Proceedings of the Fifth Canadian Mathematical Congress: University of Montreal, 1961 9781487584443

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FIFTH MATHEMATICAL CONGRESS

COMPTES RENDUS du

CINQUIEME CONGRES CANADIEN DE MATHEMATIQUES Universite de Montreal, 1961 Edite par

E. M. ROSENTHALL Professeur de Mathematiques, McGill University

UNIVERSITY OF TORONTO PRESS

Toronto, Canada 1963

PROCEEDINGS of the

FIFTH CAN AD IAN MATHEMATICAL CONGRESS University of Montreal, 1961 Edited by

E. M. ROSENTHALL

Professor of Mathematics, McGill University

UNIVERSITY OF TORONTO PRESS Toronto, Canada

1963

© University of Toronto Press 1963 Printed in Canada

Reprinted in 2018 ISBN 978-1-4875-7330-0 (paper)

PREFACE LE PRESENT 0UVRAGE constitue un compte rendu du huitieme seminaire biennal et du cinquieme congres de la Societe Mathematique du Canada tenus a l'Universite de Montreal durant l'ete 1961. Le congres de fondation de notre Societe eut lieu a Montreal en 1945 a I'Universite McGill. Dans la preface des Actes de ce premier congres, nous pouvons lire : «Nous souhaitons vivement que cette Societe soit a l'origine d'importants developpements mathematiques au Canada.» Le present rapport demontre amplement que ce voeu a ete realise bien au-dela des plus optimistes esperances. Des le debut, la Societe s'est propose de couvrir tout le domaine de l'activite mathematique. Une aide importante pour y parvenir fut la collaboration de la Societe de Physique Theorique clans !'organisation de sessions conjointes. Les details du programme de cette session 1961 revelent l'etendue du succes obtenu. Comme aux sessions precedentes, des conferences au niveau de la recherche sont donnees par d'eminents professeurs etrangers. II y a aussi des series de cours superieurs donnes generalement par des canadiens. Le programme offre egalement plusieurs points d'interet pour Jes professeurs de l'enseignement secondaire. On y trouve en particulier un symposium sur la formation clans Jes universites des futurs professeurs d'enseignement secondaire ainsi que des discussions sur le curriculum en mathematiques et en physique au niveau secondaire. Ce programme, a l'instar des programmes des sessions anterieures, reflete les multiples activites de la Societe et temoigne du niveau de l'activite mathematique generale au Canada. Parmi Jes activites de la Societe mentionnons : l'octroi de bourses aux etudiants candidats a la maitrise ou au doctorat ainsi qu'aux etudiants post-doctoraux, l'Institut de recherches d'ete, Jes sessions de cours d'ete a !'intention des professeurs de mathematiques du niveau secondaire, Jes concours clans la plupart des provinces du Canada avec un programme de prix et de bourses a !'intention des etudiants finissants des ecoles secondaires. Durant ces sessions, des conferences et des cours sont donnes a V

vi

PREFACE

tous les niveaux clans les deux langues officielles. Des le debut, la Societe s'est propose de tirer parti du caractere bi-culturel de notre pays. Les representants des deux cultures fran~aise et anglo-saxonne ont coopere avec interet et satisfaction a cette fin. L'Executif et les membres de la Societe ainsi que tous les invites et visiteurs profitent de l'occasion pour exprimer leur gratitude envers les autorites de l'Universite de Montreal pour l'invitation a tenir la session de 1961 sur leur campus. Cette expression de gratitude est egalement transmise au Comite local d'organisation dont le travail soigneux et efficace contribua pour une large part au succes de cette session. R.L.J.

PREFACE THESE PROCEEDINGS report the fifth seminar and general assembly of the Canadian Mathematical Congress, with the University of Montreal as host. The first such meeting was in 1945, and it too was in Montreal with McGill University as host. In the preface to the Proceedings of the first Congress assembly there is the following statement: "It is the hope and belief of many that this Congress will be the beginning of important mathematical developments in Canada." In the present report, which covers the return meeting to Montreal in 1961, there is ample evidence that this hope has been realized far beyond the most optimistic expectations. From the first it has been the aim of the Congress to encompass the full range of mathematical activity. An important aid in accomplishing this has been the co-operation of the Theoretical Physics Society in arranging joint sessions. The extent to which the over-all aim has been reached is revealed in the details of the programme of the session under review. In both Mathematics and Theoretical Physics there are, as in previous sessions, top level research lectures given by prominent scientists from abroad; there are series of lectures at the postgraduate level, usually given by Canadians; always there are features of interest to those teaching in secondary schools which, in the session under review, take the form of panel discussions on the training at the University given to high school teachers, and discussions on curriculum content in mathematics and physics at the secondary school level. This programme, like that of previous sessions, reflects the many interim activities of the Congress, which in turn reflects the level of activity in mathematics throughout the country.These activities include the granting of fellowships and travelling allowances for graduate students, and for post-doctoral study; the summer research institute; summer sessions for the advancement of high school teachers in subject matter; a nationwide programme of prize and scholarship competitions at the matriculation level. At all levels there are lectures in both official languages. From the vii

Vlll

PREFACE

first it has been the aim of the Congress to make the most of our good fortune as a nation in having our social structure based on two cultures, French and Anglo-Saxon. The representatives of both cultures have co-operated with interest and satisfaction towards this end. The Executive and members of the Congress, and all invited guests and visitors, take this opportunity to express gratitude to the Officers of the University of Montreal for the invitation to hold the 1961 session on their campus. This expression of appreciation and gratitude is also extended to the Committee on Arrangements for the careful planning which added so much to the success of the session.

