ε and η, if needed, to get a trivial smooth fibration G : S m 1 ε G 1 (B p η) B p η. (2.6) Therefore, by Ehresmann theorem for manifold with boundary one gets that the restriction

50 48 Chapter 2. Classical Milnor Fibrations map is a locally trivial smooth fibration. G : M ε,η,g S p 1 η (2.7) Denote by ρ(x) = x 2 the square of distance function to the origin and g(x) = G(x) 2. Since Sing G = {0} the vector g(x) is not null on B m ε \V G. Moreover, one can use the Curve Selection Lemma to show that on each point of B m ε \V G the vectors g(x) and ρ do not point in opposite direction. Therefore, one can always construct a non-singular smooth bisector vector fields which satisfies the properties: (i) v(x), ρ(x) > 0; (ii) v(x), g(x) > 0. ν(x) = g(x) g(x) + ρ(x) ρ(x) (2.8) By construction one sees that its flow inflates the Milnor tube M ε,η,g to a compact smooth manifold Sε m 1 \ G 1 (Bη) p on the sphere, keeping the boundary of the tube (M ε,η,g ) pointwise fixed. This flow produces a smooth diffeomorphism ξ : M ε,η,g S m 1 ε \ G 1 (B p η) which restricts to the identity map on the boundary (M ε,η,g ). Step 2: The composition map is also a locally trivial smooth fibration. δ := G ξ 1 : Sε m 1 \ G 1 (Bη) p Sη p 1 (2.9) By restriction, the fibration (2.6) give rise to the fibrations G : S m 1 ε G 1 (B p η \ {0}) B p η \ {0} and G : (M ε,η,g ) S p 1 η,

51 2.2. Real settings 49 which can be composed, respectively, with the projection s/ s : R p \ {0} S p 1 providing the following locally trivial smooth fibrations and After composing (2.9) with the s/ s S p 1 η G G : S m 1 ε G 1 (B p η \ {0}) S p 1, (2.10) G G : (M ε,η,g ) S p 1. (2.11) one gets the fibration δ/ δ : S m 1 ε \ G 1 (B p η) S p 1. (2.12) Since ξ restricts to identity map on (M ε,η,g ), hence one has that δ/ δ (Mε,η,G ) is exactly the map (2.11). Finally, since both fibrations (2.10) and (2.12) agree along its common boundaries, it can be glued nicely 1 to get a locally trivial smooth fibration which is independent of ε > 0 up to diffeomorphism type. S m 1 ε \ K ε S p 1, (2.13) Milnor s example Milnor also noted that in general the map projection of the fibration (2.13) fails to be the canonical map G/ G on the whole space Sε m 1 \K ε, like in the above cited case of holomorphic function germs. Actually, in [Mi, p. 99], Milnor considered the application G := (G 1,G 2 ) : (R 2,0) (R 2,0) given by G(x,y) = (x,x 2 + y(x 2 + y 2 )) which satisfies Sing G = V G = {0} and consequently has an isolated singular point. By Theorem the tube fibration exists. However, the map G/ G cannot be the projection of a locally trivial smooth fibration on S 1 ε, because it is not a submersion for ε small enough. In fact, considering v := (x,y) and the matrix ( G 1 (v) G 2 (v) G 2 (v) G 1 (v) A(v) = v ) 1 nicely in the sense that we use the identity map to glue the common boundary.

52 50 Chapter 2. Classical Milnor Fibrations one can see that there exists a non-degenerate curve C of critical points of the map G/ G : Sε 1 S 1, through the origin. As we will see in more details in the next section, the curve C represent the set of ρ-nonregular points of G/ G. Consequently, c.f. Definition 2.2.7, the map G is not ρ-regular and this is precisely the reason of the map G/ G fails to be the projection of a locally trivial smooth fibration. Figure 2 Curve of critical points of G/ G restricted to the sphere Remark The phenomenon described above in the Milnor example can be reproduced in higher dimensions considering the isolated singularity map G : (R m+2,0) (R 2,0) given by G(x,y,z 1,...,z m ) = (x,x 2 + y(x 2 + y 2 + z z2 m)) Non-isolated singular case: tube fibration Both fibrations, the Milnor tube fibration and the sphere fibration, in the real case were extended later for non-isolated singular map germs. In order to state properly these results we need to provide new definitions and notations. Let us consider U R m an open subset such that 0 U and let ρ : U R 0 be a non-negative proper function which defines the origin. Definition Let G : (R m,0) (R p,0) be an analytic map germ. We denote by M ρ (G) := {x U ρ x G} the set of ρ-nonregular points of G, sometimes also called the Milnor set of G. The transversality of the fibres of a map G to the levels of ρ is called ρ-regularity and we will see below that it is a condition for the existence of a locally trivial smooth

53 2.2. Real settings 51 fibration. It was used in the local (stratified) setting by Thom, Milnor, Mather, Looijenga, Bekka, e.g. [Th1, Th2, Mi, Lo, Be] and more recently in [AT1, AT2, ACT1], and [CSS1, CSS2, CSS3] under a different name d-regularity, as well as at infinity in the references [NZ, Ti2, Ti3, ACT2, DRT]. It follows from definition that the Milnor set M ρ (G) is the set of points x U such that the vectors { ρ(x), G 1 (x),..., G p (x)} are linearly dependent over R, i.e., M ρ (G) is the singular locus Sing (G,ρ) of the pair of map (G,ρ) : U R p+1. Hence, the singular set Sing G M ρ (G). For the sake of simplicity, in what follows we will consider ρ as the Euclidean distance function to the origin ρ(x) = x 2, and we write M(G) := M ρ (G) for short. However, all results carry out easily over any other function ρ as considered above. Consider the following condition: M(G) \V G V G {0} (2.14) where the closure of the set M(G) \V G is thought as a germ of set at the origin. The condition (2.14) was used in [AT1, AT2, Ma, ACT1] for the case Disc G = {0}, where it was shown that it insures the existence of the called Milnor tube fibration. In [Ti2, Ti3] M. Tibăr considered this condition under the name ρ-regularity to ensure the topological triviality at infinity. In [Ma] D. Massey also considered this condition but with different notation and called it a Milnor condition (b) to prove the existence of the Milnor tube fibration in the local setting, as in Theorem below. Here we shall use the same notation of [ACT1]. Theorem [Ma, Existence of the (full) Milnor s tube fibration] Let G : U R p as above and assume that it has isolated critical value at origin, i.e. Disc G = {0}, and satisfies the condition (2.14). Then, there exists ε 0 > 0 such that, for each ε, 0 < ε ε 0, there exists η, 0 < η ε, such that G : B m ε G 1 (B p η \ {0}) B p η \ {0} (2.15) is the projection of a locally trivial smooth fibration.

54 52 Chapter 2. Classical Milnor Fibrations Corollary [Ma, Existence of the tube fibration] Given G as above, for any small enough ε > 0, there exists η, 0 < η ε, such that G : B m ε G 1 (Sη p 1 ) Sη p 1 is the projection of a locally trivial smooth fibration. tube. In this case we also denote M ε,η,g = B m ε G 1 (Sη p 1 ) and also call it the Milnor Example Let G : (R 3,0) (R 2,0) given by G(x,y,z) = (xy,xz). Consider v := (x,y,z). One has that [ JG(v) = y x 0 z 0 x ] and [ JG(v)[JG(v)] t = x 2 + y 2 yz yz x 2 + z 2 ] where JG(v) and [JG(v)] t denote the Jacobian matrix of G in v and its transpose, respectively. We know that Sing G = {det(jg(v)[jg(v)] t ) = 0} thus Sing G = {x = 0}. Since V G = {x = 0} {y = z = 0} one gets that Disc G = {0}. Now to compute the Milnor set M(G) let us consider the matrix B(v) := y x 0 z 0 x x y z The Milnor set M(G) = {v R 3 det(b(v)) = 0}. Consequently, M(G) = {x = 0} {x 2 y 2 z 2 = 0}. We claim that G satisfies the condition (2.14). Indeed, let p 0 = (x 0,y 0,z 0 ) M(G) \V G V G. There exists a sequence p n := (x n,y n,z n ) M(G) \ V G such that p n p 0 and x 2 n = y 2 n + z 2 n with y 2 n + z 2 n 0. Since p 0 V G, we need to consider the following two cases: Case 1: p 0 = (0,y 0,z 0 ). Hence. 0 = limx 2 n = lim(y 2 n + z 2 n) = y z 2 0.

55 2.2. Real settings 53 Case 2: p 0 = (x 0,0,0). Hence x0 2 = limxn 2 = lim(y 2 n + z 2 n) = y z 2 0 = 0. In any case we get that p 0 = 0 and G satisfies the condition (2.14). Therefore, by Theorem G has a Milnor tube fibration. In the Figure 3 below one can see that the Milnor tube M ε,η,g consist of two connected components. Compare with Figure 1. Figure 3 The Milnor tube for G(x,y,z) = (xy,xz) Existence of the Sphere fibration Several author have worked on the problem of fibration over spheres in the real settings, for isolated and non-isolated singularities, e.g. [Ja, S1, S2, S3, RSV, RA, A1, P, PS3, CSS1, CSS2, CSS3, AT1, AT2, ACT1]. below. map In [AT1, AT2, ACT1] the authors generalized all previous results as we describe In order to explain the main results, denote the set of ρ-nonregular points of the Ψ := G/ G : U \V G S p 1

56 54 Chapter 2. Classical Milnor Fibrations as the set M(Ψ) = {x U \V G ρ x Ψ}. Definition We will say that G is ρ-regular when M(Ψ) = /0, as a germ of set at the origin. The set M(Ψ) was characterized as follows. A similar result can be found in [S]. Lemma [AT1, AT2, ACT1] Let G : (R m,0) (R p,0) be an analytic map germ. Then, on the open set {G 1 0} one has that Ω 2 (x) M(Ψ) = x U \V G rank. Ω p (x) < p, ρ(x) where Ω k = G 1 G k G k G 1, for any k = 2,..., p. Remark We notice that for any x / V G if ρ x G then ρ x Ψ. Hence, M(Ψ) M(G) \V G. The vectors Ω 2 (x),...,ω p (x) are the generators of the normal space of the fibers X y = Ψ 1 (y), y = Ψ(x). Hence, Lemma does not depend on the particular choice of the open set {G 1 (x) 0}. See for instance [S] for more details. It follows also from [AT1] that the condition M(Ψ) = /0 is equivalent to saying that for small enough ε > 0, the projection Ψ : Sε m 1 \K ε S p 1 is a smooth submersion. However, since the map is not proper (unless the link is empty), it might not be a fibration. In [ACT1] the authors used the Condition (2.14) to ensure that the map Ψ is a projection of a locally trivial smooth fibration. In this setting, their result can be read as: Theorem [ACT1, Theorem 1.3] Let G : U R p, m > p 2 be an analytic map germ such that codimv G = p. Suppose G satisfies the condition (2.14), i.e., M (G) \V G V G {0}. If M(Ψ) = /0 as germ of set, then for any ε, 0 < ε ε 0, the map projection Ψ : S m 1 ε \ K ε S p 1 (2.16)

57 2.2. Real settings 55 is a locally trivial smooth fibration, independent (up to isotopies) of small enough ε > 0. Let us point out below some important facts. In the paper [S1] published in 1997, the author used the method known by Pencil in order to construct examples of real analytic map germs with isolated singular point at the origin, which induces the so-called Open book decomposition on the sphere and hence the Milnor fibration on sphere. Such construction was also used by the authors in [RSV]. In the paper [RA] published in 2005, the authors used this technique and tools from Stratification theory to ensure the existence of the Milnor fibration for real map germ G : (R m,0) (R 2,0) with m > 2. Inspired by [RA], in the paper [AT1] published in 2008 and the paper [AT2] published in 2010 the authors used the technique of Blow-up to provide a generalization of the method for map germs G : (R m,0) (R p,0) with m > p 2, and with that, they were able to prove the following results: Theorem [AT1, Theorem 5.3 p. 10] Let G : (R m,0) (R p,0), m > p 2 be an analytic map germ with isolated critical value. Suppose that, for any small enough ε > 0 and 0 < η ε, the map: G : S m 1 ε G 1 (B p η \ {0}) B p η \ {0} (2.17) is a locally trivial fibration in a thin tube. If the map (2.16) is a submersion for any ε > 0 small enough, Then it is the projection of a locally trivial smooth fibration. Theorem [AT2, Theorem 2.2 p. 179] Let G : (R m,0) (R p,0), m > p 2 be an analytic map germ with Sing G V G = {0}. Then the following are equivalent: (a) there is ε 0 > 0 such that the map (2.16) is the projection of a locally trivial smooth fibration, for any 0 < ε ε 0. (b) M(Ψ) = /0. We notice that, the map (2.17) is a submersion for any small enough ε > 0 and 0 < η ε, if, and only if G satisfies the condition (2.14). See a proof, for instance, in Theorem in Chapter 4. On the other hand, as the proof of Proposition shows, if G satisfies Sing G V G = {0} then G also satisfies the condition (2.14). Thus,

58 56 Chapter 2. Classical Milnor Fibrations as explained in [ACT1, p. 819], the condition (2.14) allows isolated and non-isolated singularities for G since Sing G M(G). It implies that Theorem represents a simultaneous extension of Theorem and of Theorem Moreover, it turns out that the condition (2.14) can not be removed from Theorem , hence it is sharp! In fact, in his master thesis [Han] Hansen presented the example G(x,y,z) = (x 2 + y 2,(x 2 + y 2 )z), showing that Sing G V G and M (Ψ) = /0, but the topological type of the fibers of G changes along S 1. Beside that, by hand calculation one gets that M(G) = R 3, M (G) \V G V G = V G {0} and condition (2.14) breaks down! In order to produce new class of purely real examples, the authors in [ACT1] used the theory of mixed functions (see Chapter 3 of the thesis) and proved the following result. Theorem [ACT1, Theorem 1.4] Let f : C n C be a non-constant mixed polynomial which is polar weighted-homogeneous, n 2, such that codim R V f = 2. Then for any ε, 0 < ε ε 0, the map projection f / f : S 2n 1 ε \ K ε S 1 is a locally trivial smooth fibration, independent (up to isotopies) of small enough ε > 0. Moreover, they proved the result below but now there is no control on the fibration projection outside a neighbourhood of the link in the sphere. For further details see [ACT1]. Theorem [ACT1, Theorem 2.1] Let G : U R p, m > p 2 be an analytic map such that codimv G = p, Sing G V G which satisfies the condition (2.14). Then there exists a locally trivial smooth fibration Sε m 1 \ K ε S p 1 which is independent of small enough ε > 0, up to isotopies Revising the sphere fibration for holomorphic functions Let f : ( C n+1,0 ) (C,0) be a germ of holomorphic function. Let us see that the hypothesis of the previous Theorem are naturally satisfied if we consider f as

59 2.2. Real settings 57 a real map germ from R 2n+2 to R 2. Indeed, by Remark it is well known that any holomorphic function satisfies the Łojasiewicz inequality (see Definition ) f (z) θ c f (z), where 0 < θ < 1, c > 0, and for any z in a small neighborhood of the origin. So, the isolated critical value condition is already satisfied. Moreover, as we have seen in Chapter 1, Hamm and Lê in [HL, Theorem p. 322] have proved that the Łojasiewicz inequality implies that f is Thom regular at V f and hence f satisfies the condition (2.14). See Theorem Finally, by [Mi, Lemma 4.3, p. 35], one gets that for all ε > 0 small enough M( f / f ) = /0, as germ of set. Therefore, from above Theorem the Milnor fibration on sphere follows Fibration on sphere under Thom regularity condition In the sequence of papers [CSS1, CSS2, CSS3] the authors considered maps germs G : (R m,0) (R p,0), m > p 2, with isolated critical value and satisfying a condition called d-regularity which together with the Thom regularity at V G, was used to ensure the existence of the sphere fibrations. To do that, they also associated to G a pencil, as we explain below, following the notations and the construction as described in the paper [CSS2] published in For each l RP p 1 consider the line L l R p throughout the origin and set X l = {x U G(x) L l }. They noticed that each X l is a real analytic variety that contain V G, and each X l \ V G is a (n p 1)-dimensional smooth submanifolds of U, or empty. The family { Xl, l RP p 1} is called the canonical pencil of G. Definition [CSS1, CSS2, Definition of d-regularity] The map G is said to be d-regular at 0 if there exist a metric d induced by some positive-definite quadratic form and an ε > 0 such that every sphere (for the metric d) of radius ε centered at 0 meets each X l \V transversely, whenever the intersection is not empty. We shall also say that G is d-regular with respect to the metric ρ.

60 58 Chapter 2. Classical Milnor Fibrations In order to study the existence of Milnor fibrations associated to a map G, the authors introduced an auxiliary function G : B m ε \V G B p ε called Spherefication map of G. This function was defined by G(x) = x G(x) G(x) and it was used to characterize the d-regularity as following. Proposition [CSS2, Proposition 3.2] Let G : (R m,0) (R p,0) be an analytic map germ with isolated critical value at the origin. The following are equivalent: (i) The map G is d-regular at 0. (ii) For each sphere Sε m 1 of radius enough small ε > 0, the restriction map G : Sε m 1 \V G Sε p 1 is a submersion. (iii) The spherefication map G is a submersion at each x B m ε \V G. (iv) The map Ψ : Sε m 1 \K e Sε p 1 is a submersion for any enough small sphere Sε m 1. This Proposition shows us also that when the metrics d and ρ coincides, then d-regularity is equivalent to ρ-regularity of G. The main result of [CSS2] is the following. Theorem [CSS2, Theorem 5.3] Assume either V G is a point or dimv G > 0 and G has the Thom regularity. The following statements are equivalent: (i) The map G is d-regular at 0. (ii) One has a commutative diagram of smooth fibers bundles on Sη m 1 \ K ε for any sphere Sε m 1 : S m 1 ε \ K ε φ ψ RP p 1 S p 1 π

61 2.2. Real settings 59 where ψ := (G 1 (x) : : G p (x)) and φ := G/ G : Sη m 1 \K ε S p 1 is the Milnor fibration on G. (iii) For any enough small sphere Sε m 1 the restriction G : Sε m 1 \V G Sε p 1 is a smooth fiber bundle and this is the Milnor fibration φ up to multiplication by a constant Comparing the fibration structure on spheres under Thom regularity at V G and the condition (2.14) As can be seen, Proposition in Chapter 4 ensure that if a germ map G is Thom regular at V G then G satisfies the condition (2.14). Example below shows that the converse in not true in general. Therefore, Theorem is more general than Theorem Example [Han, Example 1.4.9] Consider G(x,y,z) = (x,y(x 2 + y 2 ) + xz 2 ) in three real variables. One has that Sing G = V G = {x = y = 0} and M(G) = {x = y = 0} {z = 0}. Hence, M(G) \V G V G = {0} and condition (2.14) holds. We claim that M(Ψ) = /0. Indeed, let v = (x,y,z) R 3 and consider the matrix [ ] Ω 2 (v) B(v) :=, v where, Ω 2 (v) = (x(2xy + z 2 ) y(x 2 + y 2 ) xz 2,x(x 2 + 3y 2 ),2x 2 z). By Lemma 2.2.8, M(Ψ) = {v B 3 ε \V G det(b(v)[b(v)] t ) = 0}. Since det(b(v)[b(v)] t ) = (x 2 + y 2 )(x 6 + 3x 4 y 2 + 5x 4 z 2 8x 3 yz 2 + 3x 2 y 4 + 6x 2 y 2 z 2 + y 6 + y 4 z 2 ) and M(Ψ) M(G) \ V G then M(Ψ) = /0. By Theorem we get the sphere fibration Ψ : S m 1 ε \ K ε S p 1. On the other hand, for any value z 0 consider the point p = (0,0,z), T p V G = span{(0,0,1)}, and the sequence p n = ( 1 n,0,z) which converge ) to p. One has that T pn G 1 2z (G(p n )) = span{v n }, where v n = 0,, 1, hence v 4z z 2 n 2 n 2 +1 n (0,±1,0), where plus and minus depends on the sign of z. Therefore, lim n (T pn G 1 (G(p n ))) = span{(0,1,0)} and G is not Thom regular at V G.

62

63 61 CHAPTER 3 MIXED SINGULARITIES Mixed singularities has been systematically studied by Mutsuo Oka in the sequence of papers [Oka1, Oka2, Oka3] published since 2008, thenceforward the term mixed function was coined and it has been used to identify function f (z, z) = Σ ν,µ c ν,µ z ν z µ, where c ν,µ C, z ν and z µ are multi-monomials in the variables z j, z j, with j = 1,...,n. Before that, this class of singularities had been sporadically studied, for instance [A] and in the sequence of papers [S1, RSV, P, PS1, PS3, C]. In the last years, several authors has been working on that under several different aspects, e.g. [PS3, PS2, BPS, Ti4, C1, CT, C2, ACT1, ACT2, FM, PT]. The variety of results found in these publications shows that the mixed functions are an excellent class of singularities to look for examples and properties which better express some interesting contrast between the complex and the real settings. In this sense, we study this class in order to find new examples of maps with local Milnor s fibrations with and without Thom regularity, as well as, to produce examples of maps with the more general fibration structures (Milnor-Hamm fibrations) as defined in the next chapter. For further details, we invite the reader for a quick look to Sections 3.3, and Example In this chapter we introduce a class of mixed functions which has the Thom regularity and that will be important in constructing examples with the so-called Milnor- Hamm fibration. Moreover, we extend some main results previously known for the class

64 62 Chapter 3. Mixed Singularities of singularities f ḡ, with f and g being holomorphic functions, and for a product f g with f and g mixed functions. We also introduce an algorithm which permits us to build a new class of functions called MSL, which has isolated critical value, the Thom regularity and as we will see in Chapter 6, the ρ-regularity. Finally, we construct class of analytic maps germs which admits the Milnor tube fibration without the Thom regularity. This last results extends the researches started in section 5 of the paper [PT]. 3.1 Mixed functions We consider the following definition. Definition [Oka2] A polynomial function f : C n C is called mixed polynomial function if f (z) = f (z, z) = Σ ν,µ c ν,µ z ν z µ where c ν,µ 0, z ν := z ν 1 1 zν n n and z µ := z µ 1 1 zµ n n for n-uples ν = (ν 1,...,ν n ), µ = (µ 1,..., µ n ) N n. Consider a mixed polynomial function f. We may view f as a real analytic map of 2n variables (x,y) from R 2n to R 2 by identifying C n with R 2n, (z 1,...,z n ) = z (x,y) = (x 1,y 1,...,x n,y n ), writing z=x+iy C n, where z j = x j + iy j C with x j,y j R for j = 1,...,n. In [Oka3], the author introduced the real mixed polynomial function which is a mixed polynomial function H : C n R. Under the above identification, H is a real analytic function on R 2n thus, we can consider both notations H(z, z) and H(x,y). In case of any risk of confusion we will emphasize saying complex mixed polynomial function for a mixed polynomial function f : C n C. For a real mixed function, we define the real gradient vector field as H(x,y) = ( H (x,y), H (x,y),..., H (x,y), H ) (x,y) R 2n. x 1 y 1 x n y n

65 3.1. Mixed functions 63 The holomorphic and anti-holomorphic gradients for a real mixed function H(z, z) is defined as where dh(z, z) := for all j = 1,,n. ( H (z, z),..., H ) ( H (z, z) and dh(z, z) := (z, z),..., H ) (z, z), z 1 z n z 1 z n H = 1 ( H i H ) and H = 1 ( H + i H ) z j 2 x j y j z j 2 x j y j These definitions above were originally introduced in [Oka3]. at V H. Next we will present a class of real mixed functions H which are Thom regular Proposition Let h : C n C be a holomorphic function. Then the real mixed function H : C n R given by H = h h is Thom regular at V H. First, we notice that from Lemma 3.1.7, the normal vector space of the fibers of H at the point z C n is generated by vectors n λ = 2λdH where λ R. Moreover, since H = h 2 = u 2 + v 2 one has that 2dH = H = 2u u + 2v v = h 2 = 2hdh, where the last equality follows from Lemma Therefore, the Milnor set of H is given by M(H) = { z C n λ R s.t. λz = hdh } and Sing H = { (x,y) R 2n H(x,y) = 0 } = { z C n hdh = 0 } = V h Sing h = V h. Proof of Proposition For any p 0 Sing H \ {0}, there exists a small neighborhood of p 0 where the following Lojasiewicz inequality holds c h θ dh for some θ (0,1) and some c > 0. Since the normal space to the fibers of H is spanned by n λ = λ hdh, where λ R. After dividing it by λ h 1+θ, one has that n λ have the same directions to hdh/ h 1+θ, which is bounded away from zero because hdh dh = h 1+θ h θ c > 0. Therefore, ( ) h lim n λ = lim z p 0 z p 0 h 1+θ dh

66 64 Chapter 3. Mixed Singularities dh which has the direction of lim z p0 dh. By Theorem , the zero locus V h admits a Whitney stratification such that h is Thom regular at V h. Since Sing H \ {0} = V h \ {0} we may endow this set with such a stratification with complex strata. This stratification verifies the Thom a H -condition by above reason. The next remark shows the differences between a real mixed function and a complex mixed function. Remark Consider now a mixed function f : C n C given by f = u + iv where v 0. Then, one has that the normal vector space of the fibers of f at the point z C n is spanned by the vectors n µ := 2Re(µ)d f with µ C, µ. Indeed, we notice that d f = d f, and since n µ = µd f + µ d f one gets n µ = (µ + µ)d f = 2Re(µ)d f. Now, by Remark 3.1.8, z Sing f if and only if there exists µ C, with µ = 1 such that d f (z, z) = µd f (z, z). Thus, if one considers µ = 1, the previous equation is satisfied for all z C n, which one concludes Sing f = C n. Finally, since Sing f M( f ) then M( f ) = C n. It follows from above equations that dh = dh and, if we consider C n R 2n, then H R 2n corresponds to the complex vector 2dH C n. Let f : C n C be a mixed function such that f = u + iv, where u,v : C n R 2n R. The real and imaginary part u and v are also (real) mixed functions and we will consider f = u + i v and f = u + i v, z j z j z j z j z j z j for all j = 1,,n. Finally, we can define the holomorphic and anti-holomorphic gradients of f as follows: d f := ( f,..., f ) ( f and d f :=,..., f ). z 1 z n z 1 z n Remark If f is a holomorphic (or anti-holomorphic) function, then f is a mixed polynomial function. Moreover, if f is holomorphic, then d f = grad f (z) as was considered by Milnor in [Mi]. Given f : C n C a complex mixed function we remind that the singular locus is Sing f = {z C n µ C, µ = 1 s.t. µd f (z, z) + µ d f (z, z) = 0}.

