Lecture Notes in Mathematics

Edited by A. Dold and B. Eckmann

652

I Nil IIIIII Differential Topology, Foliations and Gelfand-Fuks Cohomology Proceedings of the Symposium held at the Pontiffca Universidade Catolica do Rio de Janeiro, 5-24 January, 1976

Edited by Paul A. Schweitzer, s.j.

Springer-Verlag Berlin Heidelberg New York 1978 Editor Paul A. Schweitzer Departamento de Matem~tica PUC/RJ ZC-19 Rio de Janeiro Brazil

Supported by FINEP with the collaboration of CAPES, CNPq and FAPESP

Library of Congress Cataloging in Publication Data

Symposium on Differential and Algebraic Topology, Pontlflca Universidade Catolica do Rio de Janeiro, 1976. Differential topology, foliations, an@ Gelfand- Fuks cohomology.

(Lecture notes in mathematics ; 652) Bibliography: p. Includes index. i. Differential topology--Congresses. 2. Algebraic topology --Congresses. 5. Homology theory--Congresses. I. Schweitzer~ Paul A.~ 1937- If. Title. III. Series: Lecture notes in mathematics (Berlin) ; 652 u QA3. L28 no. 652 [QA613.6] ~ 510'. 8s ISBN O-387-07868-i [514'.7] 78-8464

AMS Subject Classifications (1970): Primary: 57D20, 57D30, 22E65, 57E15 Secondary: 55 B35, 57 D45, 58D05, 55 E15, 55 F40, 54H25

ISBN 3-540-07868-1 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-07868-1 Springer-Verlag NewYorkHeidelberg Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, re- printing, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210 Preface

This volume is essentially the Proceedings of the Symposium on Differential and Algebraic Topology (Escola de Topologia) held at the Pontificia Universidade Cat61ica do Rio de Janeiro from January 5 to 24, 1976. The central theme of the Symposium was the interaction of these two branches of topology, with special attention to foliations and the cohomology of the Lie algebra of vector fields on manifolds,following the pioneering work of I.M. Gelfand and D.B. Fuks.

A. Haefliger and R. Bott presented the two principal courses of lectures, on continuous cohomology, Gelfand-Fuks cohomology and charac- teristic classes of foliations. Aside from Haefliger's general intro- duction, "Cohomology of Lie algebras and foliations" (the first paper in this volume), only the new material from their courses is published here (in the second and third papers). References to earlier publica- tions and an account of the full scientific program of the Symposium are given below.

A list of open problems, which has grown beyond its origins in the Symposium, appears at the end of the volume. The survey article by G. Reeb also suggests future lines of research on foliations. The papers of Langevin-Rosenberg, Reeb-Schweitzer, and Schachermayer were written and submitted subsequent to the Symposium, but the other papers repre- sent lectures presented during the Symposium.

On behalf of the participants, I would like to extend heartfelt thanks to the many people who contributed to the success of the Sympo- sium, especially my colleagues on the Organizing Committee, Gilberto Loibel, Joao Bosco Pitombeira de Carvalho, and William Whitley; the Sym- posium's excellent secretary, Isabel Viveiros de Castro Coffin; Jos~ Luis Arraut, former chairman of the Department of Mathematics, and many other personnel of PUC/RJ, too numerous to cite here. The advice, en- couragement and assistance of colleagues from IMPA was indispensable. For the generous financial support which made the Symposium possible, we are grateful to Financiadora de Estudos e Projetos (FINEP) of the Brazilian government, the principal sponsor, and to the collaborating sponsors CAPES, CNPq, FAPESP, and PUC/RJ.

Paul A. Schweitzer,s.j. 2. , Vector fields on polyhedra, Trans. Amer. Math.Soc.232 (1977], 1-31.

B. Reinhart - Riemannian geometry and Godbillon-Vey classes. Ref. B. Reinhart, The second fundamental form of a plane field, J. Differential Geometry (to appear].

P. Schweitzer - Pontryagin polynomial residues of isolated foliation singularities (joint work with A.Whitman).*

G. Segal - Thurston's theorem BDiffc _ ~n~ n n Ref. D. McDuff and G. Segal, A theorem of Thurston about the classi- fying space for foliations (to appearS.

G. Segal - Proof of the Bott conjecture for Gelfand-Fuks chomology. Ref. R. Bott and G. Segal, The cohomology of the vector fields on a manifold, Topology 16 (197?~, 285-298.

