UNIVERSIDADE DE SÃO PAULO

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Doctoral Dissertation of the Graduate Program in Mathematics (PPG-Mat)

Maico Felipe Silva Ribeiro

Singular Milnor fibrations

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SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP

Data de Depósito:

Assinatura: ______________________

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

Maico Felipe Silva Ribeiro

USP – São Carlos April 2018

Advisor: Prof. Dr. Raimundo Nonato Araújo dos Santos

Co-advisor: Prof. Dr. Mihai Marius Tibăr

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Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassi e Seção Técnica de Informática, ICMC/USP,

com os dados inseridos pelo(a) autor(a)

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

R484s

Tese (Doutorado - Programa de Pós-Graduação em Matemática) -- Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 2018.

1. 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.

173 p.

Ribeiro, Maico Felipe Silva

Singular Milnor fibrations / Maico Felipe Silva Ribeiro; orientador Raimundo Nonato Araújo dos Santos; coorientador Mihai Marius Tibăr . -- São Carlos, 2018.

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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

Maico Felipe Silva Ribeiro

Fibrações de Milnor singulares

USP – São Carlos Abril de 2018

Orientador: Prof. Dr. Raimundo Nonato Araújo dos Santos

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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 Tib˘ar, 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.

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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.

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

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ABSTRACT

RIBEIRO, M. Singular Milnor fibrations. 2018. 173 f. Doctoral dissertation (Doc-torate 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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RESUMO

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.

RIBEIRO, M. Fibrações de Milnor singulares. 2018. 173 f. Tese de Doutorado (Candidato ao Programa de Doutorado em Matemática) – Instituto de Ciências

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LIST OF FIGURES

Figure 1 – The Milnor tube for G(x, y, z) = (x, y(x2+ y2+ z2)). . . 47

Figure 2 – Curve of critical points of G/kGk restricted to the sphere . . . 50

Figure 3 – The Milnor tube for G(x, y, z) = (xy, xz). . . 53

Figure 4 – Geometric representation of condition (4.2) . . . 85

Figure 5 – Milnor-Hamm fibration for G = (xy, z2). . . 88

Figure 6 – Stratification of G−1(Disc G) . . . 107

Figure 7 – Fibers of restrictions ρ₁s, Gs_1,3and its intersections . . . 108

Figure 8 – Fibers of restrictions ρ₁c, Gc_1,1and its intersections . . . 109

Figure 9 – Fibers of restrictions ρ₅c, Gc_5,4and its intersections . . . 109

Figure 10 – Fibers of restrictions ρ₉c, Gc_9,3and its intersections . . . 110

Figure 11 – Milnor-Hamm sphere fibration for G = (xy, z2) . . . 122

Figure 12 – Milnor tube blowing to sphere . . . 137

Figure 13 – x 6∈ M(G) ∪VG. . . 139

Figure 14 – x ∈ M(G) \VG, a(x) > 0. . . 140

Figure 15 – x ∈ M(G) \VG, a(x) < 0. . . 140

Figure 16 – Diagram . . . 141

Figure 17 – M(G) ∪VGfor G(x, y, z) = (xy, xz) . . . 148

Figure 18 – Equivalent fibrations for G(x, y, z) = (xy, z2) . . . 159

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LIST OF SYMBOLS

Bm_ε — m-dimensional open ball of radius ε centered at the origin.

Bm_ε — m-dimensional closed of radius ε centered at the origin.

∂ M — boundary or frontier of the Manifold M.

Sm−1_ε _{— (m − 1)-dimensional sphere of radius ε centered at the origin of R}m. Sp−1— (p − 1)-dimensional sphere of radius 1 centered at the origin of Rp. 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.} kvk — 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. VG— the zero locus of the map G.

h, i — the Euclidean dot product. JG(x) — jacobian matrix of G in x. JG(x)t — transpose of the JG(x).

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INTRODUCTION

In the famous Princeton notes of 1968 [Mi], John Milnor established the fun-daments for studying fibration structures for germs of complex analytic functions

f _{: (C}n+1_{, 0) → (C, 0) with dim Sing f ≥ 0, and for real analytic map germs} G : (Rm, 0) → (Rp, 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 Chapter1we 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

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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 Chapter2we 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 Theorem2.1.1and Theorem2.1.3. In the second section, we present the classic result by Milnor in [Mi, Theorem2.2.1] for real isolated singularity, and the Milnor method (see Section2.2.2) 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 Chapter6. In the case of non-isolated singularities in Theorem2.2.4, Theorem2.2.10and Theorem2.2.17, 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 Example2.2.18, which shows that in the classical case (Disc G = {0}), the Theorem 2.2.10] from [ACT1] is the most general result found in the literature concerning existence of Milnor’s fibration on spheres.