67 3.1. Mixed functions 65 Sometimes we will consider the set of critical point of f as the critical locus of a real map R 2n R 2. Lemma Let f : C n C be a mixed polynomial function such that f = u + iv. Then: (i) u(x,y) = d f (z, z) + d f (z, z), (ii) v(x,y) = i ( d f (z, z) d f (z, z) ). Proof. One has that f = u i v and f = u + i v, hence by adding the two z j z j z j z j z j z j equalities, we obtain 2 u = f + f. z j z j z j Thus, one has 2du = d f + d f. Since 2du = u, then u = d f + d f and the item (i) is proved. The item (ii) follows in the same way. In the result below we denote by Ω f := u v v u. It will play an important rule to provide information about the normal directions of the fibers of the f / f. Lemma Let f : C n C be a mixed polynomial function such that f = u + iv. Then: (i) f 2 = 2( f d f + f d f ) ; (ii) Ω f = i( f d f f d f ) Proof. From f 2 = 2(u u + v v), u = f + f 2 and v = f f 2i. One concludes by Lemma 3.1.5, ( ) f + f 2 f (d = 2 f + d f ) ( ) f f (d + 2i f d f ) = 2( f d f + f d f ). 2 2i To prove the item (ii), we use the equality Ω f = u v v u and the same idea as above.

68 66 Chapter 3. Mixed Singularities The next result is very important to control the limits of the tangent spaces of the fibers of f and it will be used in the following sections. It was used by several authors, e.g. [Oka2, CT] and [C2] but with different enunciate. Here, we will use the statement and the proof of [PT]. Lemma (Lemma 2.1 [PT]). Let f : C n C be a mixed function. The normal vector space of the fibers of f at the point z C n is generated by the following vectors: n µ := µd f (z, z) + µ d f (z, z), for all µ C, µ = 1. Proof. One considers f = (u,v) : R 2n R 2, as a real map under identification z = (x,y). One knows that the normal space N (x,y) of the fibers of f at the point (x,y) R 2n is generated by { u(x,y), v(x,y)}. Therefore, if v N (x,y), there exist α,β R such that v = α u(x,y) + β v(x,y). By Lemma 3.1.5, v = α(d f (z, z) + d f (z, z)) + β(i ( d f (z, z) d f (z, z) ) ) = (α + iβ)d f (z, z) + (α iβ) d f (z, z). If v 0 one can choose µ := (α + iβ)/ α + iβ which finish our proof. Remark It follows from Definition and Lemma that M( f ) = {z C n λ R, µ C, µ = 1 s.t. λz = µd f (z, z) + µ d f (z, z)}. Moreover, it follows from Lemma that z M( f / f ) if, and only if there exists λ := λ(z) R such that λz = Ω f (z). Therefore, by Lemma one has that M( f / f ) = {z C n \V f λz = i( f d f f d f )} and Sing ( f / f ) = {z C n \V f 0 = f d f f d f }, as has been shown by Chen in [C2]. 3.2 Product of mixed functions In this section we will study some properties of a mixed function F : C n C which is a product F := f g of two mixed functions f,g : C n C, in analogy with the singularities of type f ḡ where f,g are holomorphic functions. These studies will be important to pave the way to state new results and examples along this Thesis. We will use it specially in the next section for the studies of mixed singularities with separable variables. As an easy application of Lemma one has:

69 3.2. Product of mixed functions 67 Corollary Given F = f g product of mixed functions. Then, (i) The normal vector space of the fibers of F at the point z C n is generated by the vectors n µ = µḡd f + µg d f + µ f dg + µ f dg, where µ C and µ = 1; (ii) A point z M(F) if, and only if there exist λ R, µ C s.t. µ = 1 and λz = µḡd f + µg d f + µ f dg + µ f dg; (iii) The singular set Sing F = { z C n µ C s.t. µḡd f + µg d f + µ f dg + µ f dg = 0 } ; ( ) (iv) A point z M F F if, and only if there exists, λ R such that λz = g 2 Ω f + f 2 Ω g ; ( ) (v) A point z Sing F F if, and only if the one has g 2 Ω f + f 2 Ω g = 0. Lemma If F = f g, then: (i) F 2 = g 2 f 2 + f 2 g 2, (ii) Ω F = g 2 Ω f + f 2 Ω g. Proof. One has that F 2 = 2 ( FdF + F df. ). Thus, F 2 = 2 [ f g ( ḡd f + f dg ) + f g ( g d f + f dg )] = 2 g 2 ( f d f + f d f ) + 2 f 2 ( gdg + ḡdg ) = g 2 f 2 + f 2 g 2. Analogously, Ω F = i(fdf F df) = i [ f g ( ḡd f + f dg ) f g ( g d f + f dg )] = g 2 i( f d f f d f ) + f 2 i(gdg ḡ dg) = g 2 Ω f + f 2 Ω g. We notice that for any mixed function f : C n C, we have d f = d f and d f = d f. Moreover, since f 2 = f 2 hence f 2 = f 2 and Ω f = i( f d f f d f ) = i( f d f f d f ) = Ω f. Therefore, considering the special case of f,g : C n C holomorphic functions and F = f ḡ, by item (i) of Corollary one has that the normal

70 68 Chapter 3. Mixed Singularities vector space of the fibers of F is generated by n µ = µgd f + µ f dg, where µ C and µ = 1. In addition, Ω F = g 2 Ω f f 2 Ω g Mixed product in separable variables In this section, we will consider f : C n C and g : C m C mixed functions in separable variables. Consider the extended mixed functions f e : C n C m C and g e : C n C m C defined by f e (x,y) = f (x) and g e (x,y) = g(y). One has that d f e (x,y) = ( d f (x),0 ), d f e (x,y) = ( d f (x),0 ), dg e (x,y) = ( 0,dg(y) ) and dg e (x,y) = ( 0, dg(u) ). The normal vector space of the fibers of f e and g e at the point (x,y) C n C m is generated by the vectors: n fe,µ = ( µd f (x) + µ d f (x),0 ) and n ge,ξ = ( 0,ξ dg(y) + ξ dg(y) ), respectively, where µ,ξ C and µ = ξ = 1. Moreover, the singular sets are Sing f e = Sing f C m and Sing g e = C n Sing g, and the Milnor sets are M( f e ) = M( f ) C m and M(g e ) = C n M(g), respectively. By an abuse of notation we write simply f e = f, g e = g and F := f g : C n C m C, F(x,y) = f (x)g(y), the product of separable variable functions. Remark Let F := f g : C n C m C be a product of separable variable functions. The Corollary now reads as: (i) The normal vector space ( of the fibers of F at the point (x,y) C n C m is spanned ) by the vector n µ (x,y) = µg(y)d f (x) + µg(y) d f (x), µ f (x)dg(y) + µ f (x) dg(y), where µ C, µ = 1. (ii) A point (x,y) ( M(F) if, and only if there exist λ R and µ C, µ ) = 1, such that λ(x,y) = µg(y)d f (x) + µg(y) d f (x), µ f (x)dg(y) + µ f (x) dg(y). (iii) A point ((x,y) Sing F if, and only if there exist µ C, µ ) = 1, such that (0,0) = µg(y)d f (x) + µg(y) d f (x), µ f (x)dg(y) + µ f (x) dg(y).

71 3.2. Product of mixed functions 69 (iv) A point (x,y) M(F/ F ) if, and only if there exists λ R such that λ(x,y) = ( g(y) 2 Ω f (x), f (x) 2 Ω g (y) ). (v) A point (x,y) Sing (F/ F ) if, and only if (0,0) = ( Ω f,ω g ). In particular, it follows from items (iv) and (v), respectively, that M(F/ F ) M( f / f ) M(g/ g ) and Sing (F/ F ) = Sing ( f / f ) Sing (g/ g ). Therefore, if M( f / f ) = /0 or M(g/ g ) = /0, then M(F/ F ) = /0. In another words, in order for F be ρ-regular, it is enough that f or g be ρ-regular. Lemma Let F := f g be a product of separable variable functions. Then: (i) F(x,y) 2 = ( g(y) 2 f (x) 2, f (x) 2 g(y) 2). (ii) Ω F (x,y) = ( g(y) 2 Ω f (x), f (x) 2 Ω g (y) ). Proof. It follows from Lemma 3.2.2, one has that F(x,y) 2 = g e (x,y) [ 2 f e (x,y) 2 + f e (x,y) 2 g e (x,y) 2 = g(y) 2 2 ( f (x)d f (x),0 ) ( )] [ + f (x) d f (x),0 + f (x) 2 2 ( 0,g(y)dg(y) ) ( )] + 0, g(y) dg(y) = ( g(y) 2 f (x) 2, f (x) 2 g(y) 2) The item (ii) is analogous. Lemma Let F = f g in separable variables. Then, one has Sing F = Sing f Sing g ( V f Sing f ) C m C n (V g Sing g) V f V g. In particular, if Sing f V f, or Sing g V g, one has that Sing F V F. Proof. Let (x,y) Sing F. By item (iii) of Remark 3.2.3, there exists µ C such that { µg(y)d f (x) + µg(y) d f (x) = 0 (3.1) µ f (x)dg(y) + µ f (x) dg(y) = 0 Consequently, one has x Sing f or y V g and x V f or y Sing g. Therefore, (x,y) Sing f Sing g ( V f Sing f ) C m C n (V g Sing g) V f V g.

72 70 Chapter 3. Mixed Singularities Assume without lost of generality that Sing f V f and suppose that there exists (x,y) Sing F \V F. Since V F = V f C m C n V g one has that f (x) 0 and g(y) 0. Consequently, from equation (3.1), (x,y) ( ) Sing f \V f (Sing g \Vg ) = /0, which is a contradiction. Therefore Sing F \V F = /0 and the result follows. Remark Given F = f g in separable variables, it follows from Lemma that: (i) Sing F V F = [(Sing f V f ) C m ] [C n (Sing g V g )] (V f V g ). (ii) Sing F \V F ( Sing f \V f ) (Sing g \Vg ). We finish this section with an important relationship between the Milnor sets of F, f and g, as below. Proposition Let F = f g in separable variables. Then M(F)\V F ( M( f ) \V f ) (M(g) \V g ). Proof. Let (x,y) M(F) \ V F. By item (ii) of Remark 3.2.3, there exist λ R and µ C, µ = 1 such that { λx λy = µg(y)d f (x) + µg(y) d f (x) = µ f (x)dg(y) + µ f (x) dg(y). If λ = 0 one has that (x,y) Sing F \V F ( Sing f \V f ) (Sing g \Vg ) ( M( f ) \V f ) (M(g) \V g ). Suppose λ 0 and choose Γ 1 := µg(y) and Γ 2 := µ f (y). Since (x,y) V F, one has that g(y) 0 and f (x) 0, hence Γ 1,Γ 2 C and { λx = Γ 1 d f (x) + Γ 1 d f (x) λy = Γ 2 dg(y) + Γ 2 dg(y). Consequently, (x,y) ( M( f ) \V f ) (M(g) \Vg ). 3.3 Algorithm for MSL class In what follows we will consider some definitions and results in order to built classes of mixed functions which are Simple Ł-Maps in the sense defined by D. Massey in [Ma], and consequently satisfying the so-called strong Łojasiewics inequality. As

73 3.3. Algorithm for MSL class 71 proved by Massey these functions has isolated critical values and the Thom regularity. Therefore, it provides a wider class of examples. Definition [Ma, Definition 3.5] Let G := (G 1,...,G p ) : (R m,0) (R p,0) be analytic map germ. We say that G is a Simple Ł-Map, if G i, G j = 0 for any i, j = 1,..., p with i j and G i = G j for any i, j = 1,..., p. Example Consider the real map germ G := (G 1,G 2 ) : (R 8,0) (R 2,0) given by G 1 (x,y,z,w,a,b,c,d) = w 2 x 2 + w 2 y 2 + 4wxyz + x 2 z 2 y 2 z 2 + ac + bd G 2 (x,y,z,w,a,b,c,d) = 2w 2 xy 2wx 2 z + 2wy 2 z + 2xyz 2 ad + bc. Denote by e j the j-th vector of the canonical base of R 8. Hence one has that G 1 = ( 2w 2 x + 4wyz + 2xz 2 ) e 1 +(2w 2 y + 4wxz 2yz 2 ) e 2 +(4wxy + 2x 2 z 2y 2 z) e 3 +( 2wx 2 + 2wy 2 + 4xyz) e 4 +c e 5 +d e 6 +a e 7 and +b e 8 G 2 = ( 2w 2 y 4wxz + 2yz 2 ) e 1 +( 2w 2 x + 4wyz + 2xz 2 ) e 2 +( 2wx 2 + 2wy 2 + 4xyz) e 3 +( 4wxy 2x 2 z + 2y 2 z) e 4 d e 5 +c e 6 +b e 7 a e 8 Therefore, G 1, G 2 = 0 and G 1 2 = 4w 4 x 2 + 4w 4 y 2 + 4w 2 x 4 + 8w 2 x 2 y 2 + 8w 2 x 2 z 2 + 4w 2 y 4 + 8w 2 y 2 z 2 = +4x 4 z 2 + 8x 2 y 2 z 2 + 4x 2 z 4 + 4y 4 z 2 + 4y 2 z 4 + a 2 + b 2 + c 2 + d 2 = G 2 2

74 72 Chapter 3. Mixed Singularities Thus, G is a Simple Ł-Map. In what follows, a mixed function germ f = (u + iv) : (C m,0) (C,0) which is a Simple Ł-Map will be called a Mixed Simple Ł-Map, or for short, a MSL. Given u and v vectors in C m we denote its hermitian product by u, v C. Lemma A mixed function germ f : (C m,0) (C,0) is a MSL if, and only if d f, d f C = 0, i.e., Re d f, d f C = Im d f, d f = 0. In particular, in a critical point C z one has that d f (z) = d f (z) = 0. Proof. By Lemma 3.1.5, we have that u 2 v 2 = 4Re d f, d f and u, v = C 2Im d f, d f C Constructing MSL functions In this section we present an algorithm to construct MSL functions in a very easy way. This provides us a way to build examples of maps with isolated critical value and the Thom regularity. As we will see in Chapters 6 and 8, an MSL function also provides examples of maps where the Milnor fibrations on the tube and on the sphere exist and they are equivalent. As can be seen in Corollary and Example , we use the algorithm to construct an infinite number of examples which have the Milnor tube fibration but without the Thom regularity. In addition to that, in Section 4.5, we also use the algorithm to construct a variety of maps with the Milnor-Hamm fibration but without the -Thom regularity. Therefore, this algorithm plays a fundamental rule in this chapter in order to provide an easy way to construct examples and counter-examples verifying the main regularity conditions considered in this work. This makes clear the importance of the next proposition. Proposition (Algorithm to MSL). Consider the following algorithm: Step (1) Fix a copy of C n, n 2, and a coordinate system z = (z 1,...,z n ).

75 3.3. Algorithm for MSL class 73 Step (2) For each 1 k < n choose natural numbers i 1,...,i k {1,2,...,n} with i 1 < i 2 <... < i k and fix the coordinates (z i1,...,z ik ). For the complementary ordered list q 1,...,q n k {1,2,...,n} \ {i 1,...,i k }, q 1 <... < q n k, consider the reminding coordinates (z q1,...,z qn k ). Step (3) For any natural numbers j, t and p, choose arbitrary holomorphic functions f j (z i1,...,z ik ), r t (z i1,...,z ik ), g j (z q1,...,z qn k ) and h p (z q1,...,z qn k ). Step (4) Define the mixed function germ f : (C n,0) (C,0) by f (z 1,...,z n ) = j α=1 f α (z i1,...,z ik )g α (z q1,...,z qn k ) + t β=1 r β (z i1,...,z ik ) + p γ=1 h γ (z q1,...,z qn k ). Claim: The mixed function germ f is MSL. Proof. By simplicity we will choose j = t = p = 1, n = 2k for some k N and i η = η for 1 η k. In this case, f (z 1,...,z n ) = f 1 (z 1,...,z k )g 1 (z k+1,...,z n ) + r 1 (z 1,...,z k ) + h 1 (z k+1,...,z n ). We notice that for α = 1,...,k, f z α = f 1 z α g 1 + r 1 z α and f z α = 0 f = 0 and f = g 1 f 1 + h 1 z β z β z β z β for β = k + 1,...,n. Thus, one has ( f1 d f = g 1 + r 1,..., f 1 g 1 + r ) 1,0,...,0 z 1 z 1 z k z k and d f = ( g 1 0,...,0, f 1 + h 1,..., g 1 f 1 + h ) 1. z k+1 z k+1 z n z n Therefore, d f, d f = 0 and the function f is MSL. C

76 74 Chapter 3. Mixed Singularities Remark The Proposition extends Corollary 4.2 of [PT] (The Thom-Sebastiani type result). Example Let G : (C 4,0) (C,0) given by G(z 1,z 2,z 3,z 4 ) = z 2 1 z2 2 + z 3 z 4 + z 4 1 z 3 z 2 z 3 4. Considering f 1(z 1,z 3 ) = z 2 1, f 2(z 1,z 3 ) = z 3, g 1 (z 2,z 4 ) = z 2 2, g 2(z 2,z 4 ) = z 4, r(z 1,z 3 ) = z 4 1 z 3 and h(z 2,z 4 ) = z 2 z 3 4. Then one has that G = f 1ḡ 1 + f 2 ḡ 2 + r h. By Proposition 3.3.4, G is a MSL. Example Let G : (C 6,0) (C,0) given by G(z 1,z 2,z 3,z 4,z 5,z 6 ) = z 1 z z3 3 z2 4 + z 5 z 6. Let f 1 (z 1,z 3,z 5 ) = z 1, f 2 (z 1,z 3,z 5 ) = z 2 3, f 3(z 1,z 3,z 5 ) = z 5, g 1 (z 2,z 4,z 6 ) = z 2 2, g 2 (z 2,z 4,z 6 ) = z 2 4 and g 3(z 2,z 4,z 6 )z 6. One has that G = 3 j=1 f j(z 1,z 3,z 5 )g j (z 2,z 4,z 6 ). By Proposition G is a MSL. 3.4 Mixed functions with polar action Definition [Oka1, Oka2, C1, C2, C] A mixed polynomial function f : C n C is called polar weighted-homogeneous if there are non-zero integers p 1,..., p n and d, such that gcd(p 1,..., p n ) = 1 and n ( ) p j ν j µ j = d j=1 for any monomial of the expansion f (z, z) = ν,µ c ν,µ z ν z µ. We call (p 1,..., p n ) the polar weight of f and d the polar degree of f. More precisely, f is polar weighted homogeneous of type (p 1,..., p n ;d) if, and only if it verifies the following equation for all λ S 1 : f (λ (z, z)) = λ d f (z, z), where the corresponding S 1 -action on C n is: λ (z, z) = ( λ p 1 z 1,...,λ p n z n,λ p1 z 1,...,λ pn z n ), λ S 1. We will see below that in this special class of functions some important sets which are related to our studies are preserved under the S 1 -action. Lemma Let f be a mixed polar weighted homogeneous function. Then we have λ M( f ) = M( f ), for all λ S 1. In other words, the set M( f ) is invariant under S 1 - action.

77 3.4. Mixed functions with polar action 75 Proof. Let a M( f ). As f is mixed, there exist α R and β C such that α a = β d f (a,ā) + β d f (a,ā). It means that for all j = 1,n one has αa j = β f z j (a,ā) + β f z j (a,ā) (3.2) One needs to show that there are α R and µ C such that First notice that for each j = 1,n one has α λ p j a j = µ f z j (λa) + µ f z j (λa). (3.3) f (λ a) = λ d+p f j (a,ā) (3.4) z j z j and f z j (λ a) = λ d+p j f z j (a,ā) (3.5) for each λ S 1. Now, taking µ = βλ d C and α = α, one has µ f z j (λ (a,ā)) + µ f z j (λ (a,ā)) = βλ d λ d+p j f z j (a,ā) + βλ d λ d+p j f z j (a,ā) ( = λ p j β f z j (a,ā) + β ) f z j (a,ā) = α λ p ja j, where the first equality follows from equations (3.4) and (3.5) and the last equality follows from equation (3.2). Thus we obtain (3.3) and hence λ M( f ) M( f ). Reciprocally, take a M( f ). Since 1 λ S1, then b = 1 λ a M( f ) by the first part of the proof. Now, since a = λ b λ M( f ) then M( f ) λ M( f ). Therefore, we conclude that λ M( f ) = M( f ). Lemma Let f be a mixed function which is polar weighted homogeneous. Then, (i) λ Sing f = Sing f, for all λ S 1 ; ) (ii) λ M = M ( f f ( f f ), for all λ S 1.