H. Shulman - Flat bundles and the Van Est Isomorphism Theorem. Ref. D. Tisehler and H. Shulman, Leaf invariants of foliations and the Van Est isomorphism, J. Differential Geometry ii (1976], 535 - 546.

A. Verjovsky - Minima]. flows.

* Published in these Proceedings CONTENTS I. Gelfand-Fuks Theor~ and Characteristlc Classes of Follatlons

A. Haefliger Cohomology of Lie algebras and follatlons ...... I

A. Haefllger Whitehead prc~lucts and differential forms ...... 13

R.Bott On some formulas for the characteristic classes of group-actlons ...... 25

H. Shulmanand De Rham theory for BF ...... 62 J. Stasheff

R.B. Gardner Differential geometry and foliations: the God- billon-Vey Invarlant and the Bott-Pasternaok vanlshlng-theorems ...... 75

P.A.Sohweltzer Pontrya@In polynomial residues of isolated and A.P.Whltman foliation singularities ...... 95

II. ~ualitative Theor 7 of Foliations G. Reeb Structures feulllet~es ...... 104 J. Palts Rigidity of the centralizers of dtffeomer- phtsms and structural stability of suspended follatlons ...... 114 R.Langevln and Integrable perturbations of flbratlons ~n~ a H.Hosenberg theorem of Selfert ...... 122

C.Camaoho Stz~Aotural stability of follatlons with singu- larities ...... 128 G.~Reeb and Un th6or~me de Thurston 6tebll au moyen de P.A.Schweltzer itanalyse non standard ...... 138

W.Schachermayer Addendum: Une modification standard de la d~monstratlon non standar~ de Reeb et Schweltzer ,139 Q. Hector Croissance des feuilletages presque sans holonomle ...... 141 E. Fedlda Sur la th6orle des feuilletages associ~e au repute mobile: oas des feuilletages de Lie ...... 183

III. Group Actions and Relat~Topics R.J. Kntll On the index of isolated closed tori ...... 196

F, Hegenbarth An application of the f-lnvarlant ...... 212 I.J. DeJter G-transverBallty to cpn ...... 222

IV. Open /Toblels

P.A.Sohweitzer Some problems in foliation theory and (edltor) related areas ...... 2 ~0 DIFFERENTIAL TOPOLOGY, FOLIATIONS AND GELFAND-FUKS THEORY

PUC/RJ 1976

PROGRAM OF THE SYMPOSIUM

I. COURSES

R. Bott - CONTINUOUS COHOMOLOGY. Ref. i. R. Bott, M. Mostow and J. Perchik, Gelfand-Fuchs Cohomology and Foliations, Proceedings of the New Mexico State University Sym- posium, ].973 (mimeographed). 2. R. Bott, On some formulas for the characteristic classes of group-actions, these Proceedings.

A. Haefliger - ON THE GELFAND-FUKS COHOHOLOGY OF THE LIE ALGEBRA OF

SMOOTH VECTOR FIELDS. Ref. i.A. Haefliger~ Sup la cohomologie de l'algebre de Lie des champs de vecteurs~ Ann. Scient. ~cole Norm. Sup., (4] 9 (1976]~503- 532. 2. , Sur la cohomologie de Gelfand-Fuchs, Differential Topology and Geometry (Dijon, 1974), Springer Lect. Notes Math.484 (1975), 121-161. 3. , Cohomology of Lie algebras and foliations, these Proceedings. 4. , Whitehead products and differential forms, these Proceedings.

S. Git]er - THE ADAMS SPECTRAL SEOUENCE.

II. SURVEY AND EXPOSITORY LECTURES

C. Camacho - Structural stability of foliations with singularities.*

J. Cheeger - The theory of differential characters (two lectures]. Ref. J. Cheeger and J. Simons, Differential Characters and geometric invariants (preprint - lecture notes, AMS Symposium, Stanford,1973).

D.B.A. Epstein - Foliations with all leaves compact. Ref. D.B.A. Epstein, Foliations with all leaves compact, Ann. Inst. Fourier (Grenoble), 26(1976), 265-282.

* Published in these Proceedings R. Gardner - Differential geometry and foliations: the Godbillon-Vey invariant and the Bott-Pasternack vanishing theorem.*

G. Reeb - Structures feuillet~es.*

i H. Rosenberg - The work of Arnold and Herman on Diff(S ) (two lectures). Ref. i. H. Rosenberg, Les diff~omorphismes du cercle (dTapr&s M. R. Herman%, Seminaire Bourbaki n9 476, Springer Lect. Notes Math, 567 (1977), 8-98. 2. M. Herman, Sur la conjugaison differentielle des diffeomor- phismes du cercle, thesis~ Univ. de Paris (Orsay), 1977.