R.L.J.

TABLE DES MATIERES Ill

CONTENTS

PREFACE

V

PREFACE

vii

SUMMER SEMINAR

Research Lectures Instructional Lectures Entertainment List of Those Attending the Seminar 1961 Programme Invited Lectures Contributed Papers Symposium on the Teaching of Applied Mathematics in Canada Discussion on Policies Concerning the Advancement of Research in Mathematics Symposium on the Training in Mathematics Given at the University to High School Teachers Discussion on High School Mathematics (in French) Discussion on High School Mathematics (in English) Entertainment Business Meeting List of Those Attending the Congress Acknowledgment Address of the Retiring President, R. L. Jeffery Symposium on the Training in Mathematics Given at the University to Future High School Teachers Research Policy Seminar The Teaching of Applied Mathematics in Canada, A. F. Pillow

THE QUADRENNIAL CONGRESS OF

ABSTRACTS OF CONTRIBUTED PAPERS

On Conservative Dynamical Systems and the Equation of Wave Mechanics, Nicholas Chako Multiplicative Ideal Theory and Rings of Quotients, G. D. Findlay and J. Lambek p-Algebras and Derivations, K. Hoechsmann Flow of Heat from a Wedge into a Surrounding Medium, P. Mandi Homomorphism between Groups of Contact Transformations and Corresponding Groups of Point Transformations, John Mariani A Note on Weighted Compositions, T. V. Narayana The Dynamics of the Expanding Vapour Bubble in a Boiling Liquid, P. Savic and J. W. Gosnell Some Methods for Computing Model Stellar Atmospheres, Anne B. Underhill Atmospheric Waves Produced by a Large Explosion in the Atmosphere, V. H. Weston ix

3 3 3

3 4

11 11 11 12 13

13 13 13 13 13 14 15 19

25

43 75 88 97 97 97 98 99 101 102 102

104 105

X

TABLE DES MATIERES

INVITED LECTURES

CONTENTS

Structures feuilletees, Charles Ehresmann An Extension of the Concept of Real Number, A. Erdelyi Mathematics and Logic, Leon Henkin The Application of Mathematics to Biology and Medicine, Ian N. Sneddon

109 109 173 184 201

SUMMER SEMINAR

SUMMER SEMINAR THE SEMINAR OF 1961 was held at the University of Montreal from August 13 to September 1, jointly with the Theoretical Physics Division of the Canadian Association of Physicists. In accordance with the plan followed at earlier seminars, there were series of lectures at two levels, known as the Research Lectures and the Instructional Lectures, in order that the needs of all might be met. The programme was as follows: RESEARCH LECTURES C. Ehresmann, Jnstitut Henri Poincare, Paris

Categories differentiables et geometrie differentielle A. Erdelyi, California Institute of Technology Asymptotic solutions of differential equations L. Henkin, University of California The theory of cylindrical algebras Mixed boundary value problems in I. N. Sneddon, The University, Glasgow potential theory

INSTRUCTIONAL LECTURES I. Halperin, Queen's University T. E. Hull, University of British Columbia J. Maranda, University of Montreal H. F. Trotter, Queen's University

Continuous geometry Numerical analysis Algebre homologique Probability theory

ENTERTAINMENT The city of Montreal and the surrounding district proved to be an interesting place for visitors from other centres. Among the social events were a reception for members and their wives given by the University of Montreal, a tea for the wives at McGill University Faculty Club, and a visit to the Morgan Arboretum at Ste. Anne de Bellevue. A picnic at the Arboretum was arranged through the kindness of the staff of Macdonald College at Ste. Anne de Bellevue. Members of the seminar and their wives were also entertained at a dinner given by the College Militaire Royal de Saint Jean at 3

4

SUMMER SEMINAR

Saint Jean, Quebec. This was followed by an inspection of the new buildings at the College. Coffee was served each day from Monday through Friday at the University of Montreal, providing an additional opportunity for members of the seminar to become better acquainted and to exchange ideas on their current study and research. LIST OF THOSE ATTENDING THE SEMINAR