78 76 Chapter 3. Mixed Singularities Proof. Let a Sing f. Since f is mixed, there exists β C such that β f z j (a,ā) + β f z j (a,ā) = 0 (3.6) for all j = 1,...,n. Now, taking µ = βλ d C, if follows from equation (3.4), (3.5) and (3.6) that µ f z j (λa)+ µ f z j (λa) = 0. Thus, λ Sing f Sing f and applying an argument as in the proof of Lemma 3.4.2, we conclude λ Sing f = Sing f. For item (b) we use the same argument as above Milnor tube without Thom regularity We will introduce below some families of mixed functions which has Milnor tube fibration without Thom regularity. The first example in the literature was provided in [ACT1, Ti4], see also [PS4, Oka3]. The second example is a deformation of the first ones for three complex variables and it was given by Parusinski, answering a question formulated in [Ti4]. Before we present these examples we will consider a structural fibration Theorem for polar weighted homogeneous functions. Theorem [ACT1, PT] Let f : (C n,0) (C,0) be a mixed polynomial which is polar weighted-homogeneous, n 2. Then f has Milnor tube fibration (2.15). Example [ACT1, ACT2, Ti4, Oka3] Let G : (C 2,0) (C,0) given by G(x,y) = xy x. Since G is polar weighted-homogeneous, by Theorem G has Milnor tube fibration. We claim that G does not have Thom regularity. Indeed, one can write G = f g where f (x) = x x and g(y) = y. One has that V G = {x = 0} {y = 0} and from Remark 3.2.3, the normal vector space of the fibers of G at the point (x,y) C 2 is spanned by the vectors n µ = (2Re(µȳ)x, µ x 2 ). By item (i) of Remark 3.2.6, one has that Sing G V G = {x = 0}. Let p 0 = (0,iy 0 ) Sing G V G with y 0 R \ {0}. Consider a path γ(t) = (x(t),y(t)) such that γ p 0 and γ V G. Choosing µ(t) = iy(t)/ y(t) one get that Therefore on γ, n µ(t) (γ(t)) x(t) 2 = (0,iy(t)/ y(t) ). n µ(t) x(t) 2 (0, )

79 3.4. Mixed functions with polar action 77 where means some non zero complex number. By Lemma one notices that fixed any p Sing G V G one has λ p Sing G V G where λ S 1. Therefore, for any Whitney stratification of V G the y-axis minus the origin, {x = 0} \ {0} must be a single stratum. Consequently, G does not have Thom regularity. Example [PT, Parusinski s example] For any k > 1, consider the mixed function germ G k : (C 3,0) (C,0) given by G k (x,y,z) = (x + z k ) xy. One has that G k is a polar weighted homogeneous of type (k,d,1;d) with d 1. Hence, by Theorem G has Milnor tube fibration. The idea to show that this example does not have Thom regularity follows the same steps as one given in the above Example 3.4.5, which was originally presented in [PT], see section 5.1. For the special case k = 2 another proof was given by Parusinski and can be found in [PS4]. Next we introduce the main result of this section which helps us to construct classes of mixed functions with the classical Milnor tube fibration and without the Thom regularity. Proposition Let f : (C n,0) (C,0) be mixed function such that Sing f V f. Let g : (C m,0) (C,0) be a MSL such that f and g have separable variables. Then G = f + g : (C n C m,0) (C,0) has Thom regularity if, and only if f does. Remark In the case where G = f + g : C n C one has that the normal vector space of the fiber of G at the point z C n is spanned by the vectors: n µ = µd f + µ d f + µdg + µ dg, (3.7) where µ C and µ = 1. Moreover, if we assume that f and g have separable variables, then n µ = (µd f + µ d f, µdg + µ dg) and Sing ( f + g) Sing f Sing g. Proof of Proposition Since g is MSL then g has Thom regularity. Therefore, if f is Thom regular at V f, it follows from Proposition 5.2 of [ACT1] that G is Thom regular at V G. Now assume that f is not Thom regular at V f. We claim that G is not Thom regular at G as well.

80 78 Chapter 3. Mixed Singularities Indeed, from Remark one has that n µ,g = (n µ, f,n µ,g ) where n µ, f = µd f + µ d f, n µ,g = µdg + µ dg, and Sing G Sing f Sing g. On the other hand, if (x,y) Sing f Sing g, there exists µ C such that n µ, f (x) = 0 and dg(y) = dg(y) = 0, hence (0,0) = (n µ, f (x),n µ,g (y)) = n µ,g (x,y). Therefore, (x,y) Sing G and one has Sing G = Sing f Sing g. One has that V G = {(x,y) C n C m f (x) + g(y) = 0} and V f V g V G. Since Sing G = Sing f Sing g V f V g V G, hence G has isolated critical value. Consequently, V G Sing G = Sing G = Sing f Sing g. Consider any Whitney stratification W = {W α } of V G such that its extension to a Whitney stratification of C n C m provides, respectively, an Whitney stratifications W 1 and W 2 on C n and C m, which are stratifications for V f and V g. Since f is not Thom regular at V f, there exist points p 0 V f Sing f such that for any positive dimensional strata W 1 α of W 1, W 1 α Sing f containing point p 0, there exist analytic curves γ p0 C n \V f, such that γ p0 p 0 and with v n 0 and µ(t) C \ {0}. n µ(t), f (γ p0 (t)) v n / [T p0 W 1 α] For each p 0 consider any point q 0 Sing g and the pair z 0 := (p 0,q 0 ) Sing f Sing g. Let W 2 β be the stratum of W 2 which contains q 0. Define the path γ z0 := (γ p0,γ q0 ) where γ q0 q 0 and γ z0 V G and assume that n µ(t),g (γ q0 (t)) converge for some v m [T q0 W 2 β ]. Thus, one has that n µ(t),g (γ z0 (t)) = (n µ(t), f (γ p0 (t)),n µ(t),g (γ q0 (t))) (v n,v m ). Consider W α,β := W 1 α W 2 β. Hence, ) T z0 W α,β = T z0 (W α 1 Wβ 2 = T p0 Wα 1 T q0 Wβ 2. Given any (w,0) T z0 W α,β one has Re (v n,v m ),(w,0) C = Re v n,w C. Since v n / [T p0 W 1 α], there exists some w 0 T p0 W 1 α such that Re v n,w 0 C 0. Therefore, there exists v 0 = (w 0,0) T z0 W α,β such that Re (v n,v m ),v 0 C 0 which implies (v n,v m ) /

81 3.4. Mixed functions with polar action 79 [T z0 W α,β ]. Consequently, the Thom a G -condition is not satisfied along γ z0 for the pair (B 2(m+n) ε \V G,W α,β ) and G does not have Thom regularity. Next result will be important to construct examples of functions with tube fibrations without Thom regularity. See for instance Example below. Corollary Let f : (C n,0) (C,0) be mixed functions with Sing f V f without Thom regularity. Let g : (C m,0) (C,0) a MSL such that f and g have separable variables, and G = f + g is polar weighted homogeneous. Then, G has Milnor tube fibration without Thom regularity. Proof. Since G is polar weighted homogeneous, by Theorem it has isolated singular value and satisfies the condition (4.2). Therefore it has Milnor tube fibration. By Proposition 3.4.7, G is no Thom regular at V G. Next, let us show that the techniques developed in Examples and can be used to find more maps with the same properties as in the two previous examples. Remark Let f,g : C n C be holomorphic functions with common component h : C n C. Let f,g : C n C be holomorphic functions such that f = f h and g = g h. Then the normal vector space of the fibers of f ḡ at the point z C n is generated by the vectors: n µ = 2Re(µ L)hdh + h h ( µg d f + µ f dg ) (3.8) where µ C, with µ = 1. On points outside V h one can divide the above equation (3.8) by h h to obtain n µ / h 2 = 2Re(µ L)(dh/ h) + µg d f + µ f dg. Now, after expand it one get: ( n µ h 2 = 2Re(µ L) h ) h + µ g f + µ f g,..., 2Re(µ L) h + µ g z 1 z 1 z 1 h f + µ f g z n z n z n We notice that z Sing f ḡ if, and only if 2Re(µ L)hdh + h h ( µg d f + µ f dg ) = 0. Consequently, one has that V h = {h = 0} Sing f ḡ. Example Let us fix any natural number k > 1 and consider the mixed map T k := f k ḡ : C 2 C where f k := y k x and g = x. One has that V Tk = {x = 0} {y = 0}.

82 80 Chapter 3. Mixed Singularities One can write T k = l k m where l k = y k and m = x x. Thus, it follows from Remark that Sing T k V Tk = {x = 0} {y = 0} = V Tk and by Lemma one gets V Tk = Sing T k. Let p 0 := (0,y 0 ) {x = 0} \ {0} V Tk such that y 0 0. Consider a path γ(t) = (x(t),y(t)) such that γ p 0 and γ V Tk. One can define h = x, g = 1 and f k = yk. Thus, g x = g y = f k = 0 and choosing µ = il/ L, by Remark , one gets x Therefore on γ, ( n µ h 2 = 0, ) il L kyk 1. n µ(t) x(t) 2 (0, ) where means some non zero complex number. Since T k is polar weighted homogeneous, by Lemma one has that for any fixed p {x = 0} \ {0}, the point λ p {x = 0} \ {0} for any λ S 1. Therefore, for any Whitney stratification of V G the y-axis minus the origin, {x = 0} \ {0} must be a single stratum. Consequently, T k does not have Thom regularity. However, by Theorem T k has Milnor tube fibration. Example Consider the mixed map f ḡ : C 2 C were f = xy y 2 and g = x y. One has that V f ḡ = {y = 0} {x y = 0}. Let f = y, g = 1 and h = x y. It follows from Remark that ( 2Re(µL)h n µ / h 2 = h 2, 2Re(µL)h ) h 2 + µ. Choosing µ = iy/ y one gets ( ) n µ h 2 = iy 0,. y Let p 0 := (x 0,x 0 ) {x y = 0}\{0} V f ḡ. Consider a path γ(t) = (x(t),y(t)) such that γ p 0 and γ V f ḡ.therefore on γ, n µ(t) h(t) 2 (0,q) =: v 0 where q C. Since (q,q) T p0 ({h = 0}) then Re v 0,(q,q) C 0 and v 0 / [T p0 {h = 0}].

83 3.4. Mixed functions with polar action 81 Since f ḡ is polar weighted homogeneous, by Lemma 3.4.3, {h = 0} \ {0} is invariant by the S 1 -action, then for the any Whitney stratification of V f ḡ one has that it is a single stratum. As we have seen, the a f -Thom condition is not satisfied along γ for pair (B 4 \V f ḡ,{h = 0} \ {0}). Consequently, f is not Thom regular at V f ḡ. Finally, by Theorem it has Milnor tube fibration. Example Consider the following mixed functions: 1) F 1 (x,y) = xy k x, for a fixed k 1, from Example , 2) F 2 (x,y,z) = (x + z k ) xy, for a fixed k 2, from Example 3.4.6, 3) F 3 (x,y) = (xy y 2 )( x ȳ), from Example , 4) F 4 (w 1,...,w n ) = w 1 ( k j=1 w j 2a j n t=k+1 w t 2a t ), from [Oka3]. As we have seen, F 1,F 2 and F 3 has Milnor tube fibration without Thom regularity. The function F 4 was also proved in [Oka3, Example 3] that it does not have Thom regularity. Since F 4 is polar weighted homogeneous of type (d p,1,...,1;d p ), it follows from Theorem that it has Milnor tube fibration. Now, consider the mixed function G j := F j + g where j = 1,2,3,4 and g is MSL with a polar weighted homogeneous action. Then, by Corollary the map G has Milnor tube fibration without Thom regularity. General comment We notice that there exist others classes of maps which are interesting to investigate the existence of Milnor Fibrations. For instance, the class of maps that preserve Laplace s equation called harmonic morphisms. See [BW], for more details. We know that a harmonic morphism defined between Euclidean spaces have isolated critical value and the Milnor fibration on the tube and on the sphere are equivalent. In differential geometry the harmonics morphisms plays an important rule. It has connections with many important concepts, such as: geodesics, foliations, Clifford systems, Hermitian structures, iso-parametric mappings, submanifolds with constant mean curvature, and Einstein metrics. We intend to explore these connections in our future works.

84

85 83 CHAPTER 4 MILNOR-HAMM FIBRATIONS As discussed in Chapter 2, Milnor in [Mi] considered briefly the studies of fibration structures for real maps with Sing G = {0}. Extensions for the case with the property Disc G = {0} has been considered after Milnor s work for several authors as in [RSV, AT1, Ma, AT2, ACT1, DA, PT] etc, and the complementary case of Disc G larger than 0 has been approached in [ACT1] for sphere fibration (see Theorem and Theorem ) and in [MS] and [CGS] for tube fibration. In this present chapter we will introduce our first generalization of the Milnor tube fibration under the condition dimdisc G > 0 and call it Milnor-Hamm fibration in reference to Theorem in Chapter 2 for complex ICIS singularities. This extension generalizes all previous results and allows us to further generalize. We introduce a sufficient and necessary condition (see Theorem and Remark 4.1.5) for the existence of fibration and several classes of maps satisfying it, as for instance, the class finitely determined under the contact group structure. As we will see, the topological type of the fiber is not unique, however in Section 4.2 we introduce a special class of maps where the base space of the Milnor-Hamm fibration is connected, hence the topological type of the fiber is uniquely defined. Moreover, we define two new regularity conditions, namely, the condition (4.2) and a weaker Thom type regularity, called -Thom regularity. We finish this chapter showing the relationship between these new regularity conditions and introducing a first extension of the main result of the paper [PT] for the case of

86 84 Chapter 4. Milnor-Hamm Fibrations non-isolated critical value. 4.1 Milnor-Hamm fibration Definition Let G : (R m,0) (R p,0) be an analytic map germ. We say that G has Milnor-Hamm fibration if for any small enough ε > 0, there exists 0 < η ε such that the restriction: G : B m ε G 1 (Bη p \ Disc G) Bη p \ Disc G (4.1) is a locally trivial smooth fibration which is independent, up to diffeomorphisms, of the choices of ε and η. In the particular case where Disc G = {0} the Milnor-Hamm fibration becomes the known Milnor tube fibration, as already introduced on Chapter 2. Moreover, in general the complement of the discriminant set of G may consist of several many connected components through the origin, and hence the base space of the fibration may not be a connected space and then the topological type of the fibres may not be unique. However, in section 4.2 we will present a special class of maps where the base space of the fibration is connected, hence with an unique topological type for the fibers. Finally, in the particular case Disc G = Im G, the Milnor-Hamm fibration is empty. The following example show several aspects of the new fibration that we will study here. Example Let h : C n C be a holomorphic function. Consider the real valued function H : C n R given by H := h h. One has that Sing H = V H. It follows from Proposition that the map H is Thom regular at V H. Consequently, by Proposition 4.3.2, H has Milnor-Hamm fibration over R with two types of Milnor fibres, one empty and one diffeomorphic to B ε h 1 (S η ), for some 0 < η ε. Now, if we consider H : C n C as a complex mixed function like in Remark 3.1.3, then Sing H = C n and Disc H = Im H = R 0, thus its Milnor-Hamm fibration is empty. We also point out that Definition does not require Thom regularity of the map G. Actually, we shall use examples to show that the Thom regularity may fail and

87 4.1. Milnor-Hamm fibration 85 the Milnor-Hamm fibration still exists. Similar types of fibrations but with the stronger assumptions of Thom regularity has been considered in [CGS]. Let us consider now the Milnor set M(G) as in Definition and the following extended condition: M(G) \ G 1 (Disc G) V G {0} (4.2) where, as before, the closure of the set M(G) \ G 1 (Disc G) is thought as a germ at the origin. Figure 4 Geometric representation of condition (4.2) We highlight that, the condition 4.2 becomes the condition 2.14 under the special case where Disc (G) = {0}. Moreover, as the example below shows, in general the condition 2.14 fails if the dimdisc (G) > 0. Example Let G : (R 3,0) (R 2,0) given by G(x,y,z) = (xy,z 2 ). One has that G satisfies the condition (4.2). Indeed, we notice that:

88 86 Chapter 4. Milnor-Hamm Fibrations V G = {x = z = 0} {y = z = 0} Sing G = {x = y = 0} {z = 0} M(G) = {x = ±y} {z = 0} Disc G = {(0,β) β 0} {(λ,0) λ R} G 1 (Disc G) = {x = 0} {y = 0} {z = 0}. Let p 0 = (x 0,y 0,z 0 ) M(G) \ G 1 (Disc G) V G. There exists a sequence p n := (x n,y n,z n ) M(G) \ G 1 (Disc G) such that p n p 0 with p 0 V G. Thus, one has that x n = ±y n 0 for any natural number n, z 0 = 0 and x 0 = 0 or y 0 = 0. Assume without lost of generality that p 0 = (x 0,0,0). One has that x 0 = limx n = ±limy n = y 0 = 0. Therefore, p 0 = 0 and G satisfies the condition (4.2). We claim that V G M(G) \V G V G. In fact, let us write V G = V 1 V 2 where V 1 = {x = z = 0} and V 2 = {y = z = 0}. Let q 1 = (0,y 1,0) V 1 such that y 1 0. Consider the sequence of point q n := (1/n,y 1,0). One has that q n q 1 and q n M(G)\V G. Thus, q 1 M(G) \V G V G, i.e., V 1 \ {0} M(G) \V G V G. Analogously, one can show that V 2 \ {0} M(G) \V G V G. Now, if q 0 = 0 one consider the sequence of points q n := (1/n,1/n,0). Thus, V G M(G) \V G V G which implies that M(G) \V G V G =V G {0}. Therefore, G does not satisfies the condition (2.14). We introduce below our main result on this section which shows a sufficient condition for the existence of a Milnor-Hamm fibrations. Theorem Let G : (R m,0) (R p,0) be an analytic map germ. If G satisfies the condition (4.2), then G has a Milnor-Hamm fibration. Proof. The map G satisfies condition (4.2) if, and only if there exists ε 0 > 0 such that, for any 0 < ε < ε 0, there exists η, 0 < η ε, such that the restriction map is a smooth submersion. G : S m 1 ε G 1 (B p η \ Disc G) B p η \ Disc G (4.3)

89 4.1. Milnor-Hamm fibration 87 Indeed, let ε 0 > 0 such that for any 0 < ε < ε 0 condition (4.2) holds. Then one can find a conical neighborhood N of V G \{0} such that (M(G)\G 1 (Disc G)) N = /0. Consequently, for all x ( N Sε m 1 ) \ G 1 (Disc G) one has Sε m 1 x G 1 (G(x)). One considers now η ε > 0 depending on ε such that the inclusion Sε m 1 G 1 (Bη p ε \ Disc G) ( N Sε m 1 ) \ G 1 (Disc G) holds. Then for any 0 < η < η ε one has that (4.3) is a submersion. Reciprocally, let ε 0 > 0 such that for any 0 < ε < ε 0, there exists η, 0 < η ε, such that the restriction map (4.3) is a smooth submersion. In this case, one has that M(G) (Sε m 1 G 1 (Bη p \ Disc G)) = /0. By making ε 0 one get a conical neighborhood N of V G \ {0} such that (M(G) \ G 1 (Disc G)) N = /0, which is equivalent to the condition (4.2). Consequently, the map G : B m ε G 1 (B p η \Disc G) B p η \Disc G is a submersion on a manifold with boundary. We claim that it is proper. In fact, consider K B p η \ Disc G a compact subset. Since G 1 (K) G 1 (B p η \ Disc G), then B m ε G 1 (B p η \ Disc G) G 1 (K) = B m ε G 1 (B p η) G 1 (K). Moreover, the map G : B m ε G 1 (B p η) B p η is proper and K B p η. Thus, B m ε G 1 (B p η \Disc G) G 1 (K) is compact. We may therefore apply Ehresmann Theorem to conclude our proof. Remark We notice that the converse of the above Theorem is true. Indeed, if there exists the Milnor-Hamm fibration (4.1), then shrinking ε and η if necessary, there is no critical point of G on the tube B m ε G 1 (B p η \ Disc G). Consequently, the restriction map (4.3) is a submersion, which is equivalent to the condition (4.2). Example Consider the analytic map germ G(x,y,z) = (xy,z 2 ), defined as in Example By Theorem 4.1.4, G has Milnor-Hamm Fibration (4.1).

90 88 Chapter 4. Milnor-Hamm Fibrations Figure 5 Milnor-Hamm fibration for G = (xy,z 2 ) The case Sing G V G = {0} Let us see that the condition Sing G V G = {0} provides a rich class of examples where the Milnor-Hamm fibration exists. The next statement is the real counterpart of Hamm s result [Ham] that a holomorphic map which defines an ICIS has Milnor fibration outside the discriminant set. Proposition Let G : (R m,0) (R p,0) be an analytic map germ. If Sing G V G = {0}, then G has the Milnor-Hamm fibration (4.1) Proof. Since Sing G V G = {0}, there exists ε 0 > 0 such that for any 0 < ε ε 0, the set (V G \ {0}) B n ε is a smooth manifold of dimension n p and, as observed by Milnor [Mi], there exists ε 0 > 0 such that for any 0 < ε ε 0, 0 R p is a regular value of the restriction map G : Sε m 1 R p. Since transversality is an open condition, it follows that for each ε > 0 one can find 0 < η 0 ε, such that for any 0 < η η 0 the restriction map G : Sε m 1 G 1 (Bη) p Bη p is a smooth submersion. Hence the condition (4.2) is satisfied and Theorem applies.

91 4.1. Milnor-Hamm fibration 89 Remark If Sing G V G = {0}, then for any small enough ε > 0, there exists 0 < η ε such that the restriction map G : S m 1 ε G 1 (B p η) B p η (4.4) is a trivial smooth fibration with the fibre diffeomorphic to the link K ε = S m 1 ε V G. Consequently, one has an open book decomposition (see [ACT1, Definition 1.1, p. 818]) near the binding K ε, i.e. S m 1 ε G 1 (B p η) K ε B p η. Corollary [HL, Hamm fibration, Theorem 2.1.3] Let G := (G 1,...,G p ) : (C n,0) (C p,0) be a complex ICIS at 0. Then G has a Milnor-Hamm tube fibration G : B 2n ε G 1 (B 2p η \ Disc G) B 2p η \ Disc G as has been shown by Hamm [HL], see also [Lo]. The next example shows that the hypothesis Sing G V G = {0} cannot be replaced by the weaker condition Sing G \V G V G = {0}. Example Let G : (R 3,0) (R 2,0) given by G(x,y,z) = (xy,x 2 z 2 ). One has that: V G = {x = z = 0} ( {y = 0} {x 2 z 2 = 0} ) Sing G = {x = z = 0} {x = y = 0} M(G) = {z = 0} {2x 2 y 2 = 0} Disc (G) = {(0, z 2 ) z R}) G 1 (Disc (G)) = {x = 0} ( {y = 0} {x 2 z 2 0} ). First, notice that V G Sing G = {x = z = 0} and Sing G \V G = {x = y = 0} (i.e., the singular locus of an analytic map germ can have components inside and outside of the zero locus), hence Sing G \V G V G = {0}. We claim that G do not satisfies the condition (4.2). Indeed, let V 1 = {x = z = 0} and q 1 = (0,λ,0) V 1. Consider the sequence q n := (1/n,λ + 1/n,0). One has that q n q 1 and q n M(G) \ G 1 (Disc G). Thus, q 1 M(G) \ G 1 (Disc G) and V 1 M(G) \ G 1 (Disc G). Since V 1 V G, one has M(G) \ G 1 (Disc G) V G V 1 {0}.

92 90 Chapter 4. Milnor-Hamm Fibrations Remark Let f : (R m,0) (R p,0) be real analytic map germ and g : (R n,0) (R n,0) be a germ of diffeomorphism, such that f and g are in separable variables. Consider the analytic map germ G := ( f,g) : (R m R n,0) (R p R n,0). If f has isolated critical point, then Sing G V G = {0} and G has Milnor-Hamm fibration. In fact, one has that Sing G = Sing f R n = {0} R n and V G = V f V g = V f {0}. Thus, Sing G V G = ({0} R n ) V f {0} = {0}. By Proposition 4.1.7, G has Milnor-Hamm fibration (4.1). Example Let f : R n R given by f (x 1,...,x n ) = n j=1 c jx a j j with a j 2, c j R\{0} and let g : (R n,0) (R n,0) be the identity function g(y) = y. Since Sing f = {0}, by Remark , G has Milnor-Hamm fibration. Moreover, one has that Sing G = {0} R n and Disc G = {0} R n. We now formulate another consequence of Proposition 4.1.7, this time in terms of the contact determinacy group K. Proposition Let G : (R m,0) (R p,0) be an analytic map germ and 0 r <. If G is finitely-c r -K -determined then it has a Milnor-Hamm fibration (4.1). The significance of this result comes from the fact that the class of germs finitely- K -determined is large, in the sense that finite determinacy is a generic property. For more details see [T1, T2, Wa, Co]. The proof of Proposition follows from C.T.C Wall s paper, as we explain in the sequence. In the well known paper [Wa], Wall presented a series of results about the finite determinacy of smooth map germs G : (R m,0) (R p,0). In order to begin the analysis of conditions under which a smooth map germ is finitely determined it was considered the following definition: Definition [Wa, Definition p. 482] Let E be an equivalence relation defined on map germs. We say that the map germ G is E-determined by its k-jet or, more briefly, that G is k-e-determined if any smooth map germ H with j k G = j k H is E-equivalent to G. If G is k-e-determined for some positive integer k, then it is finitely-e-determined, and the least such k is the degree of determinacy.