III. RESEARCH LECTURES

R. Barre - 0-varieties: methods to study transverse structures. Ref. R. Barre, De quelques aspects de la th~orie des 0-vari~tes dif- f~rentielles et analytiques Ann. Inst. Fourier (Grenoble) 23 (1973), 227-312.

F. Gonzalez Acuna - 3-dimensional open books.

A. Haefliger - Results on Hamiltonian vector fields.

G. Hector - Growth of foliations.*

F. Hegenbarth - The G-Signature Theorem and applications to surgery.*

J. Heitsch - Residues and characteristic classes for foliations. Ref. i. J. Heitsch, Independent variation of secondary classes (to appear). 2. , Residues, r-vector fields and foliations (to appear).

R. Knill - Stability of closed cycles and closed tori.*

A. Lins Neto - Local C 2- structural stability of integrable 1-forms. Ref. A. Lins Neto, Local structural stability of C 2 integrable l- forms, Ann. Inst. Fourier (Grenoble) 27 (1977), 197-225.

J. Palis - Diffeomorphisms and the stability of suspended foliations.*

M. Penna - Tangent bundles for polyhedra. Ref. i. M. Penna, Differential geometry on simplicial spaces, Trans. Amer. Math. Soc. 214 (i975), 303-323.

* Published in these Proceedings PARTICIPANTS

Complete addresses for the institutes indicated in abbreviated form are given at the end of the list of participants.

Placido F.deA. Andrade - PUC/RJ Eduardo A. Chlncaro E. Jose Luis Arraut - PUC/RJ Departamento de Matem~tica Universidade Federal de Minas Luis Astey Q. - CIEA/IPN Gerais - Pampulha Nelio Baldin - UNICAMP 30.000 Belo Horizonte, MG BRASIL Angelo Barone Netto - IME/USP Carlie Coats Guilhermo L. de la B. Alvarez Department of Mathematics Departamento de Matematica MIT - Room 2089 Universidad de Chile 77 Massachusetts Ave. Sede Valparaiso Cambridge, MA 02139, USA Casilla 130 Valparaiso, CHILE Sueli D.R. Costa - UNICAMP Raymond Barre Italo Dejter D~partement de MathEmatique Centro de Estudos B~sicos Centre Universitaire de Universidade Federal de Santa Valenciennes Catarina - Departamento de 59326 Valenciennes, FRANCE Matematica - Conjunto Univer- sit'rio Trindade Aristides C. Barreto - PUC/RJ 88.000 Florian6polis, SC Raoul Bott BRASIL Department of Mathematics Freddy Dumortier Harvard University Institut de MathEmatique One Oxford Street Universit~ de Brussels Cambridge, MA 02138, USA i050 Brussels, BELGIUM Cesar Camacho - IMPA David B.A. Epstein Mathematics Institute Alberto Campos S. University of Warwick Departamento de Matematicas Coventry CV4 7AL, UK Universidad Naciona! de Bogota Bogota - COLOMBIA Luiz A. F~varo - ICMSC Manuel Efrain Carbajal P. Samuel Feder- CIEA/IPN Universidad Nacional Mayor de Edmond FEdida San Marcos DEpartement de Math~matique C.U. Pabellon D (Ciencias) Universit~ de Dakar Av. Venezuela S/N Dakar, SENEGAL Lima, PERU Robert B. Gardner Carlos A. Aragao de Carvalho Department of Mathematics Instituto de Matematica da UFRJ University of North Carolina Cidade Universitaria Chapel Hill, NC 27514, USA C.P. 1835 - ZC-00 20.000 - Rio de Janeiro, RJ Samuel Gitler - CIEA/IPN BRASIL Elza Gomide - IME/USP Joao B.Pitombeira de Carvalho Francisco Gonz~lez Acuna PUC/RJ Instituto de Matem~ticas Universidad Nacional AutSnoma Jeff Cheeger de M~xico - Ciudad Universit~- Department of Mathematics ria - MExico 20, D.F.-M~XICO SUNY at Stony Brook Stony Brook~ NY 11794, USA XII