In Chapter3we 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, Proposition3.1.2, that will be used in Chapter4to 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 ¯g, f and g being holomorphic functions, to the product of mixed functions. See Corollary3.2.1, Lemma3.2.2and Proposition3.2.7. In the third section, our main result is Proposition3.3.4, which is an algorithm that permits us to build a new class of functions

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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

3.4.7and Lemma3.4.3, 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 ap-propriate 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 Theorem4.1.4which gives a sufficient and necessary condition for its existence. Moreover, we show that several classes of maps satisfies this condition, see for instance, Proposition4.1.7and Proposition4.1.13concerning finitely determined map germs under the contact group structure. In the second section, the main result is Proposition4.2.4, 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 Proposition4.3.2. This condition together with our Theorem

4.3.3allows us to construct plenty of examples. In the fourth section, in Theorem4.4.5

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 ¯galso provide a good class of functions to look for the Milnor-Hamm fibration structure. In analogy with Section3.4.1, we finish this chapter with Proposition4.5.1and Proposition4.5.4, which permit us to construct a class of examples with the Milnor-Hamm fibration without ∂ -Thom regularity.

In Chapter5 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 Theorem5.1.4. 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 Corollary5.2.1to 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 Theorem2.1.1in Chapter2. Moreover, we prove Theorem5.2.2

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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 Chapter6, 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, Theorem6.1.6, which gives sufficient conditions to ensure the existence of this fibration. This result is an extension of main result of [ACT1], namely, Theorem

2.2.10. 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 Theorem6.2.5that will be very important in Chapter7. We finish this chapter applying our main criteria developed in the previous section for map germs with Disc G = {0}.

We start Chapter7presenting 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 Theorem7.1.4, 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

7.1.9 ensures the existence of a good vector field for some class of mixed complex functions. We finish the first section presenting Theorem7.1.11, which is a refinement of the main results of Section6.2for the case Disc G = {0}. In the second section, we use tools from general topology to do a slight improvement of [Han, Lemma 3.4.2 p. 32] in Theorem7.2.3. Our main result in this section is Theorem7.2.6which gives a topological condition for the set M(G) \VGto 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 Proposition7.3.3, 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 Chapter7. In the first section our main results are Theorem8.1.2, Proposition

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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 Theorem8.2.5, for the case Disc G = {0}. In the sequel we consider the more general setting dim Disc 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 Conjecture7.0.1.

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29

CHAPTER

1

PRELIMINARIES

In what follows and throughout this work, the word smooth will mean differ-entiable of class C∞_{. We will use the notation M}n _{to mean a smooth n-dimensional} manifold which is embedded into a Euclidean space Rkfor 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 TpMand if G : M → N is a smooth map such that G(x) = y then the differential map will be denoted by dxG: TxM→ TyN.

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 1.1.1. 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_txW, if either G(x) /∈ W or G(x) ∈ W and

T_G(x)N= T_G(x)W+ dxG(TxM).

We say that G is transversal to W if G is transversal to W at x for any x ∈ M.

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30 Chapter 1. Preliminaries

Definition 1.1.2. We say that N is transversal to W at x ∈ N ∩W , N_txW, if TxN+ TxW = TxM.

If NtxW for any x ∈ N ∩W , we say that N is transversal to W and we denote Nt W . Remark 1.1.3. We remind that:

(i) N_{t W if and only if iN}_{t W , where iN}: N → M is the inclusion map.

(ii) If N t W , then N ∩ W is a submanifold of M and codim N ∩ W = codim N + codimW .

(iii) If G : M → N is a smooth map and W = {x} ⊂ N is a point, then G_{t 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_txH, if both G and H are submersions at x and G−1(G(x)) txH−1(H(x)).

1.1.1 Fiber bundles

Definition 1.1.4. 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 = π1◦ h, i.e., the below diagram is commutative p−1(V ) h 00 p V V× F π1

where π1: 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 trivializationof the bundle.

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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 Vα = 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 1.1.5. Two locally trivial smooth fibrations p : E → B and p0: E0→ B are said to be equivalent if there is a smooth diffeomorphism h : E → E0such that, p0◦ h = p, i.e., the follow diagram is commutative:

E h // p B E0 p0

Consequently, for all b ∈ B, h induces a map h_b: p−1(b) → p0−1(b) which is a diffeo-morphism.

1.1.1.1 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, Rn) 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.

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An important example of vector bundle is (T M, π, M, Rn) where the total space

T M= [

p∈M

T_pM= [ p∈M

({p} × TpM)

and the projection π : T M → M is given by π(p, v) = p. The vector bundle (T M, π, M, Rn) is called tangent bundle of a smooth n-dimensional manifold M.

Definition 1.1.6. We say that M is parallelizable if (T M, π, M, Rn) is trivial bundle. Definition 1.1.7. 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 1.1.8. Let M be a smooth manifold. A smooth vector field ν in M is a section of the vector bundle (T M, π, M, Rn).

Remark 1.1.9. If M ⊂ Rn, then we say that a vector field ν : M → Rn is tangent to M when ν(x) ∈ TxM for any x ∈ M.

