CONTEMPORARY MATHEMATICS

354

Real and Complex Singularities

Proceedings of the Seventh International Workshop on Real and Complex Singularities

July 29-August 2, 2002 ICMC-USP, Sao Carlos, Brazil

Terence Gaffney Maria Aparecida Soares Ruas

Editors

CoNTEMPORARY MATHEMATICS

354

Real and Complex Singularities

Proceedings of the Seventh International Workshop on Real and Complex Singularities

July 29-August 2, 2002 ICMC-USP, Sao Carlos, Brazil

Terence Gaffney Maria Aparecida Soares Ruas

Editors

American Mathematical Society Providence, Rhode Island

http://dx.doi.org/10.1090/conm/354

Editorial Board

Dennis DeThrck, managing editor

Andreas Blass Andy R. Magid Michael Vogelius The Seventh International Workshop on Real and Complex Singularities was held at

the Instituto de Ciencias Matematicas e de Computagao, Universidade de Sao Paulo, Sao Carlos, Brazil (ICMC-USP, Sao Carlos), from July 29 to August 2, 2002.

2000 Mathematics Subject Classification. Primary 32Sxx, 58Kxx, 37G10, 37G40, 53A05.

Library of Congress Cataloging-in-Publication Data International Workshop on Real and Complex Singularities (7th : 2002 : Instituto de Ciencias Matematicas e de Computa<;iio, Universidade de Sao Paulo)

Real and Complex Singularities : proceedings of the Seventh International Workshop on Real and Complex Singularities, July 29-August 2, 2002, ICMC-USP, Sao Carlos, Brazil / Terence Gaffney, Maria Aparecida Soares Ruas, editors.

p. em.- (Contemporary mathematics, ISSN 0271-4132; 354) Includes bibliographical references. ISBN 0-8218-3665-X (alk. paper) 1. Singularities (Mathematics)-Congresses. I. Gaffney, Terence. II. Ruas, Maria

Aparecida Soares, 1948- III. Title. IV. Contemporary mathematics (American Mathematical Society); v. 354.

QA614.58.I527 2004 516.3'5-dc22 2004040314

Copying and reprinting. Material in this book may be reproduced by any means for edu-cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg-ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math-ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permissioniDams. org.

Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.)

© 2004 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

except those granted to the United States Government. Printed in the United States of America.

§ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

Visit the AMS home page at http: I /www. ams. org/

10987654321 09 08 07 06 05 04

Dedicated to Professor Gilberto Francisco Loibel, the pioneer of Singularities of Mappings in Brazil, and the founder of the singularity group in Sao Carlos, on the occasion of his 70th birthday.

Contents

Preface vii

List of Participants IX

Zero Curves of Families of Curve Congruences 1 J. W. BRUCE, G. J. FLETCHER, AND F. TARI

Hypersurface Complements, Alexander Modules and Monodromy 19 A. DIMCA AND A. NEMETHI

Invariance of the Diagonal Contribution in a Bitangency Formula 45 D. DREIBELBIS

Bounds on Leaves of Foliations of the Plane 57 E. ESTEVES AND S. KLEIMAN

Calculation of Thorn Polynomials and Other Cohomological Obstructions for Group Actions 69

L. FEHER AND R. RIMANYI

On the Bilipschitz Triviality of Families of Real Maps 95 A. C. G. FERNANDES AND C. H. SOARES JR.

A Note on the Path Formulation for (0(2), §0(2))-Forced Symmetry Breaking Bifurcation 105

J.-E. FURTER AND A. M. SITTA

Polar Methods, Invariants of Pairs of Modules and Equisingularity 113 T. GAFFNEY

Stability of Equilibria in Equations of Hodgkin-Huxley Type 137 I. LABOURIAU AND C. M. S. G. RITO

Isolated Non-Normal Crossings 145 A. LIBGOBER

Invariants of Normal Surface Singularities 161 A. NEMETHI

Families of Pairs of Hamiltonian Vector Fields in the Plane 209 R. D. S. OLIVEIRA

Topology of Unfoldings of Singularities in the E, Z and Q Series 227 A. A. DUPLESSIS AND C. T. C. WALL

v

vi CONTENTS

Semiumbilics and Geometrical Dynamics on Surfaces in 4-Spaces M. C. ROMERO-FUSTER