Andre Haefliger Oz{ride Manzoli Neto - ICMSC Institut de Mathematique Welington de Melo - IMPA Universite de Geneve C.P. 124 Alan Mitchell - PUC/RJ 1211 Geneve, SWITZERLAND Jacob Palis - IMPA Gilbert Hector C. Frederico Palmeira - PUC/RJ UER de Mathematiques Universite de Lille I Michael Penna B.P. 36 Department of Mathematics 59650 Villeneuve d'Ascq,FRANCE Indiana University - Purdue University at Indianapolis Abramo Hefez - IMPA 1201 East 38th Street Friedrich Hegenbarth - UNICAMP Indianapolis, ~N 46205, USA Universit~t Dortmund Maria do Socorro O.Pereira Abteilung Mathematik 46 Dortmund, WEST GERMANY PUC/RJ James Heitsch Harsh Pittie Department of Mathematics Courant Institute of Mathemati- Univ. of Illinois at Chicago cal Sciences Circle - Box 4348 New York University Chicago, IL 60680, USA 251 Mercer Street Wolf Iberkleid - CIEA/IPN New York, NY 10012, USA Paulo F. da S. Porto Jr.-ICMSC J. Carlos de S. Kiihl - UNICAMP Georges Reeb Ronald J. Knill UER de Mathematiques Department of Mathematics Universite de Strasbourg Tulane University 7, rue Rene Descartes New Orleans, LA 70118, USA 67084 Strasbourg Cedex, FRANCE Maynard Kong Bruce Reinhart Depamtamento de Ciencias Department of Mathematics Pontificia Universidade Catolica University of Maryland del Peru - Apartado 12514 College Park, MD 20742, USA Lima 21, PERU Jean Roberts Brasil Terra Leme Department of Mathematical Departamento de Matematica Sciences - Oakland Univ. Universidade Federal de S. Carlos Rochester, MI 48863 - USA 13560 - S.Carlos, SP, BRASIL Paulo C.P.Rodrigues - PUC/RJ Harold Levine Department of Mathematics Harold Rosenberg Brandeis University Departement de Mathematique Waltham, MA 02154, USA Univ. de Paris VII 2, place Jussieu Alcides Lins Neto - IMPA 75005 Paris, FRANCE Gilberto Loibel - ICMSC Mario R. Saab - ICMSC Artur O. Lopes Instituto de Matem~tica Cristian S~nchez Instituto de Matem~tica, Astro- Universidade Federal do Rio Grande do Sul nomia y F{sica Univ. Nac. de C~rdoba Rua Sarmento Leite S/N Cordoba 5000, ARGENTINA 90.000 - Porto Alegre, R.S. BRASIL Nathan M. dos Santos - PUC/RJ Iaci P. Malta - IMPA Paul A. Schweitzer - PUC/RJ Solange Mancini MErcia Scialom - PUC/RJ Faculdade de Filosofia e Letras do Rio Claro 13.500 Rio Claro, S.P. - BRASIL XIII

Graeme Segal Alberto Verjovsky - CIEA/IPN Mathematics Institute William G. Whitley St. Catherine's College P~s-Graduagao em Matematica Oxford OXI 3UJ, UK Universidade Federal de Santa Herbert Shulman Catarina Department of Mathematics 88.000 Florian~polis, SC. BRASIL Belfer Graduate School of Science Andrew P. Whitman - PUC/RJ Yeshiva Univ., 2495 Amsterdam Ave. New York, NY 10033, USA H. Elmar Winkelnkemper Department of Mathematics Hugo N. Torriani - UNICAMP College Park, MD 20742, USA Ana Maria Urbina F. Departamento de Matematicas Univ. de Chile - Sede Valparaiso Casilla 130~Valparaiso, CHILE

INSTITUTIONAL ADDRESSES

CIEA/IPN: Departamento de Matematica Centro de Investigacion del I.P.N. M~xico 14, D.F., MEXICO ICMSC: Instituto de Ciencias Matematicas de Sao Carlos Av. Dr. Carlos Botelho, 1465 13.560 - Sao Carlos - S.P. - BRASIL IME/USP: Instituto de Matematica e Estat[stica Universidade de Sao Paulo C.P. 20.570 - Agencia Iguatemi 01.451 - Sao Paulo, S.P. - BRASIL IMPA: Instituto de Matematica Pura e Aplicada Rua Luiz de Camoes~ 88 20.000 - Rio de Janeiro, R.J.- BRASIL PUC/RJ: Departamento de Matematica Pontif~cia Universidade Catolica do Rio de Janeiro ZC - 19 20.000 - Rio de Janeiro~ RJ, BRASIL UNICAMP: Instituto de Matematiea Universidade de Campinas C.P. 1170 13.100 - Campinas - S.P. BRASIL