1.1.2 Ehresmann Fibration Theorem

Let M, N be smooth manifolds and G : M → N a continuous map. We say that Gis 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 1.1.10. 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(xk)k∈Nis a sequence in M such that the correspondent sequence(G(xk))k∈Nin N is convergent, then(xk)k∈Nhas 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.

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1.2. Singularity theory 33

Definition 1.1.11. 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 1.1.12. [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 : (Rm, 0) → (Rp, 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) VG= G−1(0), the zero locus of G;

(ii) Sing G := {x ∈ Rm| rank JG(x) < p}, the singular locus of G; (iii) Disc G := G(Sing G), the discriminant of G;

(iv) Im G := G(Rm), 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.

1.2.1 Stratification

Definition 1.2.1. [GLPW] Let M be a smooth manifold and V a closed subset of M. A stratification of V _{is a collection W = {Wα}}_{α ∈A}of pairwise disjoint connected smooth submanifolds of M which verifies the following:

(i) V =S

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(ii) W is locally finite, i.e., for any point x ∈ V , there exists a neighborhood U of x in V such that U ∩Wα 6= /0 just to a finite number of indexes α ∈ A.

(iii) W satisfies the frontier condition, i.e., if Wα∩W_β 6= /0, where W_β is the closure of W_β on M, then Wα ⊂ Wβ.

Each Wα is called a stratum. The condition (iii) states that the frontier of a stratum is a union of strata.

A stratification W0of V is a refinement of a stratification W of V if and only if the strata in W are unions of the strata in W0.

Let W1, . . . , Wjbe stratifications of subsets V1, . . . ,Vj of smooth manifolds. We obtain a stratification of V1× · · · ×Vjby taking its strata to be sets of the form W1× · · · × W_j with Wi∈ Wi for any 1 ≤ i ≤ j. We call this the product stratification, and denote W1× · · · × Wj.

Let W0 be a stratification of a subset V0of N, and let G : M → N be a smooth map transverse to W0(i.e., transverse to any strata of W0). We obtain a stratification W of G−1(V0) by taking the strata to be set of form G−1(Wα) with Wα ∈ W0. We call W the induced stratificationon G−1(V0_{). 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.

1.2.1.1 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.

Let M be a smooth manifold, V a subset of M and W = {Wα}_{α ∈A}a stratification of V .

Definition 1.2.2. Let (Wi,Wj) be a pair of strata such that Wi⊂ Wj. Let x ∈ Wi. We say that the pair (Wi,Wj) satisfies the Whitney (a) condition along Wiat the point x, if for any sequence (xn) of points in Wj which converges to x such the sequence of tangent spaces

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TxnWj converge to some space T in the appropriate Grassmannian bundle, then one has TxWi⊂ T.

We say that the pair (Wi,Wj) satisfies the Whitney (a) condition along Wi if it satisfies the Whitney (a) condition along Wiat any point x ∈ Wi.

Definition 1.2.3. Let (Wi,Wj) be a pair of strata such that Wi⊂ Wj. Let x ∈ Wi. We say that the pair (Wi,Wj) satisfies the Whitney (b) condition along Wiat the point x, if for any sequence (xn) of points in Wjand any sequence (qn) of points in Wiwhich converges to xsuch the sequence of tangent spaces TxnWjconverge to some space T in appropriate Grassmannian bundle and the sequence of lines xnqnconverge to some line l then one has l ⊂ T.

We say that the pair (Wi,Wj) satisfies the Whitney (b) condition along Wi if it satisfies the Whitney (b) condition along any point x ∈ Wi.

Definition 1.2.4. 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 (Wi,Wj) of strata of W such that Wi⊂ Wj, it satisfies Whitney (a) condition along Wi(respectively, Whitney (b) condition along Wi).

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 1.2.5. [GLPW_{, p.12] Let W1, . . . , Wj} be Whitney stratifications of subsets V1, . . . ,Vjof smooth manifolds. The product stratification W1× · · · × Wj is a Whitney stratification of V1× · · · ×Vj.

Theorem 1.2.6. [GLPW, p.14] Let G: M → N be the smooth map transverse to a stratification W0of a subset V0_{of N, and let W the induced stratification on G}−1(V0). If W0is 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.

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1.2.1.2 Semianalytic Whitney stratifications

In this section, K denote either R or C. Let U be a open set in Km. We remind that a map G : U → Kpis 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, ifI is a set of analytic functions, then the zero locus ofI , is defined by V(I ) := {x ∈ Kn| G(x) = 0, ∀G ∈I }. When I = {G1, . . . , Gj} we will write V (G1, . . . , Gj) instead of V (I ). The definitions and results of this sections can be found in [ML, Section 5].

Definition 1.2.7. An analytic set X is a closed set in Kn such that for all x ∈ X , there exists an open neighborhood W of x ∈ Knand a finite collection G1, . . . , Gj of analytic functions such that V (G1, . . . , Gj) = W ∩ X .