93 4.1. Milnor-Hamm fibration 91 On this survey Wall basically considered the equivalence relations defined by the actions of the following groups: (i) R, of germs at 0 of smooth diffeomorphisms of (R m,0). These act on G by composition on the right and is usually called right-equivalence; (ii) L, of germs at 0 of smooth diffeomorphisms of (R p,0). These act on G by composition on the left. It is called left-equivalence; (iii) A = R L, (direct product); (iv) C, of germs at 0 of smooth diffeomorphisms H of (R m R p,0) which (a) leave fixed the projection on R m and (b) preserve the subspace R m {0}; (v) K = R C, (semi-direct product). It is called contact-equivalence. In the second part of this paper called Topological and C r Theory, the author weakens the smooth condition replacing smooth diffeomorphisms by C r -diffeomorphisms with 0 r < and studied the notion the finite determinacy induced by C r -G -equivalence on the space of smooth germs, where G denote an unspecified group R,L,A,C or K. As a consequence of the above definition and notation, one can restate Theorem 6.1 (i) and Corollary of [Wa], for G = K in the following way: Theorem [Wa, Theorem 6.1] Let G : (R m,0) (R p,0) be a smooth map germ, and 0 r <. G is C r finitely K -determined if and only if the ideal J G + G (m p ) E m is elliptic, where J G is the ideal in E m spanned by p p minors of the jacobian matrix of G and G (m p ) is the pullback ideal in E m. Corollary [Wa, Corollary 6.1.2] Let G : (R m,0) (R p,0) be a smooth map germ, and 0 r <. The ideal J G + G (m p ) E m is elliptic if, and only if there exists a neighborhood W of 0 R m and constants c,α > 0 such that det [ JG(x)(JG(x)) t] + f (x) 2 c x α, for any x W, where JG(x) denotes the jacobian matrix of G in x and (JG(x)) t its transpose.

94 92 Chapter 4. Milnor-Hamm Fibrations Proof of Corollary Under the finite determinacy condition one can apply Theorem and Corollary to get that the inequality det [ JG(x)(JG(x)) t] + f (x) 2 c x α, holds in some neighborhood W of 0 R m with constants c,α > 0. The matrix JG(x)(JG(x)) t is the Gram matrix of the vectors { G 1 (x),..., G p (x)}, therefore these vectors are linearly independent if, and only if the Gram-determinant is not zero, see [HJ]. Consequently, if x Sing G V G then x = 0 and our claim follows from Corollary General comment It is well known that for analytic map germs G : (R m,0) (R p,0), m > p, the condition Sing G V G = {0} is equivalent to finite C 0 -K -determinacy condition. See [Wa, CB] for details. map In [CB] the authors studied this class of maps and showed that the restriction G : B m ε G 1 (Sη p 1 ) Sη p 1 (4.5) is a smooth map and its homotopy A -equivalence type is independent of the choice of ε > 0 and η > 0, small enough. They used that to characterize the topological class of the map germ. See [CB, Theorem 4.2 ]. As a consequence, they also obtained that the restriction (4.5) is not a smooth fibration. However, if they had removed the discriminant locus, they would have the Milnor-Hamm fibration, according to our Proposition We wonder whether this new class of fibration would be useful to refine the homotopy A -class of the map germ. That would provide an interesting application. 4.2 Maps with full image In this section we present a class of maps under which the fibers of the Milnor- Hamm fibration is not empty. As we will see this class contains maps with isolated critical value satisfying the condition (2.14). Definition Let G : (R m,0) (R p,0) be a map germ. We say that Im G is full if it contains an open ball Bη p for some small enough η > 0.

95 4.2. Maps with full image 93 Proposition Let G : (R m,0) (R p,0), m > p 1, be an analytic map germ. If Sing G V G = {0} and the link K ε := Sε m 1 V G /0, then Im G is full. Proof. By Corollary 4.1.7, one has that the restriction G : S m 1 ε G 1 (B p η) B p η (4.6) is a submersion. Therefore, it is a open and closed map. Since Bη p is connected and K ε := Sε m 1 V G /0, one conclude that the restriction (4.6) is surjective. Consequently, Im G is full. Remark It is not possible to remove the hypothesis K ε /0 in the above result. Indeed, in the example f (x,y) = x 2 + y 2 the condition Sing f V f = {0} is satisfied, however Im f is not full. We notice that in this case K ε = /0. Proposition Let G : (R m,0) (R p,0), m > p 2 be a real analytic map germ which satisfies the condition (4.2). If codimdisc G 2, then Im G is full. In particular, the topological type of the fiber does not change. Proof. As we have seen, the map G satisfies condition (4.2) if, and only if there exists ε 0 > 0 such that, for any 0 < ε < ε 0, there exists η, 0 < η ε, such that the restriction map (4.3) is a smooth submersion. Since restriction (4.6) is a proper map, hence the (4.3) is proper too. Thus, it is an open and closed map. By hypothesis, one has that codimdisc G 2, hence Bη p \ Disc G ( is connected and G S m 1 ε G 1 (Bη p \ Disc G) ) /0. Therefore, the restriction (4.3) is surjective which implies that Im G is full. Corollary Let f 1 : (R m 1,0) (R p 1,0),..., f k : (R m k,0) (R p k,0), analytic map germs in separable variables and consider the map germ G := ( f 1,..., f k ) : (R m 1 R m k,0) (R p 1 R p k,0). If G satisfies the condition (4.2), m j > p j 2 and codimdisc f j 2 for any j = 1,...,k, then Im G is full. In particular, the topological type of the fiber does not change. Proof. Assume without lost of generality that k = 2. One has that Disc G = Disc f Im g Im f Disc g. Thus, dimdisc G p 1 + p 2 2 and consequently, codimdisc G 2. By Proposition 4.2.4, Im G is full.

96 94 Chapter 4. Milnor-Hamm Fibrations Example Let G 1 : (R m 1,0) (R p 1,0),...,G k : (R m k,0) (R p k,0), in separable variables and consider the map germ G := (G 1,...,G k ) : (R m 1 R m k,0) (R p 1 R p k,0). If G j is a MSL for any j = 1,...,k, then ImG is full and the topological type of the fiber does not change. Corollary [Ma, Corollary 4.7] Let G : (R m,0) (R p,0), m > p 2, be a real analytic map germ which satisfies the condition (2.14). If Disc G = {0}, then Im G is full. Proof. Since Sing G V G then the conditions (4.2) and (2.14) agree. Moreover, codimdisc G 2 and the result follow from Proposition The partial Thom regularity condition It is classically known that the Thom regularity at V G insures the existence of the Milnor-Hamm fibration in the case of isolated singular value. We show that a weaker Thom type regularity is sufficient for the same aim. Definition Let G : (R m,0) (R p,0) be an analytic map germ. We say that G is -Thom regular at V G (or G has -Thom regularity) if there is a Whitney (a) stratification S = {S α } of V G such that, for any stratum S α, the pair (B m ε \ G 1 (Disc G),S α ) satisfies the Thom a G -condition. A similar notion was introduced in [Ti1], [Ti3, A.1.1.1]. Let us remark that the Thom regularity at V G (and the Thom regularity of a map G in the classical sense) obviously implies -Thom regularity at V G. Proposition Let G : (R m,0) (R p,0) be an analytic map germ. If G is -Thom regular at V G, then G has Milnor-Hamm fibration (4.1). Proof. The -Thom regularity implies that for any ε > 0 small enough, there exists a neighborhood N ε of Sε m 1 V G in Sε m 1 such that for any x N ε \ G 1 (Disc G) one has x G 1 (G(x)). S m 1 ε Indeed, since G is -Thom regular at V G, there exists a Whitney (a) stratification S = {S α } of V G such that for any stratum S α, the pair (B m ε \ G 1 (Disc G),S α ) satisfies

97 4.3. The partial Thom regularity condition 95 the Thom a G -condition. Moreover, one can consider a stratification such that for any small enough ε > 0 the spheres S m 1 ε intercepts transversely any strata S α. Let p Sε m 1 S α. If the statement is not true, it is possible to obtain a sequence of points {x n } with x n Sε m 1 Now, up to subsequence, let us consider T := lim T xn G 1 (G(x n )) in some appropriate \G 1 (Disc G) such that x n p and S m 1 ε xn G 1 (G(x n )). Grassmannian bundle. By -Thom regularity one has that T p S α T which is a contradiction since T p S α + T p S m 1 ε = R m. It follows that there is η ε > 0 such that Sε m 1 G 1 (B m η ε \ Disc G) N ε. Then the restriction (4.3) is a submersion and consequently G satisfies condition (4.2) and one may apply Theorem Theorem Let f : (R m,0) (R p,0) and g : (R n,0) (R k,0) be germs of analytic maps in separable variables. If each of them has the -Thom regularity, then so does G := ( f,g) : (R m R n,0) (R p R k,0). Proof. The fiber G 1 (c 1,c 2 ) R m R n is the product f 1 (c 1 ) g 1 (c 2 ). If S f and S g are Whitney stratifications of V f and V g, respectively, such that f and g has the - Thom regularity, then the product stratification S f S g on (R m R n,0) is such that G is -Thom regular at V G = V f V g. Indeed given any stratum S W S f S g and a point (x 0,y 0 ) S W, let (x i,y i ) (R m R n ) \ G 1 (Disc G) be any sequence of regular points for G such that (x i,y i ) (x 0,y 0 ) and (x i,y i ) / V G. First, we notice that Disc G = Disc f Im f Im f Disc g, thus G 1 (Disc G) = f 1 (Disc f ) R n R m g 1 (Disc g). If f (x i ) 0 and g(x i ) 0 for any big enough index i, then one has the equality: T (xi,y i )G 1 (G(x i,y i )) = T xi f 1 ( f (x i )) T yi g 1 (g(y i )). (4.7) Else, if say f (x i ) = 0 for all i large enough, since (x i,y i ) / V G then g(y i ) 0 for those indices i. Now, let S xi the stratum of V f such that x i S xi. One claims that T (xi,y i )G 1 (G(x i,y i )) T xi S xi T yi g 1 (g(y i )). (4.8) Indeed, let v T xi S xi and γ S xi a path such that γ(0) = x i and γ (0) = v. For any t > 0, one has f (γ(t)) = 0, thus 0 = ( f γ) = d xi f (v). t t=0

98 96 Chapter 4. Milnor-Hamm Fibrations Since (x i,y i ) / G 1 (Disc G) then y i must be a regular point of g. Thus, T yi g 1 (g(y i )) = ker(d yi g) and if (v,w) T xi S xi T yi g 1 (g(y i )) one has d (xi,y i )G(v,w) = (d xi f (v),d yi g(w)) = (0,0), i.e., (v,w) Ker ( d (xi,y i )G ) = T (xi,y i )G 1 (G(x i,y i )) which proves the inclusion (4.8). Finally, by -Thom regularity, lim xi x 0 T xi f 1 ( f (x i )) T x0 S and lim yi y 0 T yi g 1 (g(y i )) T y0 W. Taking limits in (4.7) or (4.8), we get the inclusion lim i T (xi,y i )G 1 (x i,y i ) T (x0,y 0 ) (S W), then the result follows. Remark For a map G := ( f,g) : (R m R n,0) (R p R k,0) with f : (R m,0) (R p,0) and g : (R n,0) (R k,0) germs of analytic maps in separable variables, one has V G = V f V g, Sing G = Sing f R n R m Sing g, Disc G = Disc f Im g Im f Disc g and G 1 (Disc G) = f 1 (Disc f ) R n R m g 1 (Disc g). Example Let f : (R 3,0) (R 2,0), f (x,y,z) = (y 4 z 2 x 2 x 4,xy). One has that Sing f = {x = y = 0}, hence Disc f = {0}. Moreover, f is Thom regular at V f, cf [ACT1, Example 5.3]. Let us consider the following map germ g := (g 1,g 2 ) : (R 4,0) (R 2,0) given by: g 1 (a,b,c,d) = 2d 3 ab 3d 2 a 2 c + 3d 2 b 2 c + 6dabc 2 + a 2 c 3 b 2 c 3 g 2 (a,b,c,d) = d 3 a 2 d 3 b 2 6d 2 abc 3da 2 c 2 + 3db 2 c 2 + 2abc 3. So, one has g 1 = ( 2d 3 b 6d 2 ac + 6dbc 2 + 2ac 3 ) e 1 +( 2d 3 a + 6d 2 bc + 6dac 2 2bc 3 ) e 2 ( 3d 2 a 2 + 3d 2 b dabc + 3a 2 c 2 3b 2 c 2 ) e 3 +( 6d 2 ab 6da 2 c + 6db 2 c + 6abc 2 ) e 4 and, g 2 = (2d 3 a 6d 2 bc 6dac 2 + 2bc 3 ) e 1 +( 2d 3 b 6d 2 ac + 6dbc 2 + 2ac 3 ) e 2 +( 6d 2 ab 6da 2 c + 6db 2 c + 6abc 2 ) e 3 +(3d 2 a 2 3d 2 b 2 12dabc 3a 2 c 2 + 3b 2 c 2 ) e 4.

99 4.4. Classes of maps f ḡ with -Thom regularity 97 Thus, g 1, g 2 = 0 and g 1 2 = (a 2 + b 2 )(d 2 + c 2 ) 2 (4d 2 + 9a 2 + 9b 2 + 4c 2 ) = g 2 2, which shows that g is a Simple Ł-Map and in particular has isolated critical value and is Thom regular at V g. Let us now consider the map G := ( f,g) : (R 3 R 4,0) (R 4,0). By Theorem it then follows that G is -Thom regular at V G and hence it has a Milnor-Hamm fibration. By Remark one has Disc G = {0} Im g Im f {0}, hence by Proposition 4.2.4, Im G is full and the Milnor fiber topological type does not change. Example Let G := (G 1,...,G k ), with G j a MSL for any j = 1,...,k with separable variables. Then G has a Milnor-Hamm fibration and by Example the Im G is full and the topological type of the fiber is unique. 4.4 Classes of maps f ḡ with -Thom regularity In the recent paper [PT] the authors Mihai Tibăr and A.J. Parameswaran introduced natural and convenient conditions which allowed them to produce classes of genuine real map germs with the Milnor tube fibration in the classical sense. They focused on the study of mixed functions germs of type f ḡ : (C n,0) (C,0) with n > 1 where f,g : (C n,0) (C,0), are non-constant holomorphic function germs and presented the following results: Theorem [PT, Theorem 2.3 p.4 ] Let f,g : (C n,0) (C,0), n > 1, be some non-constant holomorphic function germs. Then the discriminant Disc f ḡ of the mixed function f ḡ : (C n,0) (C,0) is either the origin or a union of finitely many real halflines. Moreover, Disc ( f ḡ) {0} if, and only if the discriminant Disc ( f,g) of the map ( f,g) contains no line component different from the axes. The above result provides a clear characterization for the isolated critical value condition, i.e., Sing f ḡ V f ḡ. Definition [PT] Let f,g : (C n,0) (C,0) be any holomorphic functions germs. The Thom irregularity locus of f ḡ is defined as follows { } NT f ḡ := x V f ḡ there is no Thom (a f ḡ)-stratification of V f ḡ such that x is on a positive dimensional stratum. (4.9)

100 98 Chapter 4. Milnor-Hamm Fibrations For the pair of map germ ( f,g) : (C n,0) (C 2,0) one defines NT ( f,g) analogously just switching in the above definition the function f ḡ by ( f,g). Under these definitions and notations they proved the following results. Theorem [PT, Theorem 3.1, p.6] Let f,g : (C n,0) (C,0) be any holomorphic germs such that Disc ( f,g) does not contain lines other than the axes. Then NT f ḡ NT ( f,g). Corollary [PT, Corollay 4.1 p.8] Suppose that Disc ( f,g) does not contain lines other than the axes. If the map ( f,g) admit a Thom stratification, then f ḡ admits a Thom stratification along V f ḡ. In particular, if ( f,g) is an ICIS then f ḡ has a Thom a f ḡ -stratification along V f ḡ. These results allowed the authors to show that the real analytic map germs of type f ḡ form an excellent class to look for examples with the Milnor tube fibration and the Thom regularity. Next section we will show how to extend the previous result on this class by considering the appropriate definition for the Thom irregular locus Mixed f ḡ: A first extension for non-isolated critical value One may formulate the definition of the -Thom irregularity set in analogy with (4.9), namely by simply replacing Thom by -Thom, as follows: NT f ḡ := { x V f ḡ there is no -Thom (a f ḡ)-stratification of V f ḡ such that x is on a positive dimensional stratum and consider the similar set NT ( f,g) associated to the map ( f,g) instead of the map f ḡ. We then state and prove our main result in this section. Theorem Let f,g : (C m,0) (C,0), m > 1, be a holomorphic function germs such that Disc ( f,g) is a union of lines. Then NT f ḡ NT ( f,g). In particular, if such a map ( f,g) has -Thom regularity, then f ḡ has -Thom regularity. }

101 4.4. Classes of maps f ḡ with -Thom regularity 99 Proof. By standard arguments one has NT f ḡ Sing f ḡ V f ḡ. One shows exactly like in the proof of Theorem (see also the first part of the proof of Lemma 5.2.3) that NT f ḡ { f = g = 0}. We claim that f ḡ has -Thom regularity on the set W := { f = g = 0} \ NT ( f,g). By the definition of NT ( f,g), there is Whitney (a) stratification S = {W i } of W such that for any i, the pair (B 2n ε \ ( f,g) 1 (Disc ( f,g)),w i ) satisfies the Thom a ( f,g) - condition. Since for the -Thom regularity it is only necessary to check the Thom condition on the smooth part, outside the discriminant set, let us consider p 0 W i and an arbitrary sequence p n p 0 such that p n B 2n ε \ ( f ḡ) 1 (Disc ( f ḡ)) for some small enough ε > 0. We claim that p n B 2n ε \ ( f,g) 1 (Disc ( f,g)) for any natural number n. Indeed, assume that there exists a number n 0 such that p n0 ( f,g) 1 (Disc ( f,g)). One has that ( f (p n0 ),g(p n0 )) Disc ( f,g), which is an union of lines. Consequently, ( f (p n0 ),g(p n0 )) C i where C i is a line through the origin. By [PT, Lemma 2.5], the image of Disc ( f,g) by the map u v : C 2 C is precisely Disc f ḡ, hence one has f ḡ(p n0 ) Disc ( f ḡ) and p n0 ( f ḡ) 1 (Disc ( f ḡ)), which is a contradiction. Since W i is a -Thom regular stratum, the Thom a ( f,g) -condition is satisfied along of sequence p n to pair ( f,g). Moreover, since any fiber of f ḡ is a union of fibres of ( f,g), there exists (c n,d n ) C 2 such that p n ( f,g) 1 (c n,d n ) ( f ḡ) 1 ( f ḡ(p n )). In addition, the fiber F ( f,g) (n) := ( f,g) 1 (c n,d n ) is a smooth manifold. Therefore, one has lim p n p 0 T pn F f ḡ (n) lim p n p 0 T pn F ( f,g) (n) T p0 W i This show that W i is a -Thom stratum for the map f ḡ. Corollary Let f : (C m,0) (C,0) and g : (C n,0) (C,0) be holomorphic germs in separable variables. Then f ḡ is -Thom regular at V f ḡ and thus has Milnor-Hamm fibration. Proof. The pair ( f,g) has the -Thom regularity condition by Theorem Moreover, one has that Disc ( f,g) = Disc f C C Disc g = {0} C C {0}, i.e., it is an union of lines. Thus, the result follows from Theorem

102 100 Chapter 4. Milnor-Hamm Fibrations Example Let G : (C 4,0) (C 2,0) given by G(z 1,z 2,z 3,z 4 ) = (z 1 z 2,z 3 z 4 ). By Corollary each component function of G has -Thom regularity. Since the map G is also separate variables, one may apply Theorem to get that G has -Thom regularity as well. By Theorem the Milnor-Hamm fibration (4.1) exists for G. One has Sing G V G {0} and Disc G = {(z,0) C 2 z C} {(0,w) C 2 w C}, which shows that this example is not of the classical type. Proposition Let f,g : (C n,0) (C,0) be holomorphic function germs. If f and g are weighted homogeneous of type (q 1,...,q n ;d) and Sing ( f,g) \ V ( f,g) /0, then Disc ( f,g) is a union of complex lines and it contain a line different from the coordinates axes. Proof. We notice that for any t > 0 one has Sing ( f,g) = t Sing ( f,g), t V f = V f and t V g = V g because f, g and hence ( f,g), are weighted homogeneous. Therefore, t (Sing ( f,g) \V ( f,g) ) = Sing ( f,g)\v( f,g). Since Sing ( f,g)\v ( f,g) /0 one can choose z in this set and, up to reprarametrization, consider the path α z : (0,1] Sing ( f,g) \V ( f,g) given by α z (t) = t z. One has that η z (t) := ( f,g)(α z (t)) = t d ( f (z),g(z)) i.e., a line in C 2 \ {u = 0} {v = 0} such that η z 0. As α z ((0,1]) Sing ( f,g) \ V ( f,g) then η z Disc ( f,g). Let Disc ( f,g) = i C i a finite decomposition of irreducible connected components C i of dimension 1 such that 0 C i. If (p,q) C i \ {(0,0)} hence there exists z i Sing ( f,g) \ V ( f,g) such that ( f,g)(z i ) = (p,q). As we have seen, the line n zi Disc ( f,g). Moreover, (p,q) n zi C i and consequently, C i = n z. Therefore, Disc ( f,g) is a union of finitely many lines through the origin and contain lines different from the coordinates axis. Corollary Let f,g : (C n,0) (C,0) be holomorphic functions germs weighted homogeneous of the same type and degree, with Sing ( f,g)\v ( f,g) /0. If ( f,g) is -Thom regular at V ( f,g), then f ḡ has the Milnor-Hamm fibration and {0} Disc f ḡ. Example Let f,g : C 2 C given by f (x,y) = x 2 +y 2 and g(x,y) = x 2 y 2. One has that V ( f,g) = {(0,0)} and Sing ( f,g) = {x = 0} {y = 0}. Thus, ( f,g) has -Thom

103 4.5. Milnor-Hamm without -Thom regularity 101 regularity and Sing ( f,g) \V f ḡ /0. Therefore, by Corollary f ḡ has non-isolated critical value and it has the Milnor-Hamm fibration. 4.5 Milnor-Hamm without -Thom regularity In this section we will concentrate on maps f : (R m,0) (R p,0) and g : (R n,0) (R k,0) in separable variables in order to construct examples of Milnor-Hamm fibration without -Thom regularity. Notice that, for the map germ G := ( f,g) : (R m R n,0) (R p R k,0) one has ImG = Im f Img. Therefore, if the image of f and g are full, then so is Im G. Proposition Let f : (R m,0) (R p,0) be a real analytic map germ and let g := (g 1,...,g n ) : (R n,0) (R n,0) where g j : (R,0) (R,0) is a germ of a polynomial function for any j = 1,...,n, such that f,g 1,...,g n has separable variables. Consider the analytic map germ G := ( f,g) : (R m R n,0) (R p R n,0). Then f satisfies the condition (4.2) if, and only if G does. The proof follows from the next two lemmas. Since we are working in separable variables one can reduce to the case where g is a single function and then the result follows by induction. Lemma Let f : (R m,0) (R p,0) be real analytic map germ and let g : (R,0) (R,0) be a germ of polynomial function, such that f and g are in separable variables. Consider the analytic map germ G := ( f,g) : (R m R,0) (R p R,0). Then M(G) = M( f ) R or M(G) = M( f ) R R m {0}. Proof. Let v := (x,y) where x = (x 1,...,x m ) R m and y R. One has that ([ ] JG(v) [ ] ) ( ) ] dg 2 det([ det [JG(v)] t v t J f (w) [ ] ) = [J f (w)] v dy t (w) t. w One knows that v M(G) if and only if the determinant on the left side is null. Consequently, from the above equality M(G) = M( f ) R R m V dg/dy. Since dg/dy : (R,0) (R,0) is germ of polinomial function, then locally one has V dg/dy = /0 or V dg/dy = {0}, which finish the proof.