Abramowich, J. Adshead, J. G. Anctil, J. A. Anctil, Mrs. J. A. Antliff, W. B. Artiaga, Lucio Ayoub, R. G. Ayoub, Mrs. Christine Baragar, Diana E. Beaudet, Gilles Benjamin, I. Bonyun, D. A. Bonyun, Mrs. D. A. Bouwer, I. Z. Brender, Allan Brossard, Roland Brossard, Mrs. Roland Buchal, R. N. Bures, D. J. Bush, G. C. Bush, Mrs. G. C. Carragher, Paul Clement, Bernard Cleroux, Robert Cole, R.H. Cole, Mrs. R. H. Courteau, Bernard Crawford, W. S. H. Crosby, D. R. Daigneault, A. Daigneault, Mrs. A. Dubuc, Serge Ehresmann, Charles Eliopoulos, H. A. Erdelyi, A. Evans, Arwel Evans, Mrs. Arwel Fieldhouse, David

Assumption University, Windsor Dalhousie University, Halifax College Militaire Royal, St. Jean, Quebec Royal Military College, Kingston University of Saskatchewan, Saskatoon Pennsylvania State University Pennsylvania State University McMaster University, Hamilton University of Montreal Loyola College, Montreal McGill University, Montreal Queen's University, Kingston University of California, Berkeley University of Montreal Argonne National Laboratory, Illinois Queen's University, Kingston Queen's University, Kingston Memorial University of Newfoundland University of Montreal University of Montreal University of Western Ontario, London University of Montreal Mount Allison University, Sackville University of Alberta, Edmonton University of Montreal University of Montreal Institut Henri Poincare, Paris Assumption University, Windsor California Institute of Technology, Pasadena McGill University, Montreal Princeton University, New Jersey

MATHEMATICIANS AT SEMINAR

Fortin, Jacques Fox, Charles Fox, Mrs. Charles Fox, G. E. N. Gagnon, F. P. Gagnon, Mrs. F. P. Gandhi, S. R. Gardner, R. B. Gauthier, Abel Gauthier, Mrs. Abel Gratzer, George Guttman, Irwin Halperin, I. Hayes, J.C. Henkin, Leon Henkin, Mrs. Leon Hershorn, Michael Holbrook, J. A. Hull, Thos. E. Jeffery, R. L. Kaller, Cecil Louis Kassimatis, C. Katz, I. N. Katz, Mrs. I. N. Kaufman, H. Kaufman, Mrs. H. Kayler, Helene Keeping, E. S. Kemp, R. R. D. Kimura, Naoki Klamkin, M. S. Kleisli, H. Klemola, M. T. Klemola, Mrs. M. T. Knight, William Knight, Mrs. William Kotze, W. J. Kobayashi, S. L' Abbe, Maurice L' Abbe, Mrs. Maurice Lachapelle, Benoit Lachapelle, Mrs. Benoit Lambek, Joachim Lambek, Mrs. Joachim Lapointe, Claude Leah, Rev. Philip J. LeBlanc, Leon LeBlanc, Mrs. Leon Leduc, Pierre-Yves Legoupil, Jean

Laval University, Quebec McGill University, Montreal University of Montreal University of Montreal University of Western Ontario, London University of California, Berkeley University of Montreal Hungarian Academy, Budapest McGill University, Montreal Queen's University, Kingston College Militaire Royal, St. Jean, Quebec University of California, Berkeley McGill University, Montreal Queen's University, Kingston University of British Columbia, Vancouver Acadia University, Wolfville University of Saskatchewan, Saskatoon University of North Carolina, Raleigh A.V.C.0., Massachusetts McGill University, Montreal University of Montreal University of Alberta, Edmonton Queen's University, Kingston University of Saskatchewan, Saskatoon A.V.C.O., Massachusetts University of Montreal Assumption University, Windsor University of New Brunswick, Fredericton McGill University, Montreal University of British Columbia, Vancouver University of Montreal University of Montreal McGill University, Montreal University of Montreal St. Paul's College, Winnipeg University of Montreal University of Montreal University of Montreal

5

6

MATHEMATICIANS AT SEMINAR

Lorch, Lee MacPhee, Rev. H.J., s.j. Mandi, Dr. Paul Maranda, Jean Maranda, Mrs. Jean Melamed, Samuel Melamed, Mrs. Samuel Melzak, Z. A. Menard, Jean Messerli, R. Miller, D. R. Mohanty, S. G. Munier, Fran~ois Normand, G. Normand, Mrs. G. O'Connor, Rev. R. E., s.j. Peck, J. E. L. Pillow, A. F. Pillow, Mrs. A. F. Pilon, R. G. Pilon, Mrs. R. G. Pouliot, Adrien Prillo, A. A. J. Ritcey, L. F. S. Ritcey, Mrs. L. F. S. Robert, Pierre Robert, Mrs. Pierre Robillard, Paul Rogers, C. A. Rogers, Mrs. C. A. Ronveaux, Andre Rothberger, Fritz Schremp, Dr. E. J. Schremp, Mrs. E. J. Schwerdtfeger, Hans Schwerdtfeger, Mrs. Hans Shklov, Nathan Shklov, Mrs. Nathan Shtern, I. H. Shtern, Mrs. I. H. Smith, P. D. P. Sneddon, I. N. Sneddon, Mrs. I. N. Stallard, S. Sussman, David Sussman, Mrs. David Svilokos, Nikola Thierrin, G. Thierrin, Mrs. G. Thomas, G. H. M.