Next we consider the definition of semianalytic set, which will be more frequent in this work.

Definition 1.2.8. A subset X ⊂ Rmis 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(G1, . . . , Gk) ∩ {x ∈ W | Hj(x) > 0, j = 1, . . . , l} where G1, . . . , Gk, H1, . . . , Hlare analytic functions on W .

Definition 1.2.9. A subset X ∈ Cnis 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(G₁, . . . , G_k) ∩ {x ∈ W | H_j(x) 6= 0, j = 1, . . . , l}

where G1, . . . , Gk, H1, . . . , Hlare holomorphic functions on W . In this case, the subset X has classically been called an analytically constructible subset.

Remark 1.2.10. Note that, by negating functions, the definition of a semianalytic subset can also contain Hα(x) < 0 and so, after taking unions, we can also obtain Hα(x) 6= 0, H_α(x) ≤ 0 and H_α(x) ≥ 0.

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Theorem 1.2.11. [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 1.2.12 (Curve Selection Lemma). Let X be a semianalytic subset of Rmand x_{∈ X. Then there exists a real analytic curve γ : [0, δ ) → R}mwith γ(0) = x and γ(t) ∈ X for any t ∈ (0, δ ).

Let X be a semianalytic subset of Cn and z∈ X. Then there exists a complex analytic curve γ : Dη → Cnwith γ(0) = z and γ(t) ∈ X for any t ∈ Dη\ {0}.

An interesting application of the Curve Selection Lemma is the following propo-sition, which say that analytic functions have isolated critical values

Corollary 1.2.13. Let G : U → K be an analytic function, where U is a open set of Km with0 ∈ U a critical point. Then G(0) is a isolated critical value.

Definition 1.2.14. Let X be a subset of Rm. A semianalytic Whitney stratification (re-spectively, semianalytic stratification) W = {Wα}α ∈A of X is a Whitney stratification (respectively, stratification) of X such that each stratum Wiis an analytic submanifold of Rmand a connected semianalytic subset of Rm.

Theorem 1.2.15. [W, Theorem 19.2] Any semianalytic set has a Whitney stratification. Any stratification has a refinement which is a Whitney stratificaton.

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Remark 1.2.16. This Follows from the proof of Theorem1.2.15that any semianalytic set has a semianalytic Whitney stratification.

Theorem 1.2.17. [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 Sm−1_ε 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 Section5.1.

Let G : (Rm, 0) → (Rp, 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 target1of 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|: Wα → Sβ is a submersion, where Wα ∈ W, and Sβ ∈ S. Definition 1.2.18. We call such pair (W, S) a regular stratification of the map germ G.

1.2.2 Thom regularity

Let G : (Rm, 0) → (Rp, 0) be an analytic map germ and W a Whitney stratifica-tions of G. Let W_α and W_β be strata of W, for which Wα ⊂ W_β and the restrictions G_|

Wα and G_|_W

β

have constant ranks. Let x ∈ Wα.

The two following definitions can be found in [GLPW] and [Mat].

Definition 1.2.19. We say that W_β is Thom regular over Wα at x relative to Gor, equiv-alently, the pair (W_β,Wα) satisfies the Thom aG-condition at x, if the condition below holds:

(∗) let {xn} ⊂ Wβ sequence, such that xnconverge to x. If ker dxn(G|W β

) converge to a plane T, in the appropriate Grassmann bundle, then ker dx(G|_Wα) ⊂ T.

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We will say that W_β is Thom regular over Wα relative to Gor, equivalently, the pair (W_β,Wα) satisfies the Thom aG-condition, when the condition (∗) is satisfied for any point x ∈ Wα.

Definition 1.2.20. Let G : (Rm, 0) → (Rp, 0) be an analytic map germ and consider a regular stratification (W, S) of the map G as in Definition1.2.18. 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 Wα,Wβ ∈ W, for which Wα ⊂ Wβ, Wβ is Thom regular over Wα 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 1.2.21. Let G : (Rm, 0) → (Rp, 0) be an analytic map germ. We say that G is Thom regular at VG(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 Definition1.2.18

such that 0 is a point stratum in S, that VG is a union of strata of W, and the Thom aG-condition is satisfied at any stratum of VG.

Let G : (Rm, 0) → (Rp, 0) be an analytic map germ such that Sing G ⊂ V_G as germ of set. Consider a stratification W := {Wα}_{α ∈A}of VGsuch that in a neighborhood U of the origin, W0:= {U \VG} ∪ {Wα∩U}α ∈Ais a Whitney stratification.

Remark 1.2.22. Let G : (Rm, 0) → (Rp, 0) be an analytic map germ such that Sing G ⊂ VG as germ of set. If there exists a Whitney stratification W0 as above such that G is Thom regular at VG, (i.e., the pair (U \VG,Wα) satisfies the Thom aG-condition for any stratum Wα), we will say that G is Thom regular at VG 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 1.2.23. [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.