104 102 Chapter 4. Milnor-Hamm Fibrations Lemma Let f : (R m,0) (R p,0) be real analytic map germ and let g : (R,0) (R,0) be a germ of a polynomial function, such that f and g has separable variables. Consider the analytic map germ G := ( f,g) : (R m R,0) (R p R,0). Then f satisfies the condition (4.2) if and only if G also satisfies the condition (4.2). Proof. Assume that f satisfies condition (4.2). Let p 0 = (x 0,y 0 ) M(G) \ G 1 (Disc G) V G. One has that p 0 V G and there exists a sequence of points p n := (x n,y n ) M(G) \ G 1 (Disc G) such that p n p 0. First, we notice that locally V g = {0}, hence V G = V f {0} and y 0 = 0. Let us consider the two following cases: Case 1:V dg/dy = {0}. From Lemma 4.5.2, one has that M(G) \ G 1 (Disc G) = (M( f ) R R m {0}) \ ( f 1 (Disc f ) R R m {0} ) = ( M( f ) R \ f 1 (Disc f ) R ) \ R m {0}. Therefore, x n M( f ) \ f 1 (Disc f ) and x 0 M( f ) \ f 1 (Disc f ) V f {0}, which implies that p 0 = (0,0). Reciprocally, one assumes that G satisfies the condition (4.2). For any x 0 M( f ) \ f 1 (Disc f ) V f there exists a sequence x n M( f ) \ f 1 (Disc f ) such that x n x 0. Since ( M( f ) \ f 1 (Disc f ) ) (R \ {0}) M(G) \ G 1 (Disc G), hence (x n,1/n) M(G) \ G 1 (Disc G) for any n N and the limit point (x 0,0) M(G) \ G 1 (Disc G) V G. Therefore, we conclude that x 0 = 0, which implies that f satisfies the condition (4.2). Case 2: V dg/dy = /0. From Lemma 4.5.2, one has p n M(G) \ G 1 (Disc G) = M( f ) R n \ f 1 (Disc f ) R n. Consequently x n M( f )\ f 1 (Disc f ) and x 0 M( f ) \ f 1 (Disc f ) V f {0} which implies p 0 = (0,0) and G satisfies the condition (4.2). Reciprocally, one assumes that G satisfies the condition (4.2). For any x 0 M( f ) \ f 1 (Disc f ) V f there exists a sequence x n M( f ) \ f 1 (Disc f ) such that x n x 0. Since one has that ( M( f ) \ f 1 (Disc f ) ) {0} M(G) \ G 1 (Disc G), hence (x n,0) M(G) \ G 1 (Disc G) for any n N and the limit point (x 0,0) M(G) \ G 1 (Disc G) V G. Therefore, we conclude that x 0 = 0, which implies that f satisfies the condition (4.2). Proof of Proposition It is sufficient to apply the Lemma for mappings G 1 := ( f,g 1 ),G 2 := (G 1,g 2 ),...,G := G n = (G n 1,g n ).

105 4.5. Milnor-Hamm without -Thom regularity 103 The next result is more one step toward the construction of examples with Milnor- Hamm fibrations without -Thom regularity. Proposition Let f : (R m,0) (R p,0) be real analytic map germ and let g := (g 1,...,g n ) : (R n,0) (R n,0) where g j : (R,0) (R,0) be a germ of a polynomial function for any j = 1,...,n, such that f,g 1,...,g n has separable variables. Consider the analytic map germ G := ( f,g) : (R m R n,0) (R p R n,0). Then f has -Thom regularity if and only if G has -Thom regularity. Proof. As we have seen, it is sufficient to consider the case g = g 1 : (R,0) (R,0) and apply the result several times. Since g is -Thom regular at V g, if f is -Thom regular at V f, then it follows from Theorem that G is -Thom regular at V G as well. For the converse, consider any Whitney stratification W = {W α {0}} of V G = V f {0} which extends to a Whitney stratification of R m R. Consider the Whitney stratification W 1 = {W α } of V f. If f is not -Thom regular at V f, there exist points x 0 V f Sing f such that for any positive dimensional strata W α of W 1, W α Sing f containing x 0, there exist analytic curves γ x0 B m ε \ f 1 (Disc f ), such that γ x0 x 0 and n c(t), f (γ x0 (t)) n c(t), f (γ x0 (t)) v / [T x 0 W α ] with v 0 and c(t) = (c 1 (t),...,c p (t)) R p \ {0} such that n c(t), f (γ x0 (t)) = c 1 (t) f 1 (t) + + c p (t) f p (t). The normal vector space to fibers of G at the point p := (x,y) R m R is spanned by the vectors { ( f 1 (x),0),...,( f p,0),(0,dg/dy) }. Define the point p 0 := (x 0,0) V G. Let us consider the following two cases: Case 1: V dg/dy = {0}. Define the path γ(t) = (γ x0 (t),t). One has that γ p 0 when t 0 and for small enough t > 0, γ B m+1 ε. Assume that γ G 1 (Disc G) with t 0. Since G 1 (Disc G) = f 1 (Disc f ) R R m {0} then γ f 1 (Disc f ) R, i.e., γ x0 f 1 (Disc f ) which is a contradiction. Therefore, one has γ B m+1 ε \ G 1 (Disc G).

106 104 Chapter 4. Milnor-Hamm Fibrations Consider the vector v(t) := (c 1 (t),...,c p (t),0) R p+1 \{0} and define n v(t),g (γ(t)) = (c 1 (t) f 1 (t) + + c p (t) f p (t),0). One has that n v(t),g (γ(t)) n v(t),g (γ(t)) = ( nc(t), f (γ x0 (t)) n c(t), f (γ x0 (t)),0 ) (v,0). Since v / [T x0 W α ], there exists some w 0 T x0 W α such that the inner product v,w 0 0. Thus, (v,0),(w 0,0) 0 with (w 0,0) T p0 (W α {0}), which implies (v,0) / [T p0 (W α {0})]. Therefore, the Thom a G -condition is not satisfied along γ for pair (B m+n ε \ G 1 (Disc G),W α {0}) i.e., G is not -Thom regular at V G. Case 2:V dg/dy = /0. The same reasoning used above. Example Let G = ( f,g) = (R 4,0) (R 3,0), where f (x,y,z) = (x,y(x 2 + y 2 ) + xz 2 ) and g(w) = w. By Example the map f has isolated critical value and satisfies the condition (4.2). By Corollary one gets that G satisfies condition (4.2), therefore G has Milnor-Hamm fibration. Moreover, Disc G = {0} R hence G has no isolated critical value. On the other hand, if follows from Example that f is not -Thom regular at V G. Therefore, by Proposition the map G cannot be -Thom regular. One may construct more examples as follows: Example Let f be any MSL function and G j = F j + f in separable variables, where F j is one of the following mixed functions below: 1) F 1 (x,y) = xy k x, for a fixed k 1, from Example , 2) F 2 (x,y,z) = (x + z k ) xy, for a fixed k 2, from Example 3.4.6, 3) F 3 (x,y) = (xy y 2 )( x ȳ), from Example , 4) F 4 (w 1,...,w n ) = w 1 ( k j=1 w j 2a j n t=k+1 w t 2a t ), from [Oka3]. From each G j one may construct the map H j := (G j,g) where g := (g 1,...,g n ) : (R n,0) (R n,0) polynomial map, with G j and g j : (R,0) (R,0) in separable variables. Therefore, by Proposition 4.5.1, the map germ H j has a Milnor-Hamm fibration without -Thom regularity.

107 105 CHAPTER 5 SINGULAR MILNOR TUBE FIBRATIONS As shown in the previous chapter, the Milnor-Hamm fibration theorem is a natural extension of the classical Milnor tube fibration theorem. In this chapter we will extend the definition of the Milnor-Hamm fibration in order to prove a more general fibration Theorem in the stratified point of view, which we call singular Milnor tube fibration, where now we will include the discriminant set on the base space of fibration. For that, we will use the results and definitions introduced in Section in order to define the stratwise Milnor set and a stratified transversality condition. We introduce an example that shows that our result is not a consequence of the Thom-Mather isotopy Theorem and we compare our stratwise transversality condition with the Thom regularity condition. In particular, one gets that our result also extend Theorem in Chapter 2, called Milnor-Lê fibration. Moreover, one gives a second extension of the main result of [PT] for functions f ḡ which shows again that this class is still a good one to look for singularities with a singular Milnor tube fibration. We finish this chapter presenting another example of a map germ that has the singular Milnor tube fibration and is not a Thom mapping. 5.1 Stratwise Milnor set Let G : (R m,0) (R p,0) be an analytic map germ and the pair (W,S) a regular stratification of the map germ G. One may define a stratwise Milnor set M (G) as the

108 106 Chapter 5. Singular Milnor tube fibrations union of the Milnor sets of the restrictions of G to each stratum. Namely, let W α W be the germ at the origin of some stratum, and let M(G Wα ) be the Milnor set, like in Definition in Chapter 2, consider: M(G Wα ) := { } x W α ρ Wα x G Wα where ρ Wα denotes the restriction of the distance function ρ to the subset W α. Definition We call M (G) := α M(G Wα ) the set of stratwise ρ-nonregular points of G with respect to the stratifications W and S. Note that M (G) is not necessarily closed. We then consider the following condition: M (G) \V G V G {0}. (5.1) which extends to this new stratified setting the similar conditions (2.14). It also extends our previous condition (4.2) in the sense that the condition (5.1) restricted to the open and dense stratum B m ε \ G 1 (Disc G) is just the condition (4.2), thus obviously (5.1) implies (4.2). Example Consider G(x,y,z) = (xy,z 2 ) as in Example Let us see that G with the pair (W,S) as defined below satisfies the condition (5.1). In the sequence we will arrange the strata of W in order to help us calculate the Milnor set on each restriction. { Group 0 = W0 s = B m ε \ G 1 (Disc G) W1 s = {(0,y,z) R 3 y > 0,z > 0} W2 s = {(0,y,z) R 3 y > 0,z < 0} Group 1 = W3 s = {(0,y,z) R 3 y < 0,z > 0} W4 s = {(0,y,z) R 3 y < 0,z < 0} W5 s = {(x,0,z) R 3 x > 0,z > 0} W6 s = {(x,0,z) R 3 x > 0,z < 0} Group 2 = W7 s = {(x,0,z) R 3 x < 0,z > 0} W8 s = {(x,0,z) R 3 x < 0,z < 0}

109 5.1. Stratwise Milnor set 107 Group 3 = W c 1 = {(x,y,0) R 3 x > 0,y > 0} W c 2 = {(x,y,0) R 3 x < 0,y < 0} W c 3 = {(x,y,0) R 3 x < 0,y > 0} W c 4 = {(x,y,0) R 3 x > 0,y < 0} Group 4 = W c 5 = {(0,y,0) R 3 y > 0} W c 6 = {(0,y,0) R 3 y < 0} W c 7 = {(x,0,0) R 3 x < 0} W c 8 = {(x,0,0) R 3 x < 0} { W9 c = {(0,0,z) R 3 z > 0} Group 5 = W10 c = {(0,0,z) R 3 z < 0} { Group 6 = W11 c = {(0,0,0)} Figure 6 Stratification of G 1 (Disc G)

110 108 Chapter 5. Singular Milnor tube fibrations and S 0 = B 2 η \ Disc G S 1 = {(u,0) R 2 u > 0} S 2 = {(u,0) R 2 u < 0} S 3 = {(0,v) R 2 v > 0} S 4 = {(0,0)}. G W c k Consider the restriction maps G s i,3 := G W s i : Wk c S 1, with k = 1,2, G c k,2 := G W c : W c k Wk c S 4, with k = 5,6,7,8, G c k,3 := G W c k W11 c S 4. : W s i S 3, with i = 0,...,8, G c k,1 := k S 2, with k = 3,4, G c k,4 := G W c : k : Wk c S 3, with k = 9,10 and G c 11,4 := G W c : 11 Let us compute the relative Milnor sets. For that, consider the restrictions ρ s i := ρ W s i, with i = 0,...,8 and ρ c k := ρ W c j, with k = 1,...,11. On the Group 1, consider by simplicity the restriction G s 1,3. One has that the fibers of ρ1 s and Gs 1,3 are given as in Figure 7 Figure 7 Fibers of restrictions ρ1 s, Gs 1,3 and its intersections The third picture on Figure 7 show us the intersection of fibers ρ s 1 and Gs 1,3. We notice that the points on which the fibers could be non-transverse were removed from the stratum W1 s, more precisely, they belong to the stratum W 9 s. Indeed, first we notice that the tangent space of the of G s 1 at the point p = (0,y,z) is spanned by vector v = (0,1,0). On the other hand, one has the fibers of ρ1 s are given by {x2 + y 2 + z 2 = ε 2 } W1 s. Consequently, the normal vector to the fibers of ρ1 s is given by N := (0,y,z). Thus, one has N,v = y. We know that the fibers of ρ1 s and Gs 1,3 are tangents if, and only if

111 5.1. Stratwise Milnor set 109 N,v = 0, hence on W1 s they cannot be tangents and M(Gs 1,3 ) = /0. The same analysis is repeated for any restrictions in Group 1 and Group 2. In Group 3, consider by simplicity the restriction G c 1,1. Figure 8 Fibers of restrictions ρ1 c, Gc 1,1 and its intersections The normal space of the fibers of G c 1,1 is spanned by v := (y,x,0). The fibers of ρ1 c are given by {x2 + y 2 + z 2 = ε 2 } W1 c. Thus the normal vectors for this fibers is spanned by N = (x,y,0). We know that p M(G c 1,1 ) if, and only if the vector product v N = 0. Since v N = (0,0,y 2 x 2 ) it follows that M(G c 1,1 ) = {(λ,λ,0) λ > 0}. Therefore, M(G c 1,1 ) \V G = {0} which implies M(G c 1,1 ) \V G V G = {0}. One can use the same idea on all Group 3. In Group 4, consider the restriction G c 5,4. Figure 9 Fibers of restrictions ρ5 c, Gc 5,4 and its intersections Since T p (ρ c 5 ) = 0, hence T p(ρ c 5 ) T p(g c 5,4 ) which implies M(Gc 5,4 ) = W c 5. Thus M(G c 5,4 ) \V G = {0} and M(G c 5,4 ) \V G V G = {0}. We can use the same idea on all Group 4. In Group 5, consider the restriction G c 9,3. In this case one has again M(Gc 9,3 ) = W c 9 and M(G c 9,3 ) \V G V G = {0}. The same occur for the restriction G c 10,3.

112 110 Chapter 5. Singular Milnor tube fibrations Figure 10 Fibers of restrictions ρ9 c, Gc 9,3 and its intersections Finally, since G satisfies the condition (4.2), one has M(G s 0,0 ) \V G V G {0}. We consider the stratwise Milnor set ( 8 M (G) = M i=0 ) ( ( ) 11 ( ) ) G W s M G W c. i j j=1 By above analysis, one has that M (G) \V G V G {0} and G satisfies the condition (5.1). Definition Let G : (R m,0) (R p,0) be an analytic map germ. We say that G has a singular Milnor tube fibration if there exist stratifications W and S as above, and if for any small enough ε > 0 there exists 0 < η ε such that the restriction: G : B m ε G 1 (B p η \ {0}) B p η \ {0} (5.2) is a stratified locally trivial fibration which is independent, up to stratified homeomorphisms, of the choices of ε and η. If such a tube fibration exists, then a Milnor-Hamm fibration exists too since Disc G is a union of strata by assumption. We prove here a similar existence result, as follows: Theorem Let G : (R m,0) (R p,0) be an analytic map germ. If G satisfies condition (5.1), then G has a singular Milnor tube fibration (5.2).

113 5.2. Relation with Thom regularity condition 111 Proof. Let us fix a regular stratification (W,S), cf Definition The condition (5.1) implies the existence of ε 0 > 0 such that, for any 0 < ε < ε 0, there exists η, 0 < η ε, such that every restriction map G : W α B m ε G 1 (B p η \ {0}) S β B p η \ {0} (5.3) is a submersion on a manifold with boundary. Indeed, since the sphere Sε m 1 is transversal to all the finitely many strata of the Whitney stratification W at 0 R n, it follows that the intersection Sε m 1 W is a Whitney stratification W S,ε which refines W. By condition (5.1), the map G is not only transversal to the stratification W but also to the stratification W S,ε, for any 0 < ε < ε 0. It then follows that the map (5.3) is a stratified submersion and it is proper, thus it is a stratified fibration by Thom-Mather Isotopy Theorem. Moreover, condition (5.1) tells that this fibration is independent of ε and η up to stratified homeomorphisms. From the above proof one immediately derives: Corollary Under the hypotheses of Theorem 5.1.4, the map G has a Milnor-Hamm fibration over B p η \ Disc G, with smooth Milnor fiber. Moreover, there exist locally trivial fibrations over each stratum S β Disc G. Example As we have seen in Example 5.1.2, G(x,y,z) = (xy,z 2 ) satisfies the condition (5.1) and from Theorem G has singular Milnor tube fibration. In this case, one has three different types of fibers: empty, hyperbolas and cross-cap (x y = 0). 5.2 Relation with Thom regularity condition In the article [CGS], the authors considered a real analytic map germ G : (U,0) (R p,0), m > p 2, with U R m open set, and with a critical point at 0 such that V G has dimension 2. They considered a fixed closed ball B m ε as a stratified set, with strata the interior B m ε and the boundary Sε m 1. With these notations they used the Thom-Mather isotopy Theorem to get that the restriction map G : B m ε G 1 (R p \ Disc G) R p \ Disc G

114 112 Chapter 5. Singular Milnor tube fibrations is a locally trivial fibration. See [CGS, Proposition 2.1]. As a consequence, they obtained the following locally trivial fibrations (see [CGS, Corollary 2.2]): G : B m ε G 1 (Bη p \ Disc G) Bη p \ Disc G. (5.4) In order to ensure that this above fibration does not depend on the particular choice of ε they considered Whitney stratifications W and S, respectively, of U and G(U), such that (W,S,G) is a Thom mapping. See [CGS, Proposition 2.4] for details. Moreover, they presented in [CGS, section 3.3 p.16] a class of examples satisfying this Thom regularity condition and with positive dimensional discriminant set. As we have seen, the Thom regularity at V G is enough to insure the condition (5.1) and thus one may derive the following statement from Theorem Corollary Let G : (R m,0) (R p,0) be an analytic map germ. If G is Thom regular at V G, then G has a singular Milnor tube fibration (5.2). Note that we do not ask that G be a Thom mapping in a strict sense. We only ask that the Thom regularity condition holds along the strata outside the zero locus of G containing strata of V G on its closure, as a germ of set at origin. Moreover, since the Thom regularity condition obviously implies a Thom regularity at V G, then it also follows from Corollary and Corollary that if G is a Thom mapping, then the Milnor-Hamm fibration exists. Consequently, [CGS, Corollary 2.2] is a particular case of our results Mixed f ḡ: A second extension for non-isolated critical value In this section we will give a second extension of the results of [PT] for the class of f ḡ functions in the case where the discriminant is larger that {0}. Let us remark that this extension provides classes of f ḡ functions which are Thom regular at V f ḡ, in the sense of Definition , and thus have singular Milnor tube fibration. Hence we can state our main result in this section as follows. Theorem If the map ( f,g) is Thom regular at V ( f,g), then f ḡ is Thom regular at V f ḡ. In particular, if ( f,g) is an ICIS then f ḡ is Thom regular at V f ḡ.

115 5.2. Relation with Thom regularity condition 113 For that we will prove the Lemma 5.2.3, which is a refinement of the Theorem of [PT]. The difference now is that we will consider also the more general case where the Disc ( f,g) contains others lines different from the coordinate axes. As one can see, in such a case it follows from Theorem that the mixed function germ f ḡ : (C n,0) (C,0) has no isolated critical value, i.e., {0} Disc f ḡ as a germ of set. Consequently, if ( f,g) is Thom regular at V ( f,g) one can combine Theorem and Theorem to get new classes of real analytic map germs of type f ḡ with singular Milnor tube fibration. Lemma Let f,g : (C n,0) (C,0), n > 1, be holomorphic function germs. Then NT f ḡ NT ( f,g), where NT f ḡ and NT ( f,g) are the Thom irregularity locus of f ḡ and ( f,g), respectively. Proof. If we assume that the discriminant of f ḡ is the origin only, then the proof follows from [PT, Theorem 3.1 p.6]. Hence, one may skip to the case where this discriminant is a finite union of real half-lines, i.e., Disc f ḡ {0}. In this case, we have that Sing f ḡ \V f ḡ /0. Hence, we can consider the decomposition V f ḡ = { V f ḡ \ ( )} Sing f ḡ V { } f ḡ Sing f ḡ V f ḡ, where the symbol denote the disjoint union. By definition, NT f ḡ V f ḡ. Thus, we can write NT f ḡ = NTf 1 ḡ NT f 2 ḡ where NTf 1 ḡ = [ V f ḡ \ ( )] Sing f ḡ V f ḡ NTf ḡ and NTf 2 ḡ = Sing f ḡ V f ḡ NT f ḡ. We claim that NTf 1 ḡ = /0. Indeed, let p 0 V f ḡ \ ( ) Sing f ḡ V f ḡ be any point and N p0 a neighborhood of p 0 such that N p0 Sing f ḡ = /0. Since we can always choose N p0 such that N p0 V f ḡ is a manifold and all fibers into N p0 are regular, then the Thom regularity condition is always satisfied at p 0. Thus, [ V f ḡ \ ( )] Sing f ḡ V f ḡ NTf ḡ = /0. Consequently, on has that NT f ḡ = NTf 2 ḡ. Step 1: NT f ḡ { f = g = 0}. One can use the same argument of the Step 1 in the proof of Theorem in [PT] since this step is not sensitive to increased generality.

116 114 Chapter 5. Singular Milnor tube fibrations In [PT] it was proved that Sing f ḡ V f ḡ = Sing f Singg { f = g = 0}. Hence, we claim that the points in the sets Sing f \{ f = g = 0} and Singg\{ f = g = 0} satisfies the Thom (a f ḡ )-condition. Indeed, let p 0 Sing f \{ f = g = 0}. Since f is a holomorphic function, one can find a small enough δ > 0 such that for any points in the ball B δ (p 0 ) the Łojasiewicz inequality holds: c f θ d f for some θ (0,1) and some c > 0. Moreover, since p 0 Sing f \ { f = g = 0} then p 0 V f and g(p 0 ) 0, and hence g must be a submersion on p 0. In addition, in this neighborhood the analytic functions g and dg are bounded and g > δ 1 for some small enough real positive δ 1. The fibers of f ḡ at a point z C n are spanned by the vectors n µ = µ gd f (z) + µ f dg(z) (5.5) with µ C and µ = 1. Consider any sequence of points p n away from V f ḡ such that p n p 0. On the sequence, after dividing (5.5) by f (p n ) θ, the component µ n g(p n )d f (p n ) f (p n ) θ is bounded away from zero: indeed, one has µ n g(p n )d f (p n ) f (p n ) θ = g(p n) d f (p n ) f (p n ) θ δ 1.c > 0. On the other hand, since p 0 Sing f \ { f = g = 0} one has f (p n ) f (p 0 ) = 0 and dg(p n ) is bounded, thus the component µ f (p n )dg(p n ) f (p n ) θ = f (p n) 1 θ dg(p n ) 0 n µ because 1 θ > 0. Consequently, lim p n p 0 f θ = lim µg d f p n p 0 f θ (p n), which has the direction of lim pn p 0 d f (p n ), for any µ with µ = 1. Therefore, one get the same limit as that for the fibers of holomorphic functions f along the sequence of points p n p 0 and by Hironaka s result its zero locus V f admits a Thom stratification with complex strata.

