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Chapter VII Real Singularities with a Milnor Fibration Progress in Mathematics Volume 241 Series Editors H. Bass J. Oesterlé A. Weinstein José Seade Andrea Malchiodi On the Topology of Isolated Singularities in Analytic Spaces Birkhäuser Verlag Basel Boston Berlin Author: José Seade Instituto de Matemáticas Unidad Cuernavaca Universidad Nacional Autónoma de México (UNAM) Av. Universidad s/n, col. Lomas de Chamilpa, Ciudad Universitaria C.P. 62210 Cuernavaca, Morelos México e-mail: [email protected] 2000 Mathematics Subject Classifi cation Primary 14Bxx, 32Sxx, 53Cxx, 55A25, 57C45; Secondary 11Fxx, 32L30, 34Mxx, 37F75 A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografi e; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. ISBN 3-7643-7322-9 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifi cally the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfi lms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2006 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced of chlorine-free pulp. TCF ∞ Printed in Germany ISBN-10: 3-7643-7322-9 e-ISBN: 3-7643-7395-4 ISBN-13: 978-3-7643-7322-1 9 8 7 6 5 4 3 2 1 www.birkhauser.ch To Swami Chidvilasananda and the Siddha lineage, whose light turned my life into a joyful path; and to Teresa, whose love is my everything. Preface Singularity theory stands at a cross-road of mathematics, a meeting point where many areas of mathematics come together, such as geometry, topology and algebra, analysis, differential equations and dynamical systems, combinatorics and number theory, to mention some of them. Thus, one who would write a book about this fascinating topic necessarily faces the challenge of having to choose what to include and, most difficult, what not to include. A comprehensive treatment of singularities would have to consist of a collection of books, which would be beyond our present scope. Hence this work does not pretend to be comprehensive of the subject, neither is it a text book with a systematic approach to singularity theory as a core idea. This is rather a collection of essays on selected topics about the topology and geometry of real and complex analytic spaces around their isolated singularities. I have worked in the area of singularities since the late 1970s, and during this time have had the good fortune of encountering many gems of mathematics concerning the topology of singularities and related topics, masterpieces created by great mathematicians like Riemann, Klein and Poincar´e, then Milnor, Hirzebruch, Thom, Mumford, Brieskorn, Atiyah, Arnold, Wall, LˆeD˜ung Tr´ang, Neumann, Looijenga, Teissier, and many more whose names I cannot include since the list would be too long and, even that, I would leave aside important names. My own research has always stood on the shoulders of all of them. In taking this broad approach I realize how difficult it is to present an overall picture of the myriad of outstanding contributions in this area of mathematics during the last century, since they are scattered in very many books and research articles. This work is a step in that direction and our first main purpose is to put together some of these gems of current mathematics in an accessible way, indicating in all cases present lines of research and appropriate references to the literature. This step occupies Chapters I to V of this monograph. I have made some contributions to the subject, either by myself or in collaboration with other colleagues, and I think they are interesting enough to be included here. On the other hand, the theory of real singularities is much less developed than that of complex singularities, for various reasons. There are however interesting families of real analytic singularities having remarkable geometric and topological properties similar to those of complex singularities. These are related to complex differential equations and defined via complex geometry, and the second main xiv Preface purpose of this monograph is to explain some aspects of their geometry and sim- ilarities with complex singularities. I believe the study of these singularities in Chapters VI–VIII and the picture of complex singularities presented in Chapters I–V, together provide interesting new insights into the geometry and topology of singularities. This study also gives new methods for constructing smooth differen- tiable manifolds equipped with a rich geometry and defined by algebraic equations, which can be of interest on its own. For instance, at the end of Chapter VI we obtain explicit algebraic equations that provide analytic embeddings of oriented surfaces of all genera in the sphere S3; this can be useful for problems in differential geometry and physics. The material that I present here is mostly contained in the literature but, as said before, it is spread over many different research papers and books, making it difficult to put together a complete picture. In some cases I have given simple proofs of well-known theorems; in other cases I have included the complete proofs in detail; yet in others I have preferred to content myself by giving only some of the key ideas to understand some specific theory. In all cases I have included appropriate references and indications for further reading. I have also added a number of open problems as extension of the text. This monograph is constructed like a fan, where each individual chapter is to some degree independent of the others and can be read on its own, and yet as a unity, spreads out to give the reader a taste of the richness of this fascinating topic. I am sure that everyone who reads it, regardless of whether he or she is a graduate student, an expert in singularity theory, or a researcher in any other field of geometry and topology, will find interesting things in it. Contents Preface ...................................... xiii Introduction ................................... 1 Acknowledgments ................................ 9 I A Fast Trip Through the Classical Theory I.1 An example: the Pham-Brieskorn polynomials . 11 I.2 Thelocalconicalstructure.................... 14 I.3 Ehresmann’sfibrationlemma................... 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. TheMilnornumber........................ 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. Thepolyhedralgroups...................... 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.Thehyperboliccase.................. 57 II.7 Brieskorn-Hamm complete intersections. ThetheoremofNeumann..................... 60 II.8 Remarks.............................. 62 x Contents III 3-dimensional Lie Groups and Surface Singularities III.1 Quasi-Homogeneoussurfacesingularities............ 65 III.2 3-manifolds whose universal covering is a Lie group . 71 III.3 Lie groups and singularities I: quasi-homogeneoussingularities................. 75 III.4 LiegroupsandsingularitiesII:thecusps............ 80 III.5 Lie groups and singularities III: the Abelian and E+(2)-cases................... 82 III.6 A uniform picture of 3-dimensional Lie groups . 85 III.7 LiealgebrasandtheGorensteinproperty............ 87 III.8 Remarks.............................. 88 IV Within the Realm of the General Index Theorem IV.1 Areviewofcharacteristicclasses................. 92 IV.2 On Hirzebruch’s theorems about the signature andRiemann-Roch........................ 97 IV.3 Spin and Spinc structures on 4-manifolds. Rochlin’stheorem......................... 102 IV.4 Spin and Spinc structures on complex surfaces. Rochlin’stheorem......................... 105 IV.5 Areviewofsurfacesingularities................. 110 IV.6 Gorenstein and numerically Gorenstein singularities . 116 IV.7 An application of Riemann-Roch: Laufer’s formula . 121 IV.8 Geometric genus, spinc structures and characteristicdivisors....................... 126 IV.9 On the signature of smoothings of surface singularities . 128 IV.10 On the Rochlin µ invariant for links ofsurfacesingularities....................... 131 IV.11 Comments on new 3-manifolds invariants andsurfacesingularities...................... 134 V On the Geometry and Topology of Quadrics in CPn V.1 The topology of a quadric in CPn ................ 138 V.2 The space CPn as a double mapping cylinder . 141 V.3 The orthogonal group SO(n +1, R)and the geometry of CPn ....................... 143 V.4 Cohomogeneity 1-actions of SO(3) on CP2 and S4 ....... 148 V.5 TheArnold-Kuiper-Masseytheorem............... 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 L´opez de Medrano-Verjovsky-Meersseman
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