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In the case of hypersurfaces in Cn+1, Lê D˜ung Tráng and H. Hamm have proved this result in [HL, Theorem 1.2.1 p.322] using the Łojasiewicz inequality, which we shall remind in the following.

Definition 1.2.24. [LZ, Łojasiewicz inequality] Let f : U → C be a holomorphic function, where U ⊂ Cn is a open set such that 0 ∈ U and f (0) = 0. We say that f satisfies the Łojasiewicz inequalityat the origin if, there exists a neighborhood W of 0 in U such that for any z ∈ W , one has

ck f (z)kθ _{≤ k ∇ f (z)k} for some θ ∈ (0, 1) and some c > 0.

Remark 1.2.25. It is well known that all holomorphic function satisfies the Łojasiewicz inequality.

Unfortunately, for real analytic map germs G : (Rm, 0) → (Rp, 0), the Thom regularity at VGis 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 VG.

The next result, which is a particular case of Proposition4.3.2in Section 4.3, shows the importance of an analytic map germ G with isolated critical value be Thom regular at VG.

Proposition 1.2.26. Let G : (Rm, 0) → (Rp, 0) be an analytic map germ with isolated critical value which is Thom regular at VG. Then there exists ε0> 0 such that, for any 0 < ε < ε0, there exists η, 0 < η ε, such that the restriction map

G_|: S_εm−1∩ G−1(B_ηp\ {0}) → B_ηp\ {0} (1.1)

is a smooth submersion.

In other words, since Sm−1_ε ∩V_G6= /0, the Thom regularity at V_Gimplies that for any ε > 0 small enough, there exists a neighborhood N_ε of S_εm−1∩V_Gin Sm−1_ε such that for any x ∈ Nε\VGone has Sm−1ε txG

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1.2.2.1 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 1.2.27. [Mat, Th2, Thom-Mather first isotopy theorem] Suppose that G: X → N is a smooth, proper, stratified submersion. Then G is a (topological) lo-cally trivial fibration, and the local trivializations can be chosen to respect the strata and to be diffeomeomorphisms when restricted to strata.

In other words, Theorem1.2.27states 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−1(U ) = π ◦ h, where π is the projection from U × G

−1_{(q) onto U .} Moreover, if S = {Sα}α ∈Ais a stratification of X , then for any stratum Sα the restriction of h is a diffeomorphism from Sα∩ G−1(U ) to U × (Sα∩ 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.

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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 ⊂ Cn+1→ C with U an open set in Cn+1_{, f (0) = 0, there exists a} small enough real number ε0> 0 such that for any 0 < ε ≤ ε0,

φ := f k f k : S

2n+1

ε \ Kε → S1 (2.1)

is a smooth projection of a locally trivial fiber bundle, where Kε = f−1(0) ∩ S2n+1ε is called the link of the singularity at the origin.

In chapters 5, 6 and 7 of [Mi], Milnor gave differentiable and topological descrip-tions of the link and the fibers Fθ = φ−1 eiθ , where eiθ ∈ S1, showing that independent of the dimension of the singular locus, the fiber is a (2n)-dimensional smooth paralleliz-able manifold with the homotopy type of a n-dimensional CW-complex.

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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 k∇ f k : S 2n+1 ε → S 2n+1 ε .

In this case it was shown that the fiber F_θ has the same homotopy type of a bouquet of n-dimensional spheresWµ ( f )

i=1 Sni, with µ( f ) spheres on the bouquet.

In 1976, Lê D˜ung Tráng in his article [Le] proved the existence of a general fibra-tion structure on a complex analytic sets. We explain below an idea of such construcfibra-tion as described in the paper [CA].

Let X be an analytic set in an open neighbourhood U of the origin 0 ∈ Cn+1. Let f : (X , 0) → (C, 0) be a germ of holomorphic function.

Theorem 2.1.1. [Le, Milnor-Lê Fibration] For any small enough ε > 0, there exists η, with0 < η ε, such that

f_|: B2n+2_ε ∩ X ∩ f−1(Dη\ {0}) → Dη\ {0} (2.2) is a topological locally trivial fibration.

Proof. (Idea) Let W be a Whitney stratification of X and small enough ε > 0 such that the closed ball B2n+2_ε intersects only a finite number of strata of X and such that the sphere S2n+1_ε , boundary of B2n+2_ε , intersects all such strata transversally. According to Theorem1.2.23, we can always choose this stratification in such way f is Thom regular at Vf. By Proposition1.2.26, this implies that for 0 < η ε the fibers of the map (2.2) intersect transversally the strata of X ∩ S2n+1_ε and thus it is a stratified submersion. Now the result follow from Theorem1.2.27.

An important point to notice here is that this topological fibration structure becomes a smooth fibration if X \Vf is a non-singular analytic set in Cn+1. See details in [Le,Ham].