University of Alberta, Edmonton Loyola College, Montreal National Research Council University of Montreal McGill University, Montreal University of British Columbia, Vancouver University of Montreal University of Montreal University of Toronto University of Alberta, Edmonton Ecole Polytechnique, Montreal College Militaire Royal, St. Jean, Quebec Loyola College, Montreal University of Alberta, Calgary University of Toronto College Militaire Royal, St. Jean, Quebec Laval University, Quebec Loyola College, Montreal Canadian Mathematical Congress, Montreal University of Montreal University of Montreal University of London, England University of Montreal Laval University, Quebec U.S. Naval Research Laboratory, Washington McGill University, Montreal University of Saskatchewan, Saskatoon McGill University, Montreal Memorial University of Newfoundland The University, Glasgow College Militaire Royal, St. Jean, Quebec McGill University, Montreal Long Island University, Brooklyn University Gf Montreal University of Saskatchewan, Saskatoon

MATHEMATICIANS AT SEMINAR

Tomiuk, Daniel Trotter, Hale Turgeon, Rev. Jean, s.j. Van Der Merwe, D. F. Venne, Marc Wasan, M. T. Watson, R. E. Watson, Mrs. R. E. Williams, W. L. G. Williams, Mrs. W. L. G. Wonenburger, Maria J. Woonton, G. A. Wyman, Max Wyman, Mrs. Max Zassenhaus, Hans

University of Ottawa Queen's University, Kingston Loyola College, Montreal McGill University, Montreal University of Montreal Queen's University, Kingston Southern Illinois University McGill University, Montreal (retired) Queen's University, Kingston McGill University, Montreal University of Alberta, Edmonton Notre Dame University, South Bend, Indiana

7

THE QUADRENNIAL CONGRESS

THE QUADRENNIAL CONGRESS OF 1961 THE CONGRESS OF 1961 was held at the University of Montreal from September 3 to September 9, in joint session with the Theoretical Physics Division of the Canadian Association of Physicists. The Congress programme was as follows: SUNDAY,

2:00 8:00

September 3 Registration. Informal Reception.

P.M. P.M.

September 4 Lecture, Henkin. A.M. Lecture, Chew. P.M. Business meeting of the Canadian Mathematical Congress. P.M. Address of the retiring president of the Canadian Mathematical Congress, R. L. Jeffery: "Derivatives and integrals with respect to a base function."

MONDAY,

9:30 11 :00 2:00 5:00

A.M.

TUESDAY, September 5 9:30 A.M. Lecture, 11 :00 A.M. Lecture, 2:30 P.M. Lecture,

4:00 8:00

P.M. P.M.

WEDNESDAY,

Erdelyi. Ehresmann. de Groot. Lecture, Sneddon. Symposium on the teaching of applied mathematics in Canada.

September 6 Picnic at Mount Saint-Hilaire.

THURSDAY, September 7 9:30 A.M. Research

2:00 6:30

P.M. P.M.

Policy Seminar. Lecture, Chew. Cocktails and dinner offered by the City of Montreal, at the Restaurant Helene de Champlain, Ile Ste-Helene.

September 8 Contributed papers. Symposium on the training in mathematics given at the university to high school teachers. 8:00 P.M. Discussion on high school mathematics (in French).

FRIDAY,

9:00 2:30

A.M. P.M.

September 9 9 :00 A.M. Lecture, de Groot. 10:15 A.M. Discussion on high school mathematics (in English).

SATURDAY,

INVITED LECTURES

G. F. Chew, University of California, Berkeley

The elementary particle concept: A principle of maximum strength for strong interactions 11

12

QUADRENNIAL CONGRESS

S. R. de Groot, University of Leiden, Holland

The causality principle and the fluctuation dissipation theorem; Rayonnements de noyaux orientes

C. Ehresmann, Institut Henri Poincare

Structures feuilletees

A. Erdelyi, California Institute of Technology

An extension of the concept of real number

Leon Henkin, University of California, Berkeley

Mathematics and logic

I. N. Sneddon, The University, Glasgow

The application of mathematics to biology and medicine

CONTRIBUTED PAPERS

Nicholas Chako, On conservative dynamical systems and the equaQueens College and tion of wave mechanics Institute of Space Studies, N. Y. G. D. Findlay and Multiplicative ideal theory of rings of quotients and J. Lambek, McGill University, Montreal K. Hoechsmann, University of Notre Dame, Indiana

p-Algebras and derivations

P. Mandi, National Research Council, Ottawa

Flow of heat from a wedge into a surrounding medium

John Mariani, Fairleigh Dickinson University

Homomorphism between groups of contact transformations and corresponding groups of point transformations