117 5.2. Relation with Thom regularity condition 115 Now, one can use the same idea after switching f by g. Consequently, one has that NT f ḡ { f = g = 0}. Step 2: NT f ḡ { f = g = 0} NT ( f,g). The increase in generality affects exactly this part of the proof of [PT, Theorem 4.4.3], here one need to consider another case as we shall see, since one may have Whitney strata in the source of f ḡ produced by its singular locus. We claim that on W := { f = g = 0} \ NT ( f,g) one has Thom regularity for f ḡ. In fact, by definition of NT ( f,g), on W one gets a Whitney stratification W = iw i of W such that W i is a ( f,g) -Thom regular for each i. Take p 0 W i fixed and an arbitrary sequence p n p 0 such that p n / V f ḡ. Since Sing f ḡ \V f ḡ /0, we need to consider two cases : Case 1: For n 1, p n / Sing f ḡ. In this case, the fibers F f ḡ (n) := ( f ḡ) 1 ( f ḡ(p n )) are locally smooth manifolds, thus we may consider the tangent space T pn F f ḡ (n). We claim that ( f ḡ) 1 ( f ḡ(z)) = (c,d) A f ḡ(z) ( f,g) 1 (c,d), where A f ḡ(z) := Im( f,g) (u v) 1 ( f ḡ(z)) for each z C n. Indeed, let x ( f ḡ) 1 ( f ḡ(z)). Define (a,b) := ( f (x),g(x)). One has that (a,b) (u v) 1 ( f ḡ(z)) Im( f,g) = A f ḡ(z). Thus, x ( f,g) 1 (a,b) (c,d) A f ḡ(z) ( f,g) 1 (c,d). Reciprocally, if x (c,d) A f ḡ(z) ( f,g) 1 (c,d), then there exists (c,d) A f ḡ(z) such that x ( f,g) 1 (c,d), which implies ( f (x),g(x)) = (c,d) and f (x)g(x) = f ḡ(z). Thus, x ( f ḡ) 1 ( f ḡ(z)). Therefore, for each n 1, there exists (c n,d n ) A f ḡ(pn ) such that p n ( f,g) 1 (c n,d n ) ( f ḡ) 1 ( f ḡ(p n )). (5.6) Consider now the following subcases: Sub-case 1: p n / Sing ( f,g). In this case, the fibers F ( f,g) (n) := ( f,g) 1 (c n,d n ) are locally smooth manifolds, thus one may consider the tangent space T pn F ( f,g) (n) := T pn ( f,g) 1 (c n,d n ). Since W i is a

118 116 Chapter 5. Singular Milnor tube fibrations Thom regular stratum for ( f,g), by (5.6) one has that Sub-case 2: p n Sing ( f,g). lim T pn F f ḡ (n) lim T pn F ( f,g) (n) T p0 W i. p n p 0 p n p 0 Since the fibers F ( f,g) (n) p n are singular, consider a stratification and take a stratum S ( f,g) (n) F ( f,g) (n) which contains p n. By (5.6) one has T pn S ( f,g) (n) T pn F f ḡ (n). Consequently, lim p n p 0 T pn F f ḡ (n) lim p n p 0 T pn S ( f,g) (n) T p0 W i. This shows that in the Case 1 one has that W i is a Thom stratum for map f ḡ. Case 2: For n 1, p n Sing f ḡ. As p n / V f ḡ, by Lemma 2.3 (a) in [PT] we get that p n Sing( f,g) \ V f ḡ. Since by assumption the pair ( f,g) is Thom regular at V ( f,g), one can consider a Whitney stratification S = {S i } of the singular locus of the map ( f,g), which is induced by the Whitney stratification of W in the ambient space satisfying the Thom a ( f,g) -condition. We claim that S also satisfies the Thom a f ḡ -condition at V f ḡ. Indeed, this follows from [PT, Lemma 2.1] applied to a point of some positive dimensional smooth stratum S S. Because, given S n a positive dimensional stratum of S such that p n S n, by (5.6) we have that F ( f,g) (n) S n F f ḡ (n) S n. Consequently, lim T pn F f ḡ (n) S n lim T pn F ( f,g) (n) S n T p0 W i. p n p 0 p n p 0 Therefore, one has that W i is a Thom stratum for map f ḡ also in Case 2. Thus, NT f ḡ NT ( f,g). Now we will prove Theorem The proof follows the same idea of the proof of [PT, Corollary 4.1, p.8], but we repeat here for completeness. Proof of Theorem If ( f,g) is Thom regular at V ( f,g), then by definition one has that NT ( f,g) = /0. Therefore, the first assertion follows from Lemma Now, by definition one has that NT ( f,g) = { f = g = 0} Sing ( f,g). Therefore, from Lemma 5.2.3, one gets the following inclusion: NT f ḡ { f = g = 0} Sing ( f,g).

119 5.2. Relation with Thom regularity condition 117 Since ( f,g) is ICIS, one has that { f = g = 0} Sing ( f,g) {0}, thus NT f ḡ = /0, which proves the second statement. Example Let f,g : C 2 C given by f (x,y) = xy + x 2 and g(x,y) = y 2. One has V ( f,g) = {(0,0)} and Sing ( f,g) = {y = 0} {y = 2x}, thus ( f,g) is obviously Thom regular at V ( f,g). However Disc ( f,g) = {(x 2,0) x C} {( x 2,4x 2 ) x C} and therefore f ḡ has non-isolated critical value. It then follows from Theorem 5.2.2, that f ḡ is Thom regular at V f ḡ hence it has a Milnor-Hamm fibration, and also a singular Milnor tube fibration by Corollary The singular Milnor tube fibration may exist without any Thom regularity condition along the zero locus of the map, as shown by the next example: Example Let us consider again the real map germ G from Example 4.5.5, where it was shown that G is not -Thom regular at V G, hence is not Thom regular at V G. Let (W,S) be the following stratification of the source and of the target space of G: W 1 := B 4 ε \G 1 (Disc G), W 2 := {(0,0,z,w) R 4 w > 0}, W 3 := {(0,0,z,w) R 4 w < 0}, W 4 := {(0,0,z,0) R 4 z > 0}, W 5 := {(0,0,z,0) R 4 z < 0}, W 6 := {(0,0,0,0)}, and S 1 := B 3 η \ Disc (G), S 2 := {((0,0,γ) R 3 γ > 0)}, S 3 := {((0,0,γ) R 3 γ < 0)}, S 4 = S 5 = S 6 := {(0,0,0)}. The restriction maps are G j := G Wj : W j S j, j = 1,...,6. As we have seen G 1 satisfies condition (4.2). The Milnor sets of the remaining maps are M(G 2 ) = M(G 3 ) = the Ow-axis, and M(G i )\V G = /0, for i = 4,5,6. Therefore, the condition (5.1) holds and thus G has a singular Milnor tube fibration by Theorem

120

121 119 CHAPTER 6 SPHERE FIBRATIONS In this chapter we will present our main result concerning the existence of spheres fibrations in the more general settings, which we call the Milnor-Hamm sphere fibration. For that, we introduce the definition of the Milnor-Hamm sphere fibration and sufficient conditions to ensure its existence. This result is an extension of a main result of [ACT1], namely, Theorem Consequently, it extends all previous results in this setting. We also explore some criteria of the ρ-regularity which also permit us to control the projection of the Milnor-Hamm sphere fibrations. These results will be important in the development of Chapter 7. We finish this chapter by applying our main criteria concerning ρ-regularity in the classical case. 6.1 Existence of Milnor-Hamm Sphere Fibration In this section we will present a result about the existence of fibration on sphere that we will call Milnor-Hamm sphere fibration. This result can be seen as a natural extension of Theorem stated in Chapter 2, but now under the more general condition (4.2) that we remind below: M(G) \ G 1 (Disc G) V G {0}. Definition Let G : (R m,0) (R p,0) a smooth map germ. We say that its discriminant locus, Disc G is radial, if as a germ of set it is the origin, or a union of real half-lines

122 120 Chapter 6. Sphere Fibrations through the origin. Example Let f,g : (C n,0) (C,0) be holomorphic functions. Then the Theorem says that the discriminant set Disc f ḡ of the mixed function f ḡ : (C n,0) (C,0) is either the origin, or a union of finitely many real half-lines through the origin. Therefore, it is radial. Let G : (R m,0) (R p,0) be an analytic map germ and G : U R p a representative, with 0 U. As before, we will consider the map Ψ : U \V G S p 1. Lemma Let G : (R m,0) (R p,0) be an analytic map germ. If Disc G is radial, then the restriction map is well defined. Ψ : S m 1 ε \ G 1 (Disc G) S p 1 \ Disc G (6.1) Proof. Let x Sε m 1 \ G 1 (Disc G). Since G has radial discriminant locus, if Ψ (x) Disc G, hence G(x) = G(x) Ψ(x) Disc G, which is a contradiction. Therefore, the map (6.1) is well defined. Definition Let G : (R m,0) (R p,0) be an analytic map germ. If the restriction map (6.1) is a locally trivial smooth fibration for any small enough ε > 0, we will call it the Milnor-Hamm sphere fibration. Now, consider the set M(Ψ) := {x U \V G ρ x Ψ} of the ρ-nonregular points of Ψ. As in the case where Disc G = {0}, we can consider the following definition. Definition We will say that an analytic map germ G is ρ-regular when the condition M(Ψ) \ G 1 (Disc (G)) = /0; (6.2) is satisfied, as a germ of set at the origin. With these definitions an extension of Theorem becomes:

123 6.1. Existence of Milnor-Hamm Sphere Fibration 121 Theorem Let G : (R m,0) (R p,0), m > p 2, be an analytic map germ with radial discriminant set and satisfying the condition (4.2). If G is ρ-regular, then the restriction (6.1) is a Milnor-Hamm sphere fibration. Proof. The proof works in the same way as in Theorem , up to adaptations. Hence, one can assume that {0} Disc G and consider the following steps below. Step 1: Under the condition (4.2) one has that the restriction G : S m 1 ε G 1 (B p η \ Disc G) B p η \ Disc G (6.3) is a locally trivial smooth fibration (over its image). Since G has a radial discriminant set one conclude that π(sη p 1 Disc G) = S p 1 Disc G, for any η > 0, where π := s/ s and hence π : B p η \ Disc G S p 1 \ Disc G (6.4) is a trivial smooth fibration. Now, one can compose the fibrations (6.3) and (6.4) to conclude that Ψ : S m 1 ε G 1 (B p η \ Disc G) S p 1 \ Disc G (6.5) is a locally trivial smooth fibration. In the same way, one get the locally trivial smooth fibration: Ψ : Sε m 1 G 1 (Sη p 1 \ Disc G) S p 1 \ Disc G. (6.6) Step 2: As we have seen in Section 2.2.4, the condition (6.2) is equivalent to the map Ψ : Sε m 1 \G 1 (Disc G) S p 1 \Disc G be a submersion (over its image). Consequently, the restriction Ψ 1 := Ψ : S m 1 ε \ {G 1 (Disc G) G 1 (B p η)} S p 1 \ Disc G (6.7) is a submersion. Moreover, it coincides with the fibration (6.6) in Sε m 1 G 1 (Sη p 1 \ Disc (G)). We claim that (6.7) is proper. Indeed, let C S p 1 \ Disc G be a compact set. Since Ψ 2 := Ψ : Sε m 1 \ G 1 (Bη) p S p 1 is a proper map, Ψ 1 2 (C) is a compact set. Moreover, the hypothesis of Disc G be radial ensure that Ψ 1 (C) G 1 (Disc G) = /0. Therefore, one can conclude that Ψ 1 1 (C) = Ψ 1 2 (C) which proof our statement. Thus, the restriction (6.7) is the projection of a locally trivial smooth fibration.

124 122 Chapter 6. Sphere Fibrations Finally, the fibrations (6.5) and (6.7) may be glued together along Sε m 1 G 1 (Sη p 1 \ Disc G) (see argument in [AT1, AT2, ACT1]) to induce the locally trivial smooth fibration Ψ : Sε m 1 \ G 1 (Disc G) S p 1 \ Disc G. Example Let G : (R 3,0) (R 2,0) given by G(x,y,z) = (xy,z 2 ), as in Examples and One has that G satisfies the condition (4.2) and Disc G = {(0,β) β 0} {(λ,0) λ R} is radial. Finally, by Theorem below, G is ρ-regular. From Theorem 6.1.6, G has Milnor-Hamm sphere fibration. Figure 11 Milnor-Hamm sphere fibration for G = (xy,z 2 ) Example Let f : (R m,0) (R p,0) be a real analytic map germ and let g : (R,0) (R,0) be a germ diffeomorphism, such that f and g have separable variables. Consider the pair of map germ G := ( f,g) : (R m R,0) (R p R,0). If Disc f = {0} and f satisfies the condition (2.14), then G satisfies the condition (4.2). Moreover, since Sing G = Sing f R one has Disc G = {0} R, i.e., it is radial. Therefore, if f is a Simple Ł-Map, for instance, it follows from item (ii) of Theorem that G has a Milnor-Hamm sphere fibration.

125 6.2. Criteria for ρ-regularity 123 Example Consider G : R n R 2 given by G(x 1,...,x n ) = (x 1,x x2 n 1 xn). 2 One has V G = {x 1 = 0} {x x2 n 1 x2 n = 0} and Sing G = {x 2 = = x n = 0}, thus, Disc G is radial. Since Sing G V G = {0} hence the condition (4.2) is satisfied and Im G is full. Moreover, one has that M(Ψ) = Ox 1 -axis, i.e., M(Ψ) = Sing G. Consequently, one has that M(Ψ) \ G 1 (Disc (G)) = /0 and G is ρ-regular. Note that in Example one has Sing G \V G {0}, hence the classical Milnor tube fibration is not well defined. See Corollary Moreover, as M(Ψ) /0, by [AT2, Theorem ] the projection Ψ : Sε m 1 \ K ε S p 1 is not a locally trivial smooth fibration. Therefore, the above example shows us that the classical sphere and tube fibrations are not well defined, but there exists the Milnor-Hamm tube and sphere fibrations. In the next section we develop a study for ρ-regularity of map germ. 6.2 Criteria for ρ-regularity For the sake of simplicity, given x B m ε \V G we will assume that x belongs to the open set {G 1 (x) 0}. We notice that all results do not depend on the particular choice of the open set. As we have seen in Chapter 2, one has Sing G M(G) and from Remark 2.2.9, M(Ψ) M(G) \V G. Moreover, we notice that, in the open set {G 1 (x) 0} one can express the condition x M(G) \V G by: ρ(x) = a(x) G(x) 2 + Σ p k=2 α k(x)ω k (x), (6.8) for k = 2,..., p, where Ω k = G 1 G k G k G 1 are the generators of the normal space T x Xy in R m, with T x X y the tangent space of the fiber X y = Ψ 1 (y), y = Ψ(x). Therefore, one can conclude that x M(Ψ) if, and only if a(x) = 0. The next section is devote to the study of the coefficient a(x) and criteria that allows us to find classes of map germ with ρ-regularity.

126 124 Chapter 6. Sphere Fibrations A Matrix Criterion G is ρ-regular. In this section, we will give a matricial criterion for deciding when a map germ Proposition Let G : (R m,0) (R p,0) be smooth map germ. For any x M(G) \ (V G Sing G) one has that: where a(x) = detd(x) detm(x) (6.9) D(x) = ρ(x), ( G(x) 2 ) Ω 2 (x), ( G(x) 2 ) Ω p (x), ( G(x) 2 ) ρ(x),ω 2 (x) Ω 2 (x),ω 2 (x) Ω p (x),ω 2 (x) ρ(x),ω p (x) Ω 2 (x),ω p (x) Ω p (x),ω p (x) and, M(x) = ( G(x) 2 ), ( G(x) 2 ) Ω 2 (x), ( G(x) 2 ) Ω p (x), ( G(x) 2 ) ( G(x) 2 ),Ω 2 (x) Ω 2 (x),ω 2 (x) Ω p (x),ω 2 (x) ( G(x) 2 ),Ω p (x) Ω k (x),ω p (x) Ω p (x),ω p (x). Proof. Let x M(G) \ (V G Sing G). From equation (6.8) one gets the matrix equation T (x) = M(x) L(x), where T (x) = ρ(x), ( G(x) 2 ) ρ(x),ω 2 (x)., ρ(x),ω p (x)

127 6.2. Criteria for ρ-regularity 125 M(x) = ( G(x) 2 ), ( G(x) 2 ) Ω 2 (x), ( G(x) 2 ) Ω p (x), ( G(x) 2 ) ( G(x) 2 ),Ω 2 (x) Ω 2 (x),ω 2 (x) Ω p (x),ω 2 (x) ( G(x) 2 ),Ω p (x) Ω k (x),ω p (x) Ω p (x),ω p (x), L(x) = a(x) α 2 (x). α k (x). The matrix M(x) is non-singular because x / (V G Sing G). Moreover by Lagrange s identity its determinant is given by the sum of the squares of all p p subdeterminants of the matrix ( G(x) 2 ) Ω 2 (x). Ω p (x). Hence, detm(x) > 0 and one can write L(x) = M(x) 1 T (x). By Cramer s rule the coefficient a(x) can be written as a(x) = detd(x) detm(x) Remark It follows form the previous proof that the rank of the matrix D(x) can be understood as the rank of two matrices A(x) and B(x) below considering D(x) = A(x) B(x):

128 126 Chapter 6. Sphere Fibrations A(x) = G 2 Ω 2. [ and B(x) = ρ Ω 2 Ω p ]m p. Ω p p m Hence, the rank of the matrix D(x) is maximal if, and only if the ranks of the matrices A(x) and B(x) are maximal. Nevertheless, under the condition x / (V G Sing G) the rank of D(x) is maximal if, and only if the rank of B(x) is maximal, and this amounts to saying that x / M(Ψ). The result below shows that the Milnor tube expands in the radial direction. Hence, in the case where the Disc G = {0}, in order to inflate the smooth Milnor tube to the sphere, as was made in Section 2.2.2, one can replace the bisector vector field in equation (2.8) by the direction G(x) 2. Proposition Let G = (G 1,...,G p ) : (R m,0) (R p,0) be a smooth map germ, with m > p 2. Then for any x B m ε \ (V G Sing G) one has that G(x) 2, ρ(x) > 0. Proof. We will prove the case p = 2. The general case is proved using the same idea. One can write G(x) 2,x = 2 ( m(g 1 (x)) 2 + n(g 2 (x)) 2 + G 1 (x) ) jg 1 m+ j(x) + G 2 (x) jg 2 n+ j(x), j=1 j=1 where G i k is the Taylor homogeneous expansion term of G i of degree k, for i = 1,2. Since x / (V G Sing G), one has that G(x) 2 0. Moreover, since x / V G, one obtain that m(g 1 (x)) 2 + n(g 2 (x)) 2 > 0. Therefore, for small enough ε > 0 one can conclude that G(x) 2,x > 0 and the result follows. As an application of the matricial criterion stated in Proposition one has: Corollary Let G : (R m,0) (R p,0), m > p 2 be an analytic map germ. Suppose that for any x M(G) \ (V G Sing G) one has either G(x) 2,Ω j (x) = 0, or ρ(x),ωj (x) = 0, for 2 j p. Then the coefficient a(x) is positive and G is ρ- regular.

129 6.2. Criteria for ρ-regularity 127 Proof. In fact, assume that G(x) 2,Ω j (x) = 0 for 2 j p. It follows from Proposition that detd(x) = ρ(x), ( G(x) 2 ) det ( ) Ω i (x),ω j (x) i, j > 0. Therefore, M(Ψ) \ G 1 (Disc G) = /0, i.e., G is ρ-regular. The case where ρ,ω j (x) = 0, for 2 j p follows by using the same argument Other criteria to ρ-regularity One can write Let G : (R m,0) (R p,0) be a analytic map germ with G(x) = (G 1 (x),...,g p (x)). G 1 (x) = G 1 m 1 (x) + G 1 m 1 +1 (x) +. (6.10) G p (x) = Gm p p (x) + G p m p +1 (x) + where G i m j is the homogeneous term of degree m j, for i, j = 1,..., p. We say that G has same multiplicity if m 1 = m 2 = = m p. Our main result is the following. Theorem Let G := (G 1,...,G p ) : (R m,0) (R p,0) be an analytic map germ. If G satisfies some of conditions below: (i) G has same multiplicity. (ii) G i (x), G j (x) = 0, for any i, j = 1,..., p with i j and x M(G) \ (V G Sing G). Then for any x M(G) \ (V G Sing G) the coefficient a(x) is positive and M(Ψ) Sing G \V G. In particular, M(Ψ) \ G 1 (Disc G) = /0, i.e., G is ρ-regular. Two immediate consequences of this result is the following. Corollary If G is a Simple Ł- Map, then for any x M(G)\V G the coefficient a(x) is positive and G is ρ-regular. Proof. By definition of Simple Ł-Map, one get G i (x), G j (x) = 0, for any i, j = 1,..., p with i j. The result follows from item (ii) of Theorem

130 128 Chapter 6. Sphere Fibrations Corollary Let f : C n C be a mixed functions. If for any z M( f )\(V f Sing f ), one has Im d f (z), d f (z) = 0 then the coefficient a(z) is positive and f is ρ-regular. C Proof. Indeed, as we have seen, u, v = 2Im d f, d f. The result follows from C item (ii) of Theorem results. For the proof of Theorem 6.2.5, one need of the matrix criterion and the next two Lemma Let G(x) = (G 1 (x),...,g p (x)), be an analytic map germ. For x M(G)\ (V G Sing G) one has a(x) = α(x),g(x) G(x) 2, where α(x) = (α 1 (x),...,α p (x)) R p is such that ρ(x) = Σ p k=1 α k(x) G k (x). In particular, x M(Ψ) \ (V G Sing G) if, and only if α(x),g(x) = 0. Proof. It follows from Proposition that, for any x M(G) \ (V G Sing G) the coefficient where G(x) 2 Ω A(x) = 2 (x). Ω p (x) p m a(x) = det[a(x).b(x)] detm(x) [ ] and B(x) = ρ(x) Ω 2 (x) Ω p (x) m p and M(x) = A(x).A(x) t. Since x / V G Sing G, one has from Remark that A(x) has maximal rank and, consequently, M(x) as well. On the open set {G 1 (x) 0} one can rewrite, up to multiplication of the first line of A(x) by 2:

131 6.2. Criteria for ρ-regularity 129 A(x) = G 1 (x) G 2 (x) G p (x) G 2 (x) G 1 (x) G p (x) 0 G 1 (x) G 1 (x) G 2 (x). G p (x) B(x) t = α 1 (x) α 2 (x) α p (x) G 2 (x) G 1 (x) G p (x) 0 G 1 (x) G 1 (x) G 2 (x). G p (x). Therefore, one has that A(x) B(x) = L 1 (x) L 2 (x) L 3 (x) and M(x) = L 1 (x) L 2 (x) L 1 (x) t, where L 1 (x) = G 1 (x) G 2 (x) G p (x) G 2 (x) G 1 (x) G p (x) 0 G 1 (x), G 1 (x) G L 2 (x) = 2 (x) [. G p (x) the Grassmann matrix of JG(x), and G 1 (x) G 2 (x) G p (x) ] L 3 (x) = α 1 (x) G 2 (x) G p (x) α 2 (x) G 1 (x) α p (x) 0 G 1 (x). Thus, one has that a(x) = detl 3(x) detl 1 (x).

132 130 Chapter 6. Sphere Fibrations On the other hand, on the open set {G 1 0} one has that detl 1 (x) = (G 1 (x)) p 2 G(x) 2. Analogously, applying Laplace s rule in the first column detl 3 (x) = (G 1 (x)) p 2 Σ p k=1 α k(x)g k (x). Therefore, a(x) = Σp k=1 α k(x)g k (x) G(x) 2 = α(x),g(x) G(x) 2. Lemma Let G(x) = (G 1 (x),...,g p (x)), be an analytic map germ. In the set M(G) \ (V G Sing G), one has that α(x),g(x) = (G(x) [D(m i ) A(x)],G(x), (6.11) where, A(x) = [JG(x) JG(x) t ] 1 and D(m i ) = m m m p. Proof. Since x M(G) \ (V G Sing G), there exist α 1 (x),...,α p (x) R such that ρ(x) = Σ p j=1 α j(x) G j (x). Let us define α(x) := (α 1 (x),...,α p (x)). After changing α(x) by α(x), one can consider 2 the matrix equation [ ] [x] = α 1 (x) α p (x) G 1 (x) G 1 (x) x 1 x m..... G p (x) G p (x) x 1 x m = [α(x)] [JG(x)]. Now, after multiplying both side of the matrix equation above by [JG(x) t ] one gets: [x] [JG(x) t ] = [α(x)] [JG(x)] [JG(x) t ]. (6.12)

133 6.2. Criteria for ρ-regularity 131 The p p square matrix [JG(x)] [JG(x) t ] = [JG(x) JG(x) t ] is invertible under the condition x / Sing (G) V G (because it is the Gram matrix [JG(x) JG(x) t ] = ( G i (x), G j (x) ) i, j with i, j = 1,..., p). After multiplying both side of equality (6.12) by its inverse A(x) = [JG(x) JG(x) t ] 1, one has the following equation [α(x)] = [x] [JG(x) t ] A(x). Now, one can write the scalar product (dot product) as α(x),g(x) = [x] [JG(x) t ] A(x),G(x). (6.13) It follows from equations in (6.10) that: Then by Euler s identity one finds G 1 (x) = G 1 m 1 (x) + G 1 m 1 +1(x) +. G p (x) = Gm p p (x) + G p m p +1 (x) +. G i k (x),x = kg i k (x). Hence, G i (x),x = and one can decompose G i mi + j(x),x = j=0 j=0 (m i + j)g i m i + j(x) = m i G i (x) + ( G 1 (x),x,..., G p (x),x ) = (m 1 G 1 (x),...,m p G p (x))+ ( j=1 jg i m i + j(x) j=1 jg 1 m 1 + j(x),..., jg p m p + j ). (x) j=1 (6.14) Denote the vector expression ( j=1 jg1 m 1 + j (x),..., j=1 jgp m p + j (x)) by V (x), and denote V (x) A(x), G(x) by H.O.T. From equation (6.13) one gets α(x),g(x) = (m 1 G 1 (x),...,m p G p (x)) A(x),G(x) + H.O.T.