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2.1. Complex settings 45

Corollary 2.1.2. [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 any0 < δ ε, the map

f_|: B2n+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_|: B2n+2_ε ∩ f−1 S1_η → S1_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 : (Cn+p, 0) → (Cp, 0) that is an ICIS - Isolated Complete Intersection Singularity, Hamm proved the following result.

Theorem 2.1.3. [Ham] Let G:= (G1, . . . , Gp) : (Cn+p, 0) → (Cp, 0), p ≥ 1, be an ICIS at0. Then,

G_|: B2(n+p)_ε ∩ G−1(B2p_η \ Disc G) → B2p_η \ Disc G is a locally trivial smooth fibration.

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 hypersur-face of Cp. Hence, it does not disconnect the complement B2pη \ 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 spheresWµ

i=1Sni where now µ := rank H_{n(F, Z), the rank of the homology in the middle dimension with integer} coefficient.

One of the richest sources of information on ICIS is Looijenga’s classical book [Lo].

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2.2 Real settings

2.2.1 Isolated Singular Case: tube fibration

Given a representative of G : (Rm, 0) → (Rp, 0), m > p ≥ 2, Milnor proved that if G has isolated critical point at the origin 0 ∈ Rm, then for any small enough ε > 0, there exists η, 0 < η ε, such that the restriction map

G_|: Bm_ε ∩ G−1(S_ηp−1) → S_ηp−1 (2.5)

is a smooth projection of a locally trivial fiber bundle. More precisely, it was proved that:

Theorem 2.2.1. [Mi, Theorem 11.2, p. 97] Let G: (Rm, 0) → (Rp, 0) be a real analytic map germ such that Sing G= {0} as germs of an analytic set. Then, there exists ε0> 0 such that, for each ε, 0 < ε ≤ ε0, there exists η, 0 < η ε, such that the complement of an open tubular neighborhood of (the link) K_ε = VG∩ Sm−1ε is the total space of a smooth fiber bundle over the sphere S_ηp−1. Each fiber F is a smooth compact(n − p)-dimensional manifold bounded by a copy of K_ε.

If the link Kε 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 Kε is empty then the manifold B

m ε ∩ G

−1_(Sp−1

η ) is dif-feomorphic to the sphere Sm−1_ε 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 Bm_ε ∩ G−1(S_ηp−1) is as in the Figure1 below, in the case the link Kε is not empty for any small enough ε. The boundary manifold Bm_ε ∩ G−1(S_ηp−1) looks like a “tube” surrounding the special fiber VG. For this reason several author called this space “the Milnor tube”. From now on, we will denote the Milnor tube by Mε ,η ,G.

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2.2. Real settings 47

Figure 1 – The Milnor tube for G(x, y, z) = (x, y(x2+ y2+ z2)).

2.2.2 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 Sm−1_ε \ K_ε also fibers on S_ηp−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 Theorem2.2.1. We will split the proof in some steps and later, in section6.1, we will use these steps again to extend some results of this section.

Denote by G : U → Rp, 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_|: Sm−1_ε ∩ G−1(B_ηp) → B_ηp. (2.6) Therefore, by Ehresmann theorem for manifold with boundary one gets that the restriction

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map

G_|: Mε ,η ,G→ S p−1

η (2.7)

is a locally trivial smooth fibration.

Denote by ρ(x) = kxk2the square of distance function to the origin and g(x) = kG(x)k2. Since Sing G = {0} the vector ∇g(x) is not null on Bm_ε \V_G. Moreover, one can use the Curve Selection Lemma to show that on each point of Bm_ε \V_Gthe vectors ∇g(x) and ∇ρ do not point in opposite direction. Therefore, one can always construct a non-singular smooth bisector vector fields

ν (x) = ∇g(x) k∇g(x)k+

∇ρ (x)

k∇ρ(x)k (2.8)

which satisfies the properties: (i) hv(x), ∇ρ(x)i > 0; (ii) hv(x), ∇g(x)i > 0.

By construction one sees that its flow inflates the Milnor tube M_{ε ,η ,G}to a compact smooth manifold S_εm−1\ G−1(B_ηp) on the sphere, keeping the boundary of the tube ∂ (Mε ,η ,G) pointwise fixed. This flow produces a smooth diffeomorphism

ξ : Mε ,η ,G→ Sm−1ε \ G −1_(Bp

η) which restricts to the identity map on the boundary ∂ (Mε ,η ,G). Step 2: The composition map

δ := G ◦ ξ−1: Sm−1_ε \ G−1(B_ηp) → S_ηp−1 (2.9) is also a locally trivial smooth fibration.