T. V. Narayana, University of Alberta, Edmonton

A note on weighted compositions

P. Savic and J. W. Gosnell, The dynamics of the expanding vapour bubble in National Research Council, a boiling liquid Ottawa Anne B. Underhill, Dominion Astrophysical Observatory, Victoria

Some methods for computing model stellar atmos~ pheres

V. H. Weston, University of Michigan

Atmospheric waves produced by a large explosion in the atmosphere

QUADRENNIAL CONGRESS

13

SYMPOSIUM ON THE TEACHING OF APPLIED MATHEMATICS IN CANADA

P. R. Wallace (presiding), McGill University F. H. Northover, Carleton University A. F. Pillow, University of Toronto I. N. Sneddon, University of Glasgow RESEARCH POLICY SEMINAR

G. de B. Robinson (presiding), University of Toronto SYMPOSIUM ON THE TRAINING IN MATHEMATICS GIVEN AT THE UNIVERSITY TO HIGH SCHOOL TEACHERS

I. Halperin (presiding), Queen's University D. Betts, University of Alberta A. W. Bishop, Department of Education, Ontario R. Stanton, University of Waterloo DISCUSSION ON HIGH SCHOOL MATHEMATICS (IN FRENCH)

Maurice L'Abbe (presiding), University of Montreal I. Halperin, Queen's University L. Henkin, University of California, Berkeley P. Obreanu, Queen's University DISCUSSION ON HIGH SCHOOL MATHEMATICS (IN ENGLISH)

L. F. S. Ritcey (presiding), Canadian Mathematical Congress A. J. Coleman, Queen's University L. Henkin, University of California, Berkeley R. L. Rosenberg, University of New Brunswick ENTERTAINMENT

An informal reception was held at the University of Montreal on Sunday evening, September 3. On Wednesday, all members of the Congress and their families attended a picnic at Mount St. Hilaire. This property, which

QUADRENNIAL CONGRESS

13

SYMPOSIUM ON THE TEACHING OF APPLIED MATHEMATICS IN CANADA

P. R. Wallace (presiding), McGill University F. H. Northover, Carleton University A. F. Pillow, University of Toronto I. N. Sneddon, University of Glasgow RESEARCH POLICY SEMINAR

G. de B. Robinson (presiding), University of Toronto SYMPOSIUM ON THE TRAINING IN MATHEMATICS GIVEN AT THE UNIVERSITY TO HIGH SCHOOL TEACHERS

I. Halperin (presiding), Queen's University D. Betts, University of Alberta A. W. Bishop, Department of Education, Ontario R. Stanton, University of Waterloo DISCUSSION ON HIGH SCHOOL MATHEMATICS (IN FRENCH)

Maurice L'Abbe (presiding), University of Montreal I. Halperin, Queen's University L. Henkin, University of California, Berkeley P. Obreanu, Queen's University DISCUSSION ON HIGH SCHOOL MATHEMATICS (IN ENGLISH)

L. F. S. Ritcey (presiding), Canadian Mathematical Congress A. J. Coleman, Queen's University L. Henkin, University of California, Berkeley R. L. Rosenberg, University of New Brunswick ENTERTAINMENT

An informal reception was held at the University of Montreal on Sunday evening, September 3. On Wednesday, all members of the Congress and their families attended a picnic at Mount St. Hilaire. This property, which

14

QUADRENNIAL CONGRESS

belongs to McGill University, is situated twenty miles east of Montreal. On Thursday evening a dinner was offered by the City of Montreal on St. Helene's Island in the St. Lawrence River. The speakers at this dinner were Professor M. L'Abbe (chairman), Professor R. L. Jeffery, and Professor I. N. Sneddon. BUSINESS MEETING

A meeting of the Congress was held on Monday in which reports were given and other business was transacted. A constitution, copies of which had been sent to all members, was adopted by the Congress. According to this constitution, officers will be elected every second year for a two-year period. In future years the election will be held by mail ballot, but in 1961 elections were held at the Congress meeting. The list of elected and permanent officers is as follows: Past President: R. L. Jeffery, Acadia University President: R. D. James, University of British Columbia Vice-Presidents: A. Gauthier, University of Montreal; N. S. Mendelsohn, University of Manitoba; R. L. Rosenberg, University of New Brunswick Executive Director: L. F. S. Ritcey Treasurer: W. L. G. Williams (retired), McGill University English Secretary: R. E. O'Connor, s.j., Loyola College French Secretary: J. M. A. Maranda, University of Montreal Council Members From the West: R. Blum, University of Saskatchewan; D. Derry, University of British Columbia; A. P. Guinand, University of Saskatchewan; L. Moser, University of Alberta; B. N. Moyls, University of British Columbia; B. Noonan, University of Manitoba; J. E. L. Peck, University of Alberta at Calgary; M. Wyman, University of Alberta From Ontario and Quebec: B. Banaschewski, McMaster University; R.H. Cole, University of Western Ontario; A. J. Coleman, Queen's University; H. S. M. Coxeter, University of Toronto;