On the Vanishing Cycles of a Meromorphic Function on the Complement of its Poles

D. SrERSMA AND M. TIBAR

Some Adjacencies to Cusp Singularities J. STEVENS

Elimination of Singularities by Cobordism A. Szucs

259

277

291

301

Preface

The Workshop on Real and Complex Singularities is held every other year at the Instituto de Ciencias Matematicas e de Computac;ao, of the Universidade de Sao Paulo, Sao Carlos (ICMC-USP, Sao Carlos), Brazil. Its main purpose is to bring together specialists in the vanguard of singularities and its applications. Presenters are selected for work representing important trends in the field and are encouraged to present the context for their work and useful ways of thinking about their methods and results.

This volume contains papers presented at the seventh workshop, held from the 29th of July to the 2nd of August, 2002. Consistent with Thorn's original vision of the subject, the papers represent all of the branches of singularities: equisingularity of sets and mappings, the geometry of singular complex analytic sets, singularities of mappings and their elimination, characteristic classes, applications to differential geometry and differential equations and bifurcation theory.

We are indebted to Marcelo Jose Saia, Miriam Garcia Manoel, Roberta Godoi Wik Atique, and Washington Luis Marar for their help in organizing the meeting and to the members of the scientific committee Daniel Barlet, Bill Bruce, Gert-Martin Greuel, Osamu Saeki, Dirk Siersma, and Marco Antonio Teixeira for their support. We are also very grateful to Marilia Finazzi de Andrade Marino and to all the staff of the ICMC. Without their help we could not have organized the workshop.

We gratefully acknowledge financial support from FAPESP, CNPq, CAPES, USP and IM-AGIMB.

It is a pleasure to thank the speakers and the participants. The interactions between all those present was the real success of the conference.

It is a special pleasure to acknowledge that the conference was dedicated to Professor Gilberto Francisco Loibel on the occasion of his 70th birthday.