By restriction, the fibration (2.6) give rise to the fibrations G_|: S_εm−1∩ G−1(B_ηp\ {0}) → B_ηp\ {0} and

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which can be composed, respectively, with the projection s/ksk : Rp\ {0} → Sp−1 providing the following locally trivial smooth fibrations

G kGk_|: S m−1 ε ∩ G −1_(Bp η\ {0}) → S p−1_, _(2.10) and G kGk_|: ∂ (Mε ,η ,G) → S p−1 . (2.11)

After composing (2.9) with the s/ksk_| S_ηp−1

one gets the fibration

δ /kδ k : Sm−1_ε \ G−1(B_ηp) → Sp−1. (2.12) Since ξ restricts to identity map on ∂ (Mε ,η ,G), hence one has that δ /kδ k|_{∂ (Mε,η,G)} is exactly the map (2.11).

Finally, since both fibrations (2.10) and (2.12) agree along its common bound-aries, it can be glued nicely1to get a locally trivial smooth fibration

Sm−1_ε \ Kε → Sp−1, (2.13)

which is independent of ε > 0 up to diffeomorphism type.

2.2.2.1 Milnor’s example

Milnor also noted that in general the map projection of the fibration (2.13) fails to be the canonical map G/kGk on the whole space S_εm−1\ Kε, like in the above cited case of holomorphic function germs. Actually, in [Mi, p. 99], Milnor considered the application G:= (G1, G2) : (R2, 0) → (R2, 0) given by G(x, y) = (x, x2+ y(x2+ y2)) which satisfies Sing G = VG= {0} and consequently has an isolated singular point. By Theorem2.2.1 the tube fibration exists. However, the map G/kGk cannot be the projection of a locally trivial smooth fibration on S1_ε, because it is not a submersion for ε small enough.

In fact, considering v := (x, y) and the matrix

A(v) = G1(v)∇G2(v) − G2(v)∇G1(v) v

!

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one can see that there exists a non-degenerate curve C of critical points of the map G/kGk : S_ε1→ S1_{, 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/kGk. Consequently, c.f. Defini-tion2.2.7, the map G is not ρ-regular and this is precisely the reason of the map G/kGk fails to be the projection of a locally trivial smooth fibration.

Figure 2 – Curve of critical points of G/kGk restricted to the sphere

Remark 2.2.2. The phenomenon described above in the Milnor example can be repro-duced in higher dimensions considering the isolated singularity map G : (Rm+2, 0) → (R2, 0) given by G(x, y, z1, . . . , zm) = (x, x2+ y(x2+ y2+ z2₁+ · · · + z2m)).

2.2.3 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 ⊂ Rman open subset such that 0 ∈ U and let ρ : U → R≥0be a non-negative proper function which defines the origin.

Definition 2.2.3. Let G : (Rm, 0) → (Rp, 0) be an analytic map germ. We denote by M_ρ_{(G) := {x ∈ U | ρ 6tx}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

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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 2.2.3that the Milnor set Mρ(G) is the set of points x∈ U such that the vectors {∇ρ(x), ∇G1(x), . . . , ∇Gp(x)} are linearly dependent over R, i.e., Mρ(G) is the singular locus Sing (G, ρ) of the pair of map (G, ρ) : U → Rp+1. Hence, the singular set Sing G ⊂ Mρ(G) .

For the sake of simplicity, in what follows we will consider ρ as the Euclidean distance function to the origin ρ(x) = kxk2, and we write M(G) := Mρ(G) for short. However, all results carry out easily over any other function ρ as considered above.

Consider the following condition:

M(G) \ VG∩VG⊆ {0} (2.14)

where the closure of the set M(G) \VGis 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. Tib˘ar 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 2.2.4below. Here we shall use the same notation of [ACT1].

Theorem 2.2.4. [Ma, Existence of the (full) Milnor’s tube fibration] Let G: U → Rp 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 ε0> 0 such that, for each ε, 0 < ε ≤ ε0, there exists η, 0 < η ε, such that

G_|: Bm_ε ∩ G−1(B_ηp\ {0}) → B_ηp\ {0} (2.15)

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Corollary 2.2.5. [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_|: Bm_ε ∩ G−1(S_ηp−1) → S_ηp−1 is the projection of a locally trivial smooth fibration.

In this case we also denote Mε ,η ,G= B m ε ∩ G

−1_(Sp−1

η ) and also call it the Milnor tube.

Example 2.2.6. Let G : (R3, 0) → (R2, 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 = " x2+ y2 yz yz x2+ z2 #

where JG(v) and [JG(v)]tdenote the Jacobian matrix of G in v and its transpose, respec-tively. 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 ∈ R3| det (B(v)) = 0}. Consequently, M(G) = {x = 0} ∪ {x2− y2− z2= 0}.

We claim that G satisfies the condition (2.14).

Indeed, let p0= (x0, y0, z0) ∈ M(G) \ VG∩ VG. There exists a sequence pn:= (xn, yn, zn) ∈ M(G) \ VG such that pn → p0 and xn2= y2n+ z2n with y2n+ z2n6= 0. Since

p0∈ VG, we need to consider the following two cases: Case 1: p0= (0, y0, z0). Hence

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Case 2: p0= (x0, 0, 0). Hence

x2₀= lim x2_n= lim(y2_n+ z2_n) = y2₀+ z2₀= 0.