134 132 Chapter 6. Sphere Fibrations One can now write the vector (m 1 G 1 (x),...,m p G p (x)) as a matrix product G(x) D(m i ) where D(m i ) is a diagonal matrix with entries m i, 1 i p. Hence, α(x),g(x) = (G(x).[D(m i ) A(x)],G(x) + H.O.T. Proof of Theorem item (i) Since G has same multiplicity, namely m = m 1 = = m p > 0 the matrix D(m i ) = m.i p p where the matrix I p p is the p p identity matrix. Thus, α(x),g(x) = m (G(x) [A(x)],G(x) + H.O.T. Now, since the symmetric matrix [JG(x) JG(x) t ] is a Gram matrix and { G 1 (x),..., G p (x)} are linearly independent, one has that [JG(x) JG(x) t ] is positive definite (see Theorem p. 407 in [HJ]), consequently, the inverse matrix A(x) is positive definite (see p.558 in [Me]) and m (G(x) A(x),G(x) > 0. Therefore, for any x M(G) \ (V G Sing G) close enough to the origin the scalar product α(x),g(x) must be positive. item (ii) Since G i (x), G j (x) = 0, for any i, j = 1,..., p with i j, one has that the positive definite matrix A(x) 1 = [JG(x) JG(x) t ] is diagonal. Consequently, the matrix D(m i ) A(x) must be positive definite and the result follows. Example Any pair of map germ G = ( f,g) with f and g functions in separables variables, is ρ-regular. In fact, since it has separable variables one has f (x), g(y) = 0. In addition, if we assume that G satisfies the condition (4.2), Im G is full and Disc G is radial, then it has a Milnor-Hamm sphere fibration Some examples in the classical case: Disc G = {0} In this section we will present three examples, which shows that the previous criteria are also useful for the classical case. Example Consider the functions f : C C and g : C C given by f (y) = y k, k 1, and g(x) = x x and define F := f g : C 2 C. Since F is a mixed function which is polar weighted homogeneous, one has that F satisfies the condition (2.14). Hence, F has tube and sphere fibrations.

135 6.2. Criteria for ρ-regularity 133 Example Let G(x,y,z) = (xy,xz) as in Example As we have seen Disc G = {0} and G satisfies the condition (2.14). On the other hand, one has that G 2 = (xy 2 + xz 2,x 2 y,x 2 z) and Ω 2 = (0, x 2 z,x 2 y). Therefore, ρ, Ω j = 0 and G 2, Ω j = 0. Consequently, for any x M(G) \V G, the coefficient a(x) is positive and M(Ψ) is empty i.e., G is ρ-regular. Thus, G has the tube and the sphere fibrations in the classical sense. Example Consider the real map G := (G 1,G 2,G 3,G 4 ) : (R 6,0) (R 4,0) given by G 1 (x,y,z,w,a,b) = x 2 z + y 2 z G 2 (x,y,z,w,a,b) = wx 2 + wy 2 G 3 (x,y,z,w,a,b) = ax 2 + ay 2 G 4 (x,y,z,w,a,b) = bx 2 + by 2. One has that V G = {x 2 + y 2 = 0} {a 2 + b 2 + w 2 + z 2 = 0} Sing G = {x 2 + y 2 = 0} M(G) = {x 2 + y 2 = 0} {2a 2 + 2b 2 + 2w 2 x 2 y 2 + 2z 2 = 0}. Therefore, Disc G = {0} and G satisfies the condition (2.14). Now, one has that G 2 = (2(x 2 z + y 2 z))xz + (2(wx 2 + wy 2 ))wx + (2(ax 2 + ay 2 ))xa + (2(bx 2 + by 2 ))xb e 1 +(2(x 2 z + y 2 z))yz + (2(wx 2 + wy 2 ))wy + (2(ax 2 + ay 2 ))ya + (2(bx 2 + by 2 ))yb e 2 +(x 2 z + y 2 z)(x 2 + y 2 ) e 3 +(wx 2 + wy 2 )(x 2 + y 2 ) e 4 +(ax 2 + ay 2 )(x 2 + y 2 ) e 5 +(bx 2 + by 2 )(x 2 + y 2 ) e 6 and, Ω 2 = (2(x 2 z + y 2 z))wx (2(wx 2 + wy 2 ))xz e 1 +(2(x 2 z + y 2 z))wy (2(wx 2 + wy 2 ))yz e 2 (wx 2 + wy 2 )(x 2 + y 2 ) e 3 +(x 2 z + y 2 z)(x 2 + y 2 ) e 4 +0 e 5 +0 e 6

136 134 Chapter 6. Sphere Fibrations Ω 3 = (2(x 2 z + y 2 z))xa (2(ax 2 + ay 2 ))xz e 1 +(2(x 2 z + y 2 z))ya (2(ax 2 + ay 2 ))yz e 2 (ax 2 + ay 2 )(x 2 + y 2 ) e 3 +0 e 4 +(x 2 z + y 2 z)(x 2 + y 2 ) e 5 and +0 e 6 Ω 4 = (2(x 2 z + y 2 z))xb (2(bx 2 + by 2 ))xz e 1 +(2(x 2 z + y 2 z))yb (2(bx 2 + by 2 ))yz e 2 (bx 2 + by 2 )(x 2 + y 2 ) e 3 +0 e 4 +0 e 5 Therefore, +(x 2 z + y 2 z)(x 2 + y 2 ) e 6 ρ,ωj = 0 for any j = 2,3,4. Also G 2,Ω j = 0 for j = 2,3,4. Consequently, the coefficient a(x) is positive and one has that G is ρ- regular.

137 135 CHAPTER 7 GOOD VECTOR FIELDS Let G : (R m,0) (R p,0), m > p 2 be an analytic map germ. In general, as has been seen, if the discriminant set Disc G has positive dimension it intersects all spheres of small enough radius in the target space. However, one can use the condition (4.2) to assure that the restriction map G : B m ε G 1 (Sη p 1 \ Disc G) Sη p 1 \ Disc G (7.1) is also the projection of a locally trivial smooth fibration. If Disc G is radial, the fibration (7.1) may be composed with the canonical projection π := s/ s : R p \ {0} S p 1 to get a locally trivial smooth fibration Ψ : B m ε G 1 (S p 1 η \ Disc G) S p 1 \ Disc G. (7.2) Moreover, under additional hypothesis of ρ-regularity, the Theorem ensure that the restriction Ψ : S m 1 ε \ G 1 (Disc G) S p 1 \ Disc G (7.3) is a Milnor-Hamm sphere fibration. Therefore, the following question becomes natural: When are these two fibrations equivalent? In the classical case where Disc G = {0} this problem has been approached by several authors in the last years, see for example [A2, AT1, AT2, Oka2, Han, CSS2].

138 136 Chapter 7. Good vector fields Up to now, there is no clear proof for this problem in its complete generality, and no counterexample has been found yet. The following conjecture explains what we mean by complete generality : Conjecture Let G : (R m,0) (R p,0), m > p 2 be an analytic map germ with Disc G = {0} and M(Ψ) = /0. If both fibrations Ψ : B m ε G 1 (S p 1 η ) S p 1 (7.4) and Ψ : S m 1 ε \ K ε S p 1, (7.5) exist, for any small enough ε > 0 and 0 < η ε, then they are equivalent. Next result is a consequence of Milnor s work, see [Mi] and Section for more details. Theorem [Mi] Let G : (R m,0) (R p,0), m > p 2 be an analytic map germ with Disc G = {0}. Assume that both fibration (7.4) and (7.5) exist. If there exists a vector field ν defined on B m ε \V G satisfying the following three conditions: (c 1 ) ν is tangent to the fibers X y, with y = Ψ(x), (c 2 ) ν(x), ρ(x) > 0, (c 3 ) ν(x), G(x) 2 > 0, then the Milnor tube can be inflated to the sphere along the diffeomorphism given by the flow of ν, preserving the projection Ψ. Theorem shows us that these two fibrations are equivalent. In the literature, the proofs are based on the existence of such a vector field. See Definition On the other hand, one may consider the following problem: Problem 1. Does the equivalence between the Milnor Fibrations imply the existence of a good vector field?

139 137 Figure 12 Milnor tube blowing to sphere In this chapter we will concentrate to the problem of the existence of a good vector field for G for the case Disc G = {0}. As can be seen in [Han, Lemma 3.4.1] this problem reduces to the case where M(G) \V G is transversal to the fibers of Ψ, i.e. the case where the set M(G)\V G turns around the origin. Still in [Han] the author introduced some interesting conditions based on tools from general topology [Han, Lemma 3.4.2] and Singularity Theory, [Han, Lemma and Lemma 3.4.3] to show a partial answer

140 138 Chapter 7. Good vector fields to this problem. The approach given by the authors in [CSS2] to the equivalence problem will be discussed in subsection We will present a characterization for the existence of a good vector field using the coefficient a(x) introduced in Chapter 6. We present some sufficient conditions in order to ensure the existence of such a vector field using some results of the previous chapters, as well as tools from general topology and differential topology. The main results obtained here will be applied in the next chapter, in order to show the equivalence between the Milnor-Hamm fibrations for several class of maps. 7.1 Existence We will assume, without lost of generality, that x belongs to the open set {G 1 (x) 0}. Moreover, given a vector field ω on B m ε \V G denote by proj Tx X y (ω(x)) the orthogonal projection of ω(x) to T x X y, the tangent space of X y = Ψ 1 (y), with y = Ψ(x). As we have seen in Section 6.2, in the open set {G 1 (x) 0} the normal space T x Xy in R m is spanned by {Ω 2 (x),...,ω p (x)} where Ω k = G 1 G k G k G 1, for all k = 2,..., p. Definition Let G : (R m,0) (R p,0), m > p 2 be an analytic map germ. We will call a vector field ν defined on B m ε \V G satisfying the conditions (c 1 ) (c 3 ) of Theorem a good vector field for G, or g.v.f. for short. Now consider on B m ε \V G the following vector fields: v 1 (x) := proj Tx X y ( G(x) 2 ), and v 2 (x) := proj Tx X y ( ρ(x)). (7.6) We notice that v 1 (x) never vanishes on B m ε \V G because Disc G = {0}. Moreover, the second vector field v 2 (x) has no zeros on B m ε \V G if, and only if M(Ψ) = /0, i.e., G is ρ-regular. Therefore, if we assume the additional hypothesis M(Ψ) = /0, then one can define on B m ε \V G the vector field ν(x) = v 1(x) v 1 (x) + v 2(x) v 2 (x). (7.7) Lemma For x B ε \V G, the vectors v 1 (x) and v 2 (x) are linearly independent if, and only if x M(G) V G.

141 7.1. Existence 139 Proof. For each x B ε \V G let us consider the orthogonal decomposition sum T x X y = T x G 1 (G(x)) R(η x ), where η x T x G 1 (G(x)). One decomposes v 1 (x) = v 1 1 (x)+v2 1 (x) and v 2(x) = v 1 2 (x)+v2 2 (x), for v1 1 (x),v1 2 (x) T x G 1 (G(x)), and v 2 1 (x),v2 2 (x) R(η x). Note that, x M(G) V G if, and only if on this decomposition the first component of v 2 (x) is non-zero. Moreover, by definition, the first component of v 1 (x) on the decomposition above is always zero for all x B ε \V G. Figure 13 x M(G) V G. It means that, the vector field defined in (7.7) has no singularity on B m ε \ (M(G) V G ). Hence, if we want to extend it as a g.v.f. for G on B ε \V G we need to control it on the set M(G) \V G. Remark Since for all x B ε \V G the vector v 1 (x) η x and never vanishes, then we can consider v 1 (x) = v 2 1 (x) = η x. Moreover, as we have seen in Section 6.2, for any x M(G)\V G one can write ρ(x) = a(x) G(x) 2 + p j=2 b j(x)ω j (x). It follows from Lemma that v 1 (x) and v 2 (x) are linearly dependent, hence one has that v 2 (x) = ρ(x),v 1(x) v 1 (x) 2 v 1 (x).

142 140 Chapter 7. Good vector fields In particular, a(x) = ρ(x),v 1(x) v 1 (x) 2 Figure 14 x M(G) \V G, a(x) > 0. Figure 15 x M(G) \V G, a(x) < 0. Now we are ready to state the following characterization, which is a small improvement of Theorem [Han, Theorem 3.3.1, p. 26]. Theorem Let G : (R m,0) (R p,0), m p 2, with Disc G = {0}. Then the following are equivalent: (i) there exists a g.v.f. for G on B ε \V G, for some small enough ε > 0;

143 7.1. Existence 141 (ii) a(x) > 0, for any x M(G) \V G. Proof. By condition (c 1 ) one has that ν(x),ω j (x) = 0 for any j = 2,..., p. Therefore, ρ(x),ν(x) = a(x) G(x) 2,ν(x), which by (c 2 ) and (c 3 ) implies that a(x) > 0. Reciprocally, if a(x) > 0, it follows from Lemma and Remark that the vector field (7.7) has no singularity on B m ε \V G, hence it is a g.v.f. for G. It is important to notice however that just the existence of a g.v.f. is not enough to guarantee the existence of fibration (7.4), nor (7.5). In fact, consider the example G(x,y,z) = (x 2 + y 2,(x 2 + y 2 )z). One can apply the method of Example to get a g.v.f. for G. However, as we have seen in Section the tube and sphere fibration does not exist. Therefore, the problem of existence of a good vector field and the Milnor fibrations are independent each other and one can restate Conjecture as follows. Conjecture Let G : (R m,0) (R p,0), m > p 2 be an analytic map germ with Disc G = {0}. Then M(Ψ) = /0 if, and only if a(x) > 0 for any x M(G) \V G. Now, one can summarize the above discussion and illustrate the conjecture by using the following diagram. Figure 16 Diagram

144 142 Chapter 7. Good vector fields In the paper [Oka2], Oka introduced the notion of strongly non-degenerate convenient mixed functions and proved the Lemma below. Lemma [Oka2, Lemma 34, p. 32] Let f : C n C be a mixed function and consider two vectors on C n \V f defined by ω 1 (z, z) = dlog f (z, z) + dlog f (z, z) and ω 2 (z, z) = i(dlog f (z, z) dlog f (z, z)). If f is strongly non-degenerate and convenient, then there exists a positive number r 0 so that for any z C n with z r 0 and z / V f, the three vectors z,ω 1 (z, z) and ω 2 (z, z) are either: (i) linearly independent over R; or (ii) they are linearly dependent over R and the relation can be written as z = a(z, z)ω 1 (z, z) + b(z, z)ω 2 (z, z) with a(z, z),b(z, z) R and the coefficient a(z, z) positive. Remark By hand calculation, it is possible to show that ω 1 (z, z) = f (x,y) 2 and ω 2 (z, z) = Ω(x,y). Thus, the Lemma above tell us that in this class, there always exists a g.v.f. for f. In particular, by [Oka2, Theorem 28, Theorem 33] the fibrations (7.4) and (7.5) exist and by Theorem they are equivalent. Example For G = (G 1,G 2 ) : (R 3,0) (R 2,0), under conditions Disc G = {0} and M(Ψ) = /0, we may use the cross product in R 3 to get that v 1 (x) = Ω(x) ( G 1 G 2 ), where Ω(x) = G 1 (x) G 2 (x) G 2 (x) G 1 (x). Hence, by Lagrange s formula for the cross product we get v 1 (x) = (G 1 (x) G 2 (x) 2 G 2 (x) G 1 (x), G 2 (x) ) G 1 (x) +(G 2 (x) G 1 (x) 2 G 1 (x) G 1 (x), G 2 (x) ) G 2 (x). (7.8) We may use the expression above for the map G(x,y,z) = (x,y(x 2 + y 2 ) + xz 2 ) given in Example to see that ρ(x),v 1 (x) = 2.(x 6 + 5x 4 y 2 + 7x 2 y 4 + 3y 6 ) > 0 for any x M(G) \V G = {z = 0} \ {(0,0,0)}.

145 7.1. Existence 143 For complex functions, our first result toward the existence of a g.v.f. is as follows. Proposition Let f : C n C and g : C m C be mixed functions in separable variables. Consider the product F = f g : C n C m C. If one of the conditions below holds true: (i) Disc f = {0} and there exists ε 1 > 0 such that for any x ( M( f ) \V f ) B 2n ε1 the coefficient a 1 (x) > 0. (ii) Disc g = {0} and there exists ε 2 > 0 such that for any y (M(g) \V g ) B 2m ε 2 coefficient a 2 (y) > 0. Then the vector field (7.7) is a g.v.f. for F Proof. We will prove the item (i). The item (ii) follows in the same way. By Lemma 3.2.5, Disc F = {0} thus, for any (x,y) M(F)\V F one has (x,y) Sing F. Consequently, there exist a(x, y), b(x, y) R such that (x,y) = a(x,y) F(x,y) 2 + b(x,y)ω F (x,y). (7.9) the Now, by Lemma 3.2.4, x = a(x,y) g(y) 2 f (x) 2 + b(x,y) g(y) 2 Ω f (x) y = a(x,y) f (x) 2 g(y) 2 + b(x,y) f (x) 2 Ω g (y) (7.10) On the other hand, by Proposition 3.2.7, x M( f ) \V f and y M(g) \V g. Suppose that the condition (i) is satisfied. Let (x,y) (M(F) \V F ) Bε 2(n+m) 1. Since x Sing f there exist a 1 (x),b 1 (x) R such that x = a 1 (x) f (x) 2 + b 1 (x)ω f (x). (7.11) Comparing the first equation of (7.10) and the equation (7.11), we have that ( a(x,y) g(y) 2 a 1 (x) ) f (x) 2 + ( b(x,y) g(y) 2 b 1 (x) ) Ω f (x) = 0 Since Disc f = {0} then { f (x) 2,Ω f (x) } are linearly independent on R. Therefore, a 1 (x) = a(x,y) g(y) 2 and b 1 (x) = b(x,y) g(y) 2. Now we notice that: ε 1 > (x,y) 2 = x x n 2 + y y m 2 x x n 2 = x 2,

146 144 Chapter 7. Good vector fields Consequently, x ( M( f ) \V f ) B 2n ε1. By hypothesis, a 1 (x) > 0 which implies that a(x,y) > 0. Therefore, from Theorem the vector field (7.7) is a g.v.f. for F. Remark It follows from the proof above that, if f has Disc f = {0} and it admits a g.v.f. then for any map g nonzero in separable variables the result holds true. We finish this section by proving a refinement of the main results of Section 6.2, in the case where G has isolated critical value. Theorem Let G := (G 1,...,G p ) : (R m,0) (R p,0) be an analytic map germ with Disc G = {0}. If G satisfies one of the conditions below, for any x M(G) \V G : (i) detd(x) > 0. (ii) G(x) 2,Ω j (x) = 0, or ρ(x),ω j (x) = 0, for 2 j p. (iii) G has same multiplicity. (iv) G i (x), G j (x) = 0, for any i, j = 1,..., p with i j. Then the vector field (7.7) is a g.v.f. for G. Remark In [Ma] the author addressed the case p = 2 and considered G satisfying the Milnor condition (c) at the origin if ω(x) 2 ρ(x), ( G(x) 2 ) ω(x), ( G(x) 2 ) ω(x), ρ(x) > 0, where ω(x) := Ω 2 (x). Using our notations it is equivalent to say that detd(x) > 0. Thus, our condition detd(x) > 0 can be thought as a kind of generalization of the Milnor condition (c) at the origin. Corollary Let f : C n C be a mixed function with Disc f = {0}. If for any z M( f ) \V f, one has Im d f (z), d f (z) = 0, then the vector field (7.7) is a g.v.f. for f. C Corollary If G : (R m,0) (R p,0) be an analytic map germ which is a Simple Ł- Map. Then the vector field (7.7) is a g.v.f. for G. Proof. The result follows from the definition of Simple Ł- Map and item (iv) of the Theorem

147 7.2. Topological approach 145 Example Consider G(x,y,z,w) = (x(z 2 + w 2 ),y(z 2 + w 2 )). One has that V G = {x = y = 0} {z = w = 0}, Sing G = {z = w = 0} and M(G) \V G = {z 2 + w 2 = 2(x 2 + y 2 )}. Consequently, Disc G = {0}. Moreover, G 2 = (x(w 2 +z 2 ) 2,y(w 2 +z 2 ) 2,2z(x 2 + y 2 )(w 2 + z 2 ),2w(x 2 + y 2 )(w 2 + z 2 )) and Ω 2 = ( y(w 2 + z 2 ) 2,x(w 2 + z 2 ) 2,0,0). Hence, G 2,Ω 2 = 0. On the other hand, on the M(G) \VG one has G 1 G 2 and G 1, G Topological approach In this section we will give a topological condition on the set M(G)\V G to ensure the existence of a g.v.f.. Let G : (R m,0) (R p,0), m > p 2 be an analytic map germ. Consider the projection map Ψ : U \V G S p 1 1 and y S p 1. Define L y,η := {ty t (0,η)} with 0 < η 1 and consider the fiber X y = Ψ 1 (y). By the commutative diagram below U \V G G R p \ {0} Ψ S p 1 π the fiber X y = G 1 (L y,η ). Let M(G) \V G = β M β where M β are the connected component of the M(G) \ V G such that 0 M β. Consider the following condition: (τ) for each β there exists y S p 1 G : M β R p contains the line L y,η. such that the image of restriction map Lemma Let G : (R m,0) (R p,0) be an analytic map germ. If G satisfies the condition (τ), then for each β there exist a y S p 1 and a curve γ := γ y,η : (0,δ) M β X y B m ε such that γ(0) = 0. Proof. Let y S p 1 such that L y,η (G )(M β B m ε ) for any small enough ε > 0 and 0 < η ε. We know that (G ) 1 (L y,η ) = X y M β B m ε G 1 (Bη p \ {0}), this implies

148 146 Chapter 7. Good vector fields the existence of a sequence x n M β X y B m ε such that x n converges to 0. By the Curve Selection Lemma, one has the required curve. Remark Consider any analytic path α(t) M β such that α 0 when t 0. We have that G(α(t)) c, for c constant. As a consequence of that, and the fact that the functions ρ and G 2 are positive and vanish at 0, for any small enough t > 0 one has: a) b) d dt ρ(α(t)) = ρ(t),α (t) > 0, d dt ( G(α(t) 2 ) = G(t) 2,α (t) > 0. The next result is a slight improvement of [Han, Lemma p. 32]. Theorem Let G : (R m,0) (R p,0), m > p 2 be an analytic map germ with Disc G = {0}. If G satisfies the condition (τ), then the vector field (7.7) is a g.v.f. for G. Proof. By Lemma 7.2.1, the condition (τ) implies in the existence of a curve α : (0,δ) M β X y B m ε such that α(0) = 0, for some y S p 1. It follows from Remark 7.2.2, that there exists 0 < δ 1 δ such that for any t (0,δ 1 ) one has (i) dρ(α(t)) dt = ρ(t),α (t) > 0 (ii) d G(α(t)) 2 dt = G(t) 2,α (t) > 0. Since α((0,δ 1 )) X y, then for any t (0,δ 1 ), it follows that α (t) T α(t) X y, v 1 (t),α (t) > 0 and v 2 (t),α (t) > 0. Consequently, a(t) > 0 and as M β is connected one has that a(x) > 0 for all x M β. By Theorem the vector fields (7.7) is g.v.f. for G. We have the following corollary of the proof of the previous theorem, which is an improvement of [Han, Lemma 3.4.1, p. 31]: Corollary Let α : [0,γ) M β be an analytic path with α(0) = 0. If there exists some t 0 > 0 small enough such that α (t 0 ) T α(t0 )X Ψ(α(t0 )), then v 1 (x) and v 2 (x) points in the same direction at any point x in M β.