14

QUADRENNIAL CONGRESS

belongs to McGill University, is situated twenty miles east of Montreal. On Thursday evening a dinner was offered by the City of Montreal on St. Helene's Island in the St. Lawrence River. The speakers at this dinner were Professor M. L'Abbe (chairman), Professor R. L. Jeffery, and Professor I. N. Sneddon. BUSINESS MEETING

A meeting of the Congress was held on Monday in which reports were given and other business was transacted. A constitution, copies of which had been sent to all members, was adopted by the Congress. According to this constitution, officers will be elected every second year for a two-year period. In future years the election will be held by mail ballot, but in 1961 elections were held at the Congress meeting. The list of elected and permanent officers is as follows: Past President: R. L. Jeffery, Acadia University President: R. D. James, University of British Columbia Vice-Presidents: A. Gauthier, University of Montreal; N. S. Mendelsohn, University of Manitoba; R. L. Rosenberg, University of New Brunswick Executive Director: L. F. S. Ritcey Treasurer: W. L. G. Williams (retired), McGill University English Secretary: R. E. O'Connor, s.j., Loyola College French Secretary: J. M. A. Maranda, University of Montreal Council Members From the West: R. Blum, University of Saskatchewan; D. Derry, University of British Columbia; A. P. Guinand, University of Saskatchewan; L. Moser, University of Alberta; B. N. Moyls, University of British Columbia; B. Noonan, University of Manitoba; J. E. L. Peck, University of Alberta at Calgary; M. Wyman, University of Alberta From Ontario and Quebec: B. Banaschewski, McMaster University; R.H. Cole, University of Western Ontario; A. J. Coleman, Queen's University; H. S. M. Coxeter, University of Toronto;

MATHEMATICIANS AT CONGRESS

15

Rev. D. T. Faught, Assumption University; K. D. Fryer, University of Waterloo; J. Lambek, McGill University; N. D. Lane, McMaster University; V. Linis, University of Ottawa; F. Rothberger, Laval University; E. M. Rosenthall, McGill University; H. Schwerdtfeger, McGill University From the Atlantic Provinces: W. J. Blundon, Memorial University of Newfoundland; W. S. H. Crawford, Mount Allison University; D. 0. Snow, Acadia University; A. J. Tingley, Dalhousie University Editor-in-Chief, Canadian Journal of Mathematics: W. T. Tutte, University of Toronto Managing Editor, Canadian Journal of Mathematics: G. de B. Robinson, University of Toronto Editor-in-Chief, Canadian Mathematical Bulletin: W. Moser, University of Manitoba Managing Editor, Canadian Mathematical Bulletin; P. Scherk, University of Toronto Director, Summer Research Institute: R. L. Jeffery, Acadia University LIST OF THOSE ATTENDING THE CONGRESS

Abramowich, J. Adshead, J. G. Anctil, J. A. Anctil, Mrs. J. A. Antliff, W. B. Artiaga, Lucio Ayoub, R. G. Ayoub, Mrs. Christine Baragar, Diana E. Beaudet, Gilles Benjamin, I. Blundon, W. J. Bonyun, D. A. Bonyun, Mrs. D. A. Bouwer, I. Z. Brender, Allan Brossard, Roland Brossard, Mrs. Roland Buchal, Robert N. Bures, D. J.

Assumption University, Windsor Dalhousie University, Halifax College Militaire Royal, St. Jean, Quebec Royal Military College, Kingston University of Saskatchewan, Saskatoon Pennsylvania State University Pennsylvania State University McMaster University, Hamilton University of Montreal Loyola College, Montreal Memorial University of Newfoundland McGill University, Montreal Queen's University, Kingston University of California, Berkeley University of Montreal Argonne National Laboratory, I linois Queen's University, Kingston

MATHEMATICIANS AT CONGRESS

15

Rev. D. T. Faught, Assumption University; K. D. Fryer, University of Waterloo; J. Lambek, McGill University; N. D. Lane, McMaster University; V. Linis, University of Ottawa; F. Rothberger, Laval University; E. M. Rosenthall, McGill University; H. Schwerdtfeger, McGill University From the Atlantic Provinces: W. J. Blundon, Memorial University of Newfoundland; W. S. H. Crawford, Mount Allison University; D. 0. Snow, Acadia University; A. J. Tingley, Dalhousie University Editor-in-Chief, Canadian Journal of Mathematics: W. T. Tutte, University of Toronto Managing Editor, Canadian Journal of Mathematics: G. de B. Robinson, University of Toronto Editor-in-Chief, Canadian Mathematical Bulletin: W. Moser, University of Manitoba Managing Editor, Canadian Mathematical Bulletin; P. Scherk, University of Toronto Director, Summer Research Institute: R. L. Jeffery, Acadia University LIST OF THOSE ATTENDING THE CONGRESS

Abramowich, J. Adshead, J. G. Anctil, J. A. Anctil, Mrs. J. A. Antliff, W. B. Artiaga, Lucio Ayoub, R. G. Ayoub, Mrs. Christine Baragar, Diana E. Beaudet, Gilles Benjamin, I. Blundon, W. J. Bonyun, D. A. Bonyun, Mrs. D. A. Bouwer, I. Z. Brender, Allan Brossard, Roland Brossard, Mrs. Roland Buchal, Robert N. Bures, D. J.