vii

Terence Gaffney Maria Aparecida Soares Ruas

List of Participants

R. S. E. Allan ICMC-USP

R. G. W. Atique ICMC-USP

F. C. de Avila ICMC-USP

P. H. Baptistelli ICMC-USP

G. F. Barbosa ICMC-USP

E. A. Bartolo Universidad de Zaragoza

M.A. Bertolim ICMC-USP

C. Bivia-Ausina Universitat Politecnica de Valencia

J. P. Brasselet IML- CNRS

F. S. Bringas Universidad Nacional Aut6noma de Mexico

L. F. M. Brito ICMC-USP

J. W. Bruce University of Liverpool

R. Bulajich Universidad Aut6noma del Estado de Morelos

M. Buosi ICMC-USP

C. A. Buzzi UNESP- Rio Preto

ix

E. Capelato ICMC-USP

V. L. Carbone ICMC-USP

P. Cassou-Nogues Universite Bordeaux I, Laboratoire de Maths Pures

L. S. Challapa ICMC-USP

J. C. F. Costa ICMC-USP

S. I. R. Costa IMECC-UNICAMP

C. P. Cremasco ICMC-USP

G. J. L. Cruz ICMC-USP

F. S. Dias ICMC-USP

D. L. Dreibelbis University of North Florida

E. A. C. Egusquiza Universidade Federal De Minas Gerais

C. Eyral International Centre for Theoretical Physics

S. Fantin ICMC-USP

C. 0. de Faria ICMC-USP

X PARTICIPANTS

L. Feher Eotvos University

R. E. Ferreira UNIMONTES

V. G. Ferreira ICMC-USP

L. R. A. Gabriel-Filho ICMC-USP

A. Fruehbis-Krueger Universitaet Kaiserslautern

J. E. Furter Brunei University

T. Gaffney Northeastern University

J. B. Gon<;alves University of Porto

A. G. Guimaraes ICMC-USP

L. Grossi ICMC-USP

M. R. A. Gouveia IME-USP

N. G. Grulha Jr. ICMC-USP

C. Gutierres ICMC-USP

A. W. Guzman IME- USP

M. E. Hernandes Universidade Estadual de Maringa

J. T. Hiratuka IME-USP

A. Jeddi Universite Henri Poincare and I.U.F.M de Lorraine

C. M. de Jesus UNIMONTES

A. Joets Laboratoire de Physique des Solides, University de Paris-Sud

N. K. Kakuta UNESP-Sao Jose do Rio Preto

S. L. Kleiman MIT

L. K. Schnur Facultad de Ciencias, UNAM-Mexico

I. S. Labouriau Universidade doPorto

0. A. Laudal Universitetet i Oslo

D. Lehmann Universite de Montpellier II

D. Levcovitz ICMC-USP

A. Libgober University of Illinois at Chicago

S. M. Licanic ICMC-USP

G. F. Loibel IGCE-UNESP

C. Lossen FB Mathematik, Universitiit Kaiserslautern

D. M. Louren<;o ICMC-USP

I. Luengo Universidad Complutense de Madrid

S. Mancini UNESP- Rio Claro

M. Manoel ICMC -USP

M. F. P. Mariano ICMC-USP

T. Marar USP

PARTICIPANTS

R. Martins ICMC-USP

D. B. Massey Northeastern University

S. L. de Medrano Sanchez Instituto de Matematicas,UNAM.

A.M. Hernandez Universidad Complutense de Madrid

C. C. Mendes UNIMONTES

C. M. Mendes ICMC-USP

D. Mond University of Warwick

R. M. de Siqueira ICMC-USP

R. R. Montoya ICMC-USP

S. M. de Moraes Universidade Federal de Vic;osa

A. C. Nabarro Universidade Estadual de Maringa

H. A. Navarro IGCE-UNESP

A. Nemethi The Ohio State University

J. J. N uiio-Ballesteros Universitat de Valencia

R. D. S. Oliveira Universidade Estadual de Maringa

L. P. Orosz UFES

J.P. Papa UNESP - Bauru

V. H. J. Perez Universidade Estadual de Maringa

C. G. Pessoa IME- USP

A. Pichon Institut de Mathematiques de Luminy

K. A. de Rezende IMECC-UNICAMP

J. H. Rieger Martin-Luther Universitat Halle

J. J. Risler U niversiste Paris VI

E. C. Rizziolli USP-Sao Carlos

L. A. F. Roberto ICMC-USP

M. E. Rodrigues ICMC-USP

M. C. Romero Fuster Universitat de Valencia

M.A. S. Ruas ICMC-USP

M. J. Saia ICMC-USP

E. L. dos Santos ICMC-USP

H. A. dos Santos ICMC-USP

J. Seade Instituto de Matematicas de Cuernavaca

D. Siersma Utrecht University

A. L. da Silva ICMC-USP

H. G. da Silva ICMC-USP

P. R. da Silva UNESP - Rio Preto

R. P. da Silva ICMC-USP

R. S. Siquieri ICMC-USP

xi

xii

M. R. da Silveira ICMC-USP

D. de Siqueira ICMC-USP

A.M. Sitta IBILCE-UNESP

C. H. Soares Jr. ICMC-USP

L. M. F. Soares ICMC-USP

A. A. Spadotto Unesp

J. Stevens Gi:iteborgs Universitet

A. Szucs Ei:itvi:is University

F. Tari ICMC-USP

M. A. Teixeira IMECC-UNICAMP

J. N. Tomazella UFSCar

D. J. A. Trotman Universite de Provence

R. Uribe-Vargas Universite Paris VII

A.M. Vila ICMC-USP

W. Veys Katholieke Universiteit Leuven

C. T. C. Wall University of Liverpool

R. G. Wik Atique ICMC-USP

M. Yamamoto Kyushu University

E. Zancanella ICMC-USP

PARTICIPANTS

The Workshop on Real and Complex Singularities is held every other year at the Instituto de Ciencias Matematicas e de Computa~ao (Sao Carlos, Brazil) and brings together specialists in the vanguard of singularities and its applications. This volume contains arti-cles contributed by participants of the seventh workshop. The included papers reflect Fields Medalist Rene Thorn's original vision of singularities and represent all branches of the subject: equisingularity of sets and mappings, the geom-etry of singular complex analytic sets, singularities of mappings and their elimination, characteristic classes, applications to differential geometry, differential equations, and bifurcation theory. The book is suitable for graduate students and researchers interested in singularity theory.

ISBN 0- 8218- 3665- X

9 780821 836651