On the topology of isolated singularities in analytic spaces / José Seade.

Format:

Book

Author/Creator:

Seade, J. (José)

Series:

Progress in mathematics (Boston, Mass.) ; v. 241.

Progress in mathematics ; v. 241

Language:

English

Subjects (All):

Singularities (Mathematics).

Topology.

Analytic spaces.

Geometry, Algebraic.

Physical Description:

xiv, 238 pages : illustrations ; 24 cm.

Place of Publication:

Basel ; Boston : Birkhäuser Verlag, [2006]

Summary:

The aim of this book is to give an overview of selected topics on the topology of real and complex isolated singularities, with emphasis on its relations to other branches of geometry and topology. The first chapters are mostly devoted to complex singularities and a myriad of results spread in a vast literature, which are presented here in a unified way, accessible to non-specialists. Among the topics are the fibration theorems of Milnor; the relation with 3-dimensional Lie groups; exotic spheres; spin structures and 3-manifold invariants; the geometry of quadrics and Arnold's theorem which states that the complex projective plane modulo conjugation is the 4-sphere.

The second part of the book studies pioneer work about real analytic singularities which arise from the topological and geometric study of holomorphic vector fields and foliations. In the low dimensional case these turn out to be related to fibred links in the 3-sphere defined by meromorphic functions. This provides new methods for constructing manifolds equipped with a rich geometry. The book is largely self-contained and serves a wide audience of graduate students, mathematicians and researchers interested in geometry and topology.

Contents:

I A Fast Trip Through the Classical Theory

I.1 An example: The Pham-Brieskorn polynomials 11

I.2 The local conical structure 14

I.3 Ehresmann's fibration lemma 17

I.4 Milnor's fibration theorem for real singularities 18

I.5 Open book decompositions and fibred knots 21

I.6 On Milnor's fibration theorem for complex singularities 22

I.7 The join of Pham and the topology of the Milnor fibre. The Milnor number 25

I.8 Exotic spheres and the topology of the link 30

II Motions in Plane Geometry and the 3-dimensional Brieskorn Manifolds

II.1 Groups of motions in the 2-sphere. The polyhedral groups 36

II.2 Triangle groups and the classical plane geometries 42

II.3 The 3-sphere as a Lie group and its finite subgroups 47

II.4 Brieskorn manifolds and Klein's theorem 50

II.5 The group PSL(2, R) and its universal cover SL(2, R) 55

II.6 Milnor's theorem for the 3-dimensional Brieskorn manifolds. The hyperbolic case 57

II.7 Brieskorn-Hamm complete intersections. The theorem of Neumann 60

II.8 Remarks 62

III 3-dimensional Lie Groups and Surface Singularities

III.1 Quasi-Homogeneous surface singularities 65

III.2 3-manifolds whose universal covering is a Lie group 71

III.3 Lie groups and singularities I: quasi-homogeneous singularities 75

III.4 Lie groups and singularities II: the cusps 80

III.5 Lie groups and singularities III: the Abelian and E[superscript +] (2)-cases 82

III.6 A uniform picture of 3-dimensional Lie groups 85

III.7 Lie algebras and the Gorenstein property 87

III.8 Remarks 88

IV Within the Realm of the General Index Theorem

IV.1 A review of characteristic classes 92

IV.2 On Hirzebruch's theorems about the signature and Riemann-Roch 97

IV.3 Spin and Spin[superscript c] structures on 4-manifolds. Rochlin's theorem 102

IV.4 Spin and Spin[superscript c] structures on complex surfaces. Rochlin's theorem 105

IV.5 A review of surface singularities 110

IV.6 Gorenstein and numerically Gorenstein singularities 116

IV.7 An application of Riemann-Roch: Laufer's formula 121

IV.8 Geometric genus, spin[superscript c] structures and characteristic divisors 126

IV.9 On the signature of smoothings of surface singularities 128

IV.10 On the Rochlin [mu] invariant for links of surface singularities 131

IV.11 Comments on new 3-manifolds invariants and surface singularities 134

V On the Geometry and Topology of Quadrics in CP[superscript n]

V.1 The topology of a quadric in CP[superscript n] 138

V.2 The space CP[superscript n] as a double mapping cylinder 141

V.3 The orthogonal group SO(n + 1,R) and the geometry of CP[superscript n] 143

V.4 Cohomogeneity 1-actions of SO(3) on CP[superscript 2] and S[superscript 4] 148

V.5 The Arnold-Kuiper-Massey theorem 150

VI Real Singularities and Complex Geometry

VI.1 The space of Siegel leaves of a linear flow 157

VI.2 Real singularities and the Lopez de Medrano-Verjovsky-Meersseman manifolds 163

VI.3 Real singularities and holomorphic vector fields 167

VI.4 On the topology of certain real hypersurface singularities 170

VII Real Singularities with a Milnor Fibration

VII.1 Milnor's fibration theorem revisited 175

VII.2 The strong Milnor condition 177

VII.3 Real singularities of the Pham-Brieskorn type 181

VII.4 Twisted Pham-Brieskorn singularities and the strong Milnor condition 187

VII.5 On the topology of the twisted Pham-Brieskorn singularities 191

VII.6 Stability of the Milnor conditions under perturbations 194

VII.7 Remarks and open problems 196

VIII Real Singularities and Open Book Decompositions of the 3-sphere

VIII.1 On the resolution of embedded complex plane curves 200

VIII.2 The resolution and Seifert graphs 204

VIII.3 Seifert links and horizontal fibrations 206

VIII.5 Resolution and topology of the singularities z[superscript p subscript 1]z[subscript 2] + z[superscript q subscript 2]z[subscript 1] = 0 213

VIII.6 On singularities of the form fg 216.

Notes:

Includes bibliographical references (pages [221]-233) and index.

ISBN:

3764373229

OCLC:

62325145

Publisher Number:

9783764373221