Assumption University, Windsor Dalhousie University, Halifax College Militaire Royal, St. Jean, Quebec Royal Military College, Kingston University of Saskatchewan, Saskatoon Pennsylvania State University Pennsylvania State University McMaster University, Hamilton University of Montreal Loyola College, Montreal Memorial University of Newfoundland McGill University, Montreal Queen's University, Kingston University of California, Berkeley University of Montreal Argonne National Laboratory, I linois Queen's University, Kingston

16

MATHEMATICIANS AT CONGRESS

Bush, G. C. Bush, Mrs. G. C. Carragher, Paul Clement, Bernard Cleroux, Robert Cole, R.H. Cole, Mrs. R. H. Coleman, A. J. Courteau, Bernard Coxeter, H. S. M. Crawford, W. S. H. Crosby, D. R. Csorgo, M. Daigneault, A. Daigneault, Mrs. A. De Laet, Christian Derry, D. Derry, Mrs. D. Divinsky, Nathan Dubuc, Serge Edwards, L. P. Ehresmann, Charles Eliopoulos, H. A. Erdelyi, A. Evans, Arwel Evans, Mrs. Arwel Faught, Rev. Donald T. Fieldhouse, David Fortin, Jacques Fox, Charles Fox, Mrs. Charles Fox, G. E. N. Fryer, K. D. Gagnon, F. P. Gagnon, Mrs. F. P. Gandhi, S. R. Gardner, R. B. Gauthier, Abel Gauthier, Mrs. Abel Gra tzer, George Guttman, Irwin Haley, K. D. C. Halperin, I. Hayes, J.C. Henkin, Leon Henkin, Mrs. Leon Hershorn, Michael Hoechsmann, K. Holbrook, J. A. Hull, Thos. E.

Queen's University, Kingston Memorial University of Newfoundland University of Montreal University of Montreal University of Western Ontario, London Queen's University, Kingston University of Montreal University of Toronto Mount Allison University, Sackville University of Alberta, Edmonton McGill University, Montreal University of Montreal KCS (Quebec) Limited, Montreal University of British Columbia, Vancouver University of British Columbia, Vancouver University of Montreal University of New Brunswick, Fredericton lnstitut Henri Poincare, Paris Assumption University, Windsor California Institute of Technology, Pasadena McGill University, Montreal Assumption University of Windsor Princeton University, New Jersey Laval University, Quebec McGill University, Montreal University of Montreal University of Waterloo, Ontario University of Montreal University of Western Ontario, London University of California, Berkeley University of Montreal Hungarian Academy, Budapest McGill University, Montreal Acadia University, Wolfville Queen's University, Kingston College Militaire Royal, St. Jean, Quebec University of California, Berkeley McGill University, Montreal University of Notre Dame, Indiana Queen's University, Kingston University of British Columbia, Vancouver

MATHEMATICIANS AT CONGRESS

James, R. D. Jeffery, R. L. Kaller, C. L. Kassimatis, C. Katz, I. N. Katz, Mrs. I. N. Kaufman, H. Kaufman, Mrs. H. Kayler, Helene Keeping, A. J. Keeping, E. 5. Kemp, R. R. D. Kimura, Naoki Klamkin, M. 5. Kleisli, H. Klemola, M. T. Klemola, Mrs. M. T. Knight, William Knight, Mrs. William Kotze, W. J. Kobayashi, 5. L' Abbe, Maurice L'Abbe, Mrs. Maurice Lachapelle, Benoit Lachapelle, Mrs. Benoit Lambek, Joachim Lambek, Mrs. Joachim Lapointe, Claude Laufer, P. J. Laufer, Mrs. P. J. Leah, Rev. Philip J. LeBlanc, Leon LeBlanc, Mrs. Leon Leduc, Pierre-Yves Legoupil, Jean Linis, V. Lorch, Lee MacPhee, Rev. H. J., s.j. Magee, G. R. Mandi, Dr. Paul Maranda, Jean Maranda, Mrs. Jean Mariani, John Melamed, Samuel Melamed, Mrs. Samuel Melzak, Z. A. Menard, Jean Mendelsohn, N. 5. Messerli, R. Miller, C. E.