In any case we get that p0= 0 and G satisfies the condition (2.14). Therefore, by Theorem

2.2.4Ghas a Milnor tube fibration.

In the Figure 3 below one can see that the Milnor tube Mε ,η ,G consist of two connected components. Compare with Figure1.

Figure 3 – The Milnor tube for G(x, y, z) = (xy, xz).

2.2.4 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].

In [AT1,AT2,ACT1] the authors generalized all previous results as we describe below.

In order to explain the main results, denote the set of ρ-nonregular points of the map

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as the set

M(Ψ) = {x ∈ U \VG| ρ 6txΨ} .

Definition 2.2.7. 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 2.2.8. [AT1,AT2,ACT1] Let G_{: (R}m_{, 0) → (R}p, 0) be an analytic map germ. Then, on the open set{G₁6= 0} one has that

M(Ψ) =            x∈ U \VG| rank       Ω2(x) .. . Ωp(x) ∇ρ (x)       < p            ,

where Ωk= G1∇Gk− Gk∇G1, for any k = 2, . . . , p.

Remark 2.2.9. We notice that for any x /∈ V_Gif ρ txGthen ρ txΨ. Hence, M(Ψ) ⊂ M(G) \ V_G.

The vectors Ω2(x), . . . , Ωp(x) are the generators of the normal space of the fibers X_y= Ψ−1(y), y = Ψ(x). Hence, Lemma2.2.8does not depend on the particular choice of the open set {G1(x) 6= 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 : Sm−1_ε \ Kε→ Sp−1is 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 2.2.10. [ACT1, Theorem 1.3] Let G: U → Rp, 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 ≤ ε0, the map projection

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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 : (Rm, 0) → (R2, 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 : (Rm, 0) → (Rp, 0) with m> p ≥ 2, and with that, they were able to prove the following results:

Theorem 2.2.11. [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 and0 < η ε, the map:

G_|: Sm−1_ε ∩ G−1(B_ηp\ {0}) → B_ηp\ {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 2.2.12. [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 ε0> 0 such that the map (2.16) is the projection of a locally trivial smooth fibration, for any0 < ε ≤ ε0.

(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 4.1.4in Chapter 4. On the other hand, as the proof of Proposition 4.1.7

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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 Theorem2.2.10represents a simultaneous extension of Theorem2.2.11and of Theorem2.2.12. Moreover, it turns out that the condition (2.14) can not be removed from Theorem 2.2.10, hence it is sharp! In fact, in his master thesis [Han] Hansen presented the example G(x, y, z) = (x2+ y2, (x2+ y2)z), showing that Sing G ⊂ V_Gand M (Ψ) = /0, but the topological type of the fibers of G changes along S1. Beside that, by hand calculation one gets that M_{(G) = R}3, M (G) \VG∩VG= VG6= {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 Chapter3of the thesis) and proved the following result.

Theorem 2.2.13. [ACT1, Theorem 1.4] Let f _{: C}n_{→ C be a non-constant mixed} poly-nomial which is polar weighted-homogeneous, n≥ 2, such that codim_RV_f = 2. Then for any ε, 0 < ε ≤ ε0, the map projection

f/k f k : S2n−1_ε \ Kε → S1

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 2.2.14. [ACT1, Theorem 2.1] Let G: U → Rp, m> p ≥ 2 be an analytic map such that codimV_G= p, Sing G ⊂ VG which satisfies the condition (2.14). Then there exists a locally trivial smooth fibration

Sm−1_ε \ K_ε → Sp−1

which is independent of small enough ε > 0, up to isotopies.

2.2.4.1 Revising the sphere fibration for holomorphic functions

Let f : Cn+1, 0 → (C,0) be a germ of holomorphic function. Let us see that the hypothesis of the previous Theorem2.2.10are naturally satisfied if we consider f as

UNIVERSIDADE DE SÃO PAULO

UNIVERSID

ADE DE SÃ

O P

AUL

O

Institut

o de C

iências M

at

emá

ticas e de C

omputação

Maico Felipe Silva Ribeiro

Singular Milnor fibrations

Maico Felipe Silva Ribeiro

Maico Felipe Silva Ribeiro

Fibrações de Milnor singulares

ACKNOWLEDGEMENTS

ABSTRACT

RESUMO

LIST OF FIGURES

LIST OF SYMBOLS

CONTENTS

INTRODUCTION

1

PRELIMINARIES

1.1

Differential Topology

1.1.1

Fiber bundles

1.1.2

Ehresmann Fibration Theorem

1.2

Singularity theory

1.2.1

Stratification

1.2.2

Thom regularity

2

CLASSICAL MILNOR FIBRATIONS

2.1

Complex settings

2.2

Real settings

2.2.1

Isolated Singular Case: tube fibration

2.2.2

Sphere fibration: Milnor’s method

2.2.3

Non-isolated singular case: tube fibration

2.2.4

Existence of the Sphere fibration