On the Topology of Isolated Singularities in Analytic Spaces

by Seade, Jose

• ISBN13: 9783764373221
• ISBN10: 3764373229
• Format: Hardcover
• Copyright: 2006-01-01
• Publisher: Birkhauser

Summary

This book has been awarded the Ferran Sunyer i Balaguer 2005 prize. 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.

Table of Contents

Preface	xiii
Introduction	1	(8)
Acknowledgments	9	(2)
A Fast Trip Through the Classical Theory
An example: the Pham-Brieskorn polynomials	11	(3)
The local conical structure	14	(3)
Ehresmann's fibration lemma	17	(1)
Milnor's fibration theorem for real singularities	18	(3)
Open book decompositions and fibred knots	21	(1)
On Milnor's fibration theorem for complex singularities	22	(3)
The join of Pham and the topology of the Milnor fibre. The Milnor number	25	(5)
Exotic spheres and the topology of the link	30	(6)
Motions in Plane Geometry and the 3-dimensional Brieskorn Manifolds
Groups of motions in the 2-sphere. The polyhedral groups	36	(6)
Triangle groups and the classical plane geometries	42	(5)
The 3-sphere as a Lie group and its finite subgroups	47	(3)
Brieskorn manifolds and Klein's theorem	50	(5)
The group PSL (2, R) and its universal cover SL (2, R)	55	(2)
Milnor's theorem for the 3-dimensional Brieskorn manifolds. The hyperbolic case	57	(3)
Brieskorn-Hamm complete intersections. The theorem of Neumann	60	(2)
Remarks	62	(3)
3-dimensional Lie Groups and Surface Singularities
Quasi-Homogeneous surface singularities	65	(6)
3-manifolds whose universal covering is a Lie group	71	(4)
Lie groups and singularities I: quasi-homogeneous singularities	75	(5)
Lie groups and singularities II: the cusps	80	(2)
Lie groups and singularities III: the Abelian and E+(2)-cases	82	(3)
A uniform picture of 3-dimensional Lie groups	85	(2)
Lie algebras and the Gorenstein property	87	(1)
Remarks	88	(4)
Within the Realm of the General Index Theorem
A review of characteristic classes	92	(5)
On Hirzebruch's theorems about the signature and Riemann-Roch	97	(5)
Spin and Spinc structures on 4-manifolds. Rochlin's theorem	102	(3)
Spin and Spinc structures on complex surfaces. Rochlin's theorem	105	(5)
A review of surface singularities	110	(6)
Gorenstein and numerically Gorenstein singularities	116	(5)
An application of Riemann-Roch: Laufer's formula	121	(5)
Geometric genus, spinc structures and characteristic divisors	126	(2)
On the signature of smoothings of surface singularities	128	(3)
On the Rochlin μ invariant for links of surface singularities	131	(3)
Comments on new 3-manifolds invariants and surface singularities	134	(4)
On the Geometry and Topology of Quadrics in CPn
The topology of a quadric in CPn	138	(3)
The space CPn as a double mapping cylinder	141	(2)
The orthogonal group SO (n + 1, R) and the geometry of CPn	143	(5)
Cohomogeneity 1-actions of SO(3) on CP2 and S4	148	(2)
The Arnold-Kuiper-Massey theorem	150	(7)
Real Singularities and Complex Geometry
The space of Siegel leaves of a linear flow	157	(6)
Real singularities and the Lopez de Medrano-Verjovsky-Meersseman manifolds	163	(4)
Real singularities and holomorphic vector fields	167	(3)
On the topology of certain real hypersurface singularities	170	(5)
Real Singularities with a Milnor Fibration
Milnor's fibration theorem revisited	175	(2)
The strong Milnor condition	177	(4)
Real singularities of the Pham-Brieskorn type	181	(6)
Twisted Pham-Brieskorn singularities and the strong Milnor condition	187	(4)
On the topology of the twisted Pham-Brieskorn singularities	191	(3)
Stability of the Milnor conditions under perturbations	194	(2)
Remarks and open problems	196	(4)
Real Singularities and Open Book Decompositions of the 3-sphere
On the resolution of embedded complex plane curves	200	(4)
The resolution and Seifert graphs	204	(2)
Seifert links and horizontal fibrations	206	(3)
An example	209	(4)
Resolution and topology of the singularities zp1z2 + zq2z1 = 0	213	(3)
On singularities of the form fg	216	(5)
Bibliography	221	(14)
Index	235