149 7.2. Topological approach 147 As we have discussed in the introduction, Corollary reduces the study of the equivalence of fibrations to the case where the component M β is transverse to the fibers of Ψ i.e., the set M(G) \V G is turning around the origin. Lemma Let G : (R m,0) (R p,0) be an analytic map germ such that Im G is full. Then for any small enough ε > 0 and each a Bη p \ {0} with 0 < η ε, (M(G) \V G ) G 1 (a) B m ε /0. In particular, the restriction G : M(G) Bη p (7.12) is surjective. Proof. Since Im G is full, for each ε > 0 enough small there exists 0 < η ε such that for each a B p η \ {0}, F a,ε := G 1 (a) B m ε is not empty. Since {a} is a closed subset of R p, then {a} = {a} B p η \ {0} is a relatively closed subset of B p η \ {0}. Moreover, F a,ε is closed in the closed ball B m ε, because of the continuity of G. Hence, there exists at least one point x 0 F a,ε such that the distance function d a,ε : F a,ε R 0 is minimum and d a,ε (x 0 ) < ε, thus x 0 M(G) \V G. Therefore, we have x 0 (M(G) \V G ) F a,ε and in addition, the restriction map is surjective. In the following we state our main result of this section. Theorem Let G : (R m,0) (R p,0) be an analytic map germ such that Im G is full and Disc G = {0}. If M(G) \V G is connected, then the vector field (7.7) is a g.v.f. for G. Proof. By Lemma the restriction map (7.12) is surjective. Consequently, its image contains a line L y,η for some y S p 1 and small enough η > 0. Now, as M(G) \V G is connected, one has that G satisfies the condition (τ) and the result follows from Theorem In general, the set M(G) \V G is not connected. Example Consider G(x,y,z) = (xy,xz) as in Example and A := {x 2 y 2 z 2 = 0}, B := {x = 0} and C := {y = z = 0}. As we have seen, the Milnor set M(G) = A B and V G = B C, see Figure 17 below. Therefore, M(G) \ V G is disconnected. However, G has a g.v.f., by Example

150 148 Chapter 7. Good vector fields Figure 17 M(G) V G for G(x,y,z) = (xy,xz) 7.3 Approach using differentials In this section we will address the problem of the existence of a g.v.f. by using the differential of a map. A similar approach was done in [CSS2] as we explain in the end of this section. For that, given G : (R m,0) (R p,0) an analytic map germ with Disc G = {0}, one reminds that its differential map is denoted by d x G : R m R p. Definition For a vector v R m we say that d x G(v) is transversal to the sphere S p 1 G(x) if d xg(v),g(x) = 0. Moreover, we say that d x G(v) is pointing to the exterior (respectively, to the interior) if, d x G(v),G(x) > 0 (respectively, d x G(v),G(x) < 0). Lemma G : (R m,0) (R p,0) be a analytic map germ with Disc G = {0}. Then for any x M(G) \V G one has: (i) d x G(v 1 (x)) is tranversal to the sphere S p 1, pointing to the exterior. G(x)

151 7.3. Approach using differentials 149 (ii) d x G(v 2 (x)) is tranversal to the sphere S p 1, pointing to the exterior or to the G(x) interior, since M(Ψ) = /0. Proof. Since d x G(v 1 (x)) = ( G 1 (x),v 1 (x) ),, G p (x),v 1 (x) ), then d x G(v 1 (x)),g(x) = Σ p j=1 G j(x) G j (x),v 1 (x) = 1 2 G(x) 2,v 1 (x) > 0. Thus the first statement follows. For the second statement, one has that d x G(v 2 (x)),g(x) = 1 2 G(x) 2,v 2 (x). Since v 2 (x) = a(x)v 1 (x) hence d x G(v 2 (x)),g(x) = a(x) 2 G(x) 2,v 1 (x). Moreover, if follows from hypothesis that a(x) 0. Therefore, from the first part of proof, d x G(v 2 (x)),g(x) = 0, for any x M(G) \V G. Next we will state the main result of this section. Proposition Let G : (R m,0) (R p,0) be a analytic map germ with Disc G = {0}. The following are equivalent. (i) there exists a g.v.f. for G on B ε \V G, for some enough small ε > 0; (ii) For x M(G) \V G, d x G(v 2 (x)) is transversal to the sphere S p 1, pointing to the G(x) exterior; Proof. Since there exists a g.v.f. for G, then a(x) > 0 and the necessary condition follows from the proof of Lemma 7.3.2, item (ii). Now, as v 2 (x) = a(x)v 1 (x) we have 0 < d x G(v 2 (x)),g(x) = a(x) 2 G(x) 2,v 1 (x). Consequently, a(x) > 0 and the result follows from Theorem Some Comments on the method of [CSS2] In paper [CSS2] the authors approached the equivalence problem between the Milnor fibrations. Their main result is based on the following lemma, which we remind below with its proof.

152 150 Chapter 7. Good vector fields Lemma [CSS2, Lemma 5.2] The map f : R n R p is d-regular if and only if there exists a smooth vector field w on B m ε \V f which has the following properties: (i) It is radial, i.e., it is transverse to all spheres in B m ε, centered at 0. (ii) It is tangent to each X l \V f, whenever it is not empty. (iii) It is transverse to all the tubes. Proof. [CSS2, Lemma 5.2] If f is d-regular, by Proposition the spherefication map F : B m ε \V f B p ε \ {0} is a submersion. Let u be the canonical radial vector field on R p given by u(z) = z. Using f and F we can lift u to smooth vector fields w f and w F on B m ε \V f such that for every x B m ε \V f we have: d f x (w f (x)) = u( f (x)) and d x F(w F(x) ) = u(w F (x)). The local flow associated to w f is transverse to all Milnor tubes f 1 ( B p η), while the one associated to w F is transverse to all spheres in B m ε centered at 0. The integral paths of both move along X l \V f. We claim that the vectors w f (x) and w F(x) cannot point in opposite directions for x B m ε with ε 0 > 0 sufficiently small. Suppose w f (x) and w F(x) point in opposite directions with x X l \V f. Since w f (x) T x X l, there exists a curve β : ( δ,δ) X l such that β(0) = x and β (0) = w f (x). Since w F (x) points in the opposite direction as w f (x), it is the velocity vector of a curve γ : ( δ,δ) X l of the form γ(t) = β( rt) with r > 0, that is, γ(0) = x and γ (0) = w F (x). We have that w f (x) is transverse to the sphere S m 1 x, since it is collinear to w F (x) and by definition w F (x) is tranverse to S m 1 x. Then β(t) is a strictly monotone function of t (possibly in a smaller subinterval of ( δ,δ)). Without loss of generality, assume it is a strictly increasing function, then it implies that γ(t) is a strictly decreasing function of t, and by definition of F we have that the function F(t) is also a strictly decreasing function. Since the image curve F(γ(t)) lies on the line L l, we have that D x F(w F (x)) is a radial vector pointing toward the origin. This is a contradiction, since by definition w F (x) is a lift of a radial vector pointing away from the origin.

153 7.3. Approach using differentials 151 Hence we have that the vector field w f satisfies properties (ii) and (iii) and that w F (x) satisfies properties (i) and (ii). Adding up w f and w F on B m ε \V f we get a vector field w which satisfies these three properties. The converse is obvious by properties (i) and (ii). Before we start the analysis of the arguments used by the authors in [CSS2] to prove this lemma, let us remind that they are assuming Disc f = {0}. Moreover, as we have seen in Chapter 2, the d-regularity condition is equivalent to ρ-regularity and the Spherification map is given by F(x) = x f (x) f (x). We notice that for the vector field w f one has that 1 f (x) 2,w f = dx f (w f ), f (x) = f (x) 2 > 0. 2 Thus, the local flow associate to w f is, indeed, transverse to all Milnor tubes f 1 ( B p ε ), pointing to the exterior. In the same way one can assure that w F is transverse to the spheres in B m ε as was stated. However, in general is not clear that w F is pointing to the exterior or interior to the sphere, hence one cannot ensure that the vector field defined by w := w f + w F points to the exterior of all spheres and all tubes simultaneously. In other words, it does not ensure that the vector field w satisfies the conditions (c 2 ) and (c 3 ) of Definition Compare with the Lemma and Theorem In order that the vector field w be not zero, the vectors w f and w F must not point in opposite directions. To ensure this, as can be seen in the proof, the authors considered only the case where the norm γ(t) is a strictly decreasing function of t. But this case is trivial since by the construction of γ the velocity vector at 0 is the vector w F (x) which is assumed to be the lift of a radial vector field pointing away from the origin. Therefore, the norm of γ increases by construction. The more important case is the complementary case, i.e., when: the norm of β(t) is decreasing while the norm of γ(t) is increasing, which was not approached in the proof. We notice that the preceding argument does not works for this case. Indeed, the authors noted that d x F(w F (x)) is a radial vector pointing toward the origin and since

154 152 Chapter 7. Good vector fields w F is the lifting of a radial vector pointing away from the origin, in this case there is no contradiction. Considering the above discussion, we understand that the problem of the existence of a g.v.f. under conditions Disc G = {0} and M(Ψ) = /0 is still open. This fact had been noted by Hansen in [Han, p.4]. In fact, by the time this work was in the final writing, this was confirmed to us by Professor J-L. Cisneros, one of the authors of [CSS2], during his research visit to our institution (ICMC-USP). Therefore, the equivalence problem in its complete generality is still open, as well.

155 153 CHAPTER 8 EQUIVALENCE OF MILNOR-HAMM FIBRATIONS In this chapter we show that, for some classes of maps, if the fibrations (7.4) and (7.5) exist then they are equivalent. See Definition For that we will use the main results developed in Chapter 7 which we summarize in the Theorem below. Theorem Let G : (R m,0) (R p,0) be an analytic map germ with Disc G = {0} satisfying the condition (2.14). If G satisfies one of the conditions below, for any x M(G) \V G : (i) det D(x) > 0. (See Proposition 6.2.1). (ii) G(x) 2,Ω j (x) = 0, or ρ(x),ω j (x) = 0, for 2 j p. (iii) G has same multiplicity. (See Theorem 6.2.5). (iv) G i (x), G j (x) = 0, for any i, j = 1,..., p with i j. (v) d x G(v 2 (x)) is transversal to the sphere S p 1, pointing to the exterior. (See Definition G(x) 7.3.1). (vi) G satisfies the condition (τ). (See Section 7.2 in page 145). (vii) M(G) \V G is connected.

156 154 Chapter 8. Equivalence of Milnor-Hamm fibrations Then there exists the fibrations (7.4), (7.5) and they are equivalent. As a first application we give a positive answer for the equivalence problem for the Simple Ł-Maps class introduced by Massey in [Ma] as follows. Corollary If G : (R m,0) (R p,0) be an analytic map germ which is a Simple Ł- Map. Then there exist the fibrations (7.4), (7.5) and they are equivalent. We also will do applications in the classes of mixed functions and maps with radial action. In this latter class, we do the first approach in the literature toward the equivalence problem in the case dimdisc G > 0, that provides a partial answer to the question introduced in Chapter 7. We finish the chapter, and hence the thesis, presenting an answer to an weaker version of the Conjecture Mixed functions and equivalence Lemma Let f : C n C be a mixed function germ which is polar weighted homogeneous. If M β is a connected component of M( f ) \V f such that 0 M β. Then λ M β = M β. Proof. Since the S 1 -action is continuous and M β is connected for each β, the result follows from Lemma Theorem Let f : C n C be a mixed function germ which is polar weighted homogeneous. Then f satisfies the condition (τ). Proof. It is sufficient to show that the image f (M β ) contains a small enough disk D η at 0 C. We have that M β is a semi-algebraic set and 0 M β. By the Curve Selection Lemma, there exists an analytic curve γ M β such that γ(0) = 0. Consider the image curve ˆγ := f γ. For any enough small η > 0, let a S 1 η ˆγ and z f 1 (a). It follows from Lemma that for any λ S 1, λ d a = f (λ (z, z)) f (M β ). Consequently, S 1 η f (M β ). Finally, as ˆγ 0 when t 0 and ˆγ(0) = 0, we have that D η f (M β ), hence the result follows.

157 8.1. Mixed functions and equivalence 155 The first part of next result was proved in [ACT1, PT]. The second part is a new statement regarding the equivalence of the Milnor tube and sphere fibrations and is a consequence of the previous theorem. Proposition Let f : C n C be a mixed function germ which is polar weighted homogeneous. Then there exist the fibrations (7.4) and (7.5). Moreover these fibrations are equivalent. Proof. The first part follows from Theorem and [PT, Theorem 5.2]. The equivalence between the fibrations follows from Theorem and Theorem Proposition Let f : C n C and g : C m C mixed functions in separable variables. Suppose that the mixed function F = f g : C n C m C satisfies the condition (2.14). If either the conditions below holds true: (i) Disc f = {0} and there exists ε 1 > 0 such that for any x ( M( f ) \V f ) B 2n ε1 the coefficient a 1 (x) > 0. (ii) Disc g = {0} and there exists ε 2 > 0 such that for any y (M(g) \V g ) B 2m ε 2 coefficient a 2 (y) > 0. the Then for any small enough ε > 0 and 0 < δ ε, there exist the Milnor tube fibration and the Milnor sphere fibration and they are equivalent. F F : B2(n+m) ε F 1 (S 1 δ ) S1, (8.1) F F : S2(n+m) 1 ε \ K ε S 1 (8.2) Proof. We suppose without lost of generality that the condition (i) is satisfied. By Lemma one has that Disc F = {0}. By Proposition there exists a g.v.f for F and it is ρ-regular. Consequently, the Milnor tube and sphere fibrations exist and they are equivalent.

158 156 Chapter 8. Equivalence of Milnor-Hamm fibrations Corollary Let f : C n C be a holomorphic function and g : C m C be a mixed functions in separable variables. Suppose that the mixed function F = f g : C n C m C satisfies the condition (2.14). Then there exists the Milnor tube fibration (8.1) and the Milnor sphere fibration (8.2) and they are equivalent. Proof. Since f is a holomorphic function, then the condition (i) of Theorem is satisfied. Example [ACT1] Consider the functions f : C C and g : C C given by f (y) = y and g(x) = x x and define F := f g : C 2 C. Since F is a mixed function which is polar weighted homogeneous, then F satisfies the condition (2.14). By Corollary 8.1.5, F has Tube fibration, Sphere fibration and the fibrations are equivalent. Notice that we also could apply Proposition to get the same result. 8.2 Maps with radial action Let us consider the (R >0 )-action on R m : t x := (t q 1x 1,...,t q m x m ) for t R 0 and q 1,...,q m N relatively prime positive integers. We say that G = (G 1,,G p ) : R m R p is a radial weighted-homogeneous (or radial for short) of weights (q 1,,q m ) and of degree d > 0, if G(t x) = t d G(x). In [A2] the author used the flow of Euler s vector field to create a diffeomorphism which provide the equivalence between the fibrations (7.4) and (7.5), associated to a radial weighted-homogeneous map G : R m R 2 with isolated critical point at the origin. In [AT2], the authors proved a generalization as follows: Theorem [AT2, p.184, Theorem 3.1] Let G : R m R p be a radial weightedhomogeneous map such that Sing G {0}. Then the fibrations on the tube and on the sphere are equivalent. In this section we will present an extension for non-isolated singularities of Theorem above. For that, we consider the following: Lemma Let G : (R m,0) (R p,0) be a radial analytic map germ of type (q 1,...,q m ;d). Then for any t > 0 one has:

159 8.2. Maps with radial action 157 (i) M(G) = t M(G); (ii) Sing G = t Sing G. Proof. Fix t R 0. Since G is radial, one has the following relation: G k x j (t x) = t d q j G k x j (x) (8.3) for any j = 1,...,m and k = 1,..., p. Let x M(G). One has that there exist λ,β 1,...,β p R such that λ x = β 1 G 1 (x) + + β p G p (x), which implies λx j = β 1 G 1 x j (x) + + β p G p x j (x). (8.4) Take λ = λ and β j = β jt 2q j d for j = 1,..., p. From the equations (8.3) and (8.4), one has λ t q j x j = β 1 G 1 (t x) + + β G p p (t x), x j x j for j = 1,...,m. It means that λ t x = β 1 G 1(t x) + + β p G p (t x). Consequently, t x M(G) and t M(G) M(G). Now, let x M(G) and t R 0. Since 1 t R 0 from the first part of proof, b := 1 t x M(G). Thus, a = t b t M(G) and one has the equality M(G) = t M(G). The item (ii) is proved analogously. Corollary Let G : (R m,0) (R p,0) be a radial analytic map germ of type (q 1,...,q m ;d). Then Disc G is radial. Proof. Let y Disc G = G(Sing G). There exists x Sing G such that G(x) = y. By item (ii) of Lemma 8.2.2, one has t x Sing G for any t > 0. Hence, t d G(x) Disc G for any t > 0. Therefore Disc G is radial. Corollary Let G : (R m,0) (R p,0) be a radial map germ of type (q 1,...,q m ;d). If M β is a connected component of M(G) \V such that 0 M β. Then M β = t M β, for all t R 0. Proof. Since the (R >0 )-action is continuous and M β is connected for each β, the result follows from Lemma

160 158 Chapter 8. Equivalence of Milnor-Hamm fibrations The next Theorem represents an extension for non-isolated singularities of Theorem 8.2.1: Theorem Let G : (R m,0) (R p,0) be a radial map germ of type (q 1,...,q m ;d) with Disc G = {0} and satisfying the condition (2.14). Then the fibrations (7.4), (7.5) exist and they are equivalent. Proof. Let M β a connected component of M(G) \V such that 0 M β and x M β. Follows from Corollary 8.2.4, that t x M β for any t > 0. Consequently, G satisfies the condition (τ) and the result follows from item (vi) of the Theorem Milnor-Hamm fibrations and equivalence Let us denote by γ(x) := m q j x j ( / x j ) j=1 the Euler vector field corresponding to the (R >0 )-action on R m. Thus, one has that G i (x),γ(x) = d G i (x) for any i = 1,..., p. Our main result in this subsection is: Theorem Let G : (R m,0) (R p,0) be a radial map germ of type (q 1,...,q m ;d), satisfying the condition (4.2). Then G has Milnor-Hamm tube and sphere fibrations and they are equivalent. Proof. Since G is a radial map germ one has that Disc G is also radial by Corollary Moreover, for any x B m ε \V G one has that γ(x),ω k (x) = d (G 1 (x)g k (x) G k (x)g 1 (x)) = 0, for any k = 2,..., p. Hence, γ(x) is tangent to the fiber X y and (c 2 ) γ(x), ρ(x) = 2Σ p k=1 w kxk 2 > 0, and (c 3 ) γ(x), G(x) 2 = 2d G(x) 2 > 0.

161 8.2. Maps with radial action 159 Consequently the Euler vector field is g.v.f. for G and hence G is ρ-regular. Therefore the Milnor-Hamm tube and sphere fibrations exist. Finally, since the (R >0 )-action on R m does not mix up singular and nonsingular points of G, the equivalence follows. Example Consider the map germ G(x,y,z) = (xy,z 2 ). As we have seen, G has Milnor-Hamm tube and sphere fibrations. Moreover, by Theorem they are equivalent. Figure 18 Equivalent fibrations for G(x,y,z) = (xy,z 2 ) Remark In the radial case, the existence of the g.v.f is independent of the dimension of the discriminant locus of G.

162 160 Chapter 8. Equivalence of Milnor-Hamm fibrations Corollary Let ( f,g) be a holomorphic map germ which is Thom regular at V ( f,g). If f ḡ is a radial weighted homogeneous function, then f ḡ has Milnor-Hamm tube and sphere fibrations. Moreover, they are equivalent. Proof. By hypothesis f ḡ satisfies the condition (4.2). Since f ḡ is radial the result follows from Theorem above. The remark below shows a way to construct a wide class of examples satisfying our Theorem Remark Let f and g holomorphic functions and radial weighted-homogeneous of type (q 1,...,q n ;d), such that ( f,g) is ICIS. Then f ḡ has Milnor-Hamm tube and sphere fibrations and they are equivalent. Example Let f,g : C 2 C given by f (x,y) = x 2 +y 2 and g(x,y) = x 2 y 2 as we have seen in Example , Sing ( f,g) V ( f,g) = {(0,0)}. Notice that Disc f ḡ {0}. 8.3 Diffeomorphisms between the Milnor fibers In this section we will approach a weaker version of the Conjecture We remind that for any vector field ω on B m ε \V G, the restriction of the proj Tx X y (ω(x)) on the analytic manifold X y is a tangent vector field, and one can write proj Tx X y (ω(x)) = ω(x) Σ p k=2 β k(x)ω k (x), (8.5) where the β k, k = 2, p, are defined by the system of linear equations ω(x) Σ p k=2 β k(x)ω k (x),ω j (x) = 0; for j = 2,... p. Since Disc G = 0, by Lagrange s Identity the determinant (x) of this system is strictly positive, hence the coefficients β k are uniquely defined. Moreover, for any smooth function φ on R m the restriction of the vector field proj Tx X y ( φ(x)) to X y is the gradient field φ, since the Riemannian metric on X X y y is given by the restriction of the (canonical) Riemannian metric of R m.

163 8.3. Diffeomorphisms between the Milnor fibers 161 In [Io, Lemma 1, p. 343], Iomdin proved the following result for a map with isolated singular point. Lemma [Io, Lemma 1, p. 343] If f = ( f 1,..., f p ) and g are polynomial functions on R m, such that f (0) = g(0) = 0, f Vi 0 and g Vi 0, where V i := f1 1 1 (0) fi (0), with 1 i p, then one can find a small enough ε > 0 such that for x (V i \ 0) B m ε the vectors proj Tx V i ( f (x)) and proj Tx V i ( g(x)) cannot have strictly opposite direction, except for the possibility that at least one of them is equal to zero. follows: Using the idea of the proof of Iomdin s Lemma one can restate the result as Lemma If f and g are analytic functions on R m, such that f (0) = g(0) = 0, f X y > 0 and g X y > 0, then one can find a small enough ε > 0 such that for x X y B m ε the vectors proj Tx X y ( f (x)) and proj Tx X y ( g(x)) cannot have strictly opposite direction, except for the possibility that at least one of them is equal to zero. Proof. Suppose the statement is false. Then is possible to get x X y arbitrarily near of the origin such that proj Tx X y ( f (x)) and proj Tx X y ( g(x)) are both different from zero and have strictly opposite directions. Now, for this x, consider the vectors ω 1 (x) := (x) proj Tx X y ( f (x)) and ω 2 (x) := (x) proj Tx X y ( g(x)). By equation (8.5) the coordinates of ω 1 (x) and ω 2 (x) are analytic in x 1,...,x m. Let us denote these coordinates by ω1 i (x) and ω j 2 (x), respectively, i, j = 1,...,m. Moreover, these vectors have strictly opposite directions at x iff the above coordinates at the point belongs to the intersection A B where A = {x X y ;ω1 i (x)ω j 2 (x) ω j 1 (x)ωi 2 (x) = 0} and B = {x X y ; ω 1 (x),ω 2 (x) < 0}. So, if 0 A B, if follows from the Curve Selection Lemma that there exists a real analytic curve α : [0,δ) R m such that α(0) = 0 and for all t > 0 we have α(t) A B X y and the vectors proj Tα(t) X y(t) ( f (α(t))) and proj Tα(t) X y(t) ( g(α(t))) are both different from zero and have strictly opposite direction. We notice that for

164 162 Chapter 8. Equivalence of Milnor-Hamm fibrations t > 0, f (α(t)) > 0 and g(α(t)) > 0. Since f (α(t)) and g(α(t)) are positive functions which vanish for t = 0, for any small enough t > 0 we have that d f (α(t)) > 0 and dt dg(α(t)) > 0. On the other hand, since α(t) X y : dt (i) d f (α(t)) dt (ii) dg(α(t)) dt = proj Tα(t) X y(t) ( f (α(t)),α (t) > 0, = proj Tα(t) X y(t) ( g(α(t)),α (t) > 0, which is a contradiction because the vectors proj Tα(t) X y(t) ( f (α(t)) and proj Tα(t) X y(t) ( g(α(t)) have strictly opposite directions. Therefore, the Lemma is proved. Corollary Let G : (R m,0) (R p,0) be an analytic map germ with Disc G = {0} and M(Ψ) = /0. Then for each fiber X y, there exists small enough ε > 0 such that for any x X y B ε or the vectors v 1 (x) and v 2 (x) are linearly independent, or they are linearly dependent and cannot have strictly opposite direction i.e., if v 1 (x) = a(x)v 2 (x) then a(x) > 0 for all x X y B ε. Now, we can prove our main result in this section. Theorem Let G : (R m,0) (R p,0) be an analytic map germ with Disc G = {0} and M(Ψ) = /0. If there exist the Milnor fibrations (7.4) and (7.5) then their fibers are diffeomorphic. Proof. Let ε 0 > 0 such that both fibrations (7.4) and (7.5) exist, for any 0 < ε < ε 0 and 0 < η ε ε 0. By Corollary one can fix an ε < ε 0 and a fibre X y of Ψ such that v 1 (x) and v 2 (x) cannot have strictly opposite direction on X y B m ε. Hence the vector field (7.7) has no zeros on X y B m ε. Consequently, one can used the flow of such a vector field to guarantee that the Milnor tube fiber contained in X y B m ε is diffeomorphic to the Milnor sphere fiber X y B m ε. Therefore, our statement is proved.

165 8.3. Diffeomorphisms between the Milnor fibers 163 Figure 19 Diffeomorphism between Milnor Fibers