University of British Columbia, Vancouver Acadia University, Wolfville University of Saskatchewan, Saskatoon University of North Carolina, Raleigh A.V.C.O., Massachusetts McGill University, Montreal University of Montreal University of Alberta, Edmonton University of Alberta, Edmonton Queen's University, Kingston University of Saskatchewan, Saskatoon A.V.C.O., Massachusetts University of Montreal Assumption University, Windsor University of New Brunswick, Fredericton McGill University, Montreal University of British Columbia, Vancouver University of Montreal University of Montreal McGill University, Montreal University of Montreal College Militaire Royal, St. Jean, Quebec St. Paul's College, Winnipeg University of Montreal University of Montreal University of Montreal University of Ottawa University of Alberta, Edmonton Loyola College, Montreal University of Western Ontario, London National Research Council University of Montreal Fairleigh Dickinson University McGill University, Montreal University of University of University of University of University of

British Columbia, Vancouver Montreal Manitoba, Winnipeg Montreal Saskatchewan, Saskatoon

17

18

MATHEMATICIANS AT CONGRESS

Miller, D. R. Mohanty, S. G. Moser, W. 0. Munier, Fran!;ois Narayana, T. V. Normand, G. Normand, Mrs. G. Obreanu, P. O'Connor, Rev. R. E., s.j. Peck, J. E. L. Pillow, A. F. Pillow, Mrs. A. F. Pilon, R. G. Pilon, Mrs. R. G. Pouliot, Adrien Pounder, I. R. Prillo, A. A. J. Raboy, D. Ritcey, L. F. S. Ritcey, Mrs. L. F. S. Robert, Pierre Robert, Mrs. Pierre Robillard, Paul Robinson, G. de B. Robinson, Mrs. G. de B. Rogers, C. A. Rogers, Mrs. C. A. Ronveaux, Andre Rooney, P. J. Rosenberg, R. L. Rosenthall, E. M. Rosenthall, Mrs. E. M. Rothberger, Fritz Russell, Dennis Russell, Mrs. Dennis Scherk, Peter Schirmer, Dr. H. H. Schremp, Dr. E. J. Schremp, Mrs. E. J. Schwerdtfeger, Hans Schwerdtfeger, Mrs. Hans Shklov, Nathan Shklov, Mrs. Nathan Shtern, I. H. Shtern, Mrs. I. H. Smith, P. D. P. Sneddon, I. N. Sneddon, Mrs. I. N. Snow, D. 0. Stallard, S.

University of Toronto University of Alberta, Edmonton University of Manitoba, Winnipeg Ecole Polytechnique, Montreal University of Alberta, Edmonton College Militaire, Royal St. Jean, Quebec Queen's University, Kingston Loyola College, Montreal University of Alberta, Edmonton University of Toronto College Militaire Royal, St. Jean, Quebec Laval University, Quebec York University, Toronto Loyola College, Montreal Tutorial High School, Montreal Canadian Mathematical Congress, Montreal University of Montreal University of Montreal University of Toronto University of London, England University of Montreal University of Toronto University of New Brunswick, Fredericton McGill University, Montreal Laval University, Quebec York University, Toronto University of Toronto University of New Brunswick, Fredericton U.S. Naval Research Laboratory, Washington McGill University, Montreal University of Saskatchewan, Saskatoon McGill University, Montreal Memorial University of Newfoundland The University, Glasgow Acadia University, Wolfville College Militaire Royal, St. Jean, Quebec

ACKNOWLEDGMENT

Stanton, R. G. St. Pierre, Jacques Sumner, D. B. Sussman, David Sussman, Mrs. David Svilokos, Nikola Thierrin, G. Thierrin, Mrs. G. Thomas, G. H. M. Tingley, Arnold J. Tomiuk, Daniel Trotter, Hale Turgeon, Rev. Jean, s.j. Tutte, W. T. Underhill, Anne B. Van Der Merwe, D. F. Venne, Marc Wallace, P. R. Wallace, Mrs. P. R. Wasan, M. T. \Vatson, R. E. Watson, Mrs. R. E. Williams, W. L. G. Williams, Mrs. W. L. G. Wittenberg, A. Wonenburger, Maria J. Woonton, G. A. Wyman, Max Wyman, Mrs. Max Zassenhaus, Hans

19

University of Waterloo, Ontario University of Montreal McMaster University, Hamilton McGill University, Montreal Long Island University, Brooklyn University of Montreal University of Saskatchewan, Saskatoon Dalhousie University, Halifax University of Ottawa Queen's University, Kingston Loyola College, Montreal University of Toronto Dominion Astrophysical Observatory, Victoria McGill University, Montreal University of Montreal McGill University, Montreal Queen's University, Kingston Southern Illinois University McGill University, Montreal (retired) Laval University, Quebec Queen's University, Kingston McGill University, Montreal University of Alberta, Edmonton Notre Dame University, South Bend, Indiana ACKNOWLEDGMENT

As Executive Director of the Canadian Mathematical Congress, I take pleasure in here recording, on behalf of the Congress, the names of the public bodies and companies which contributed to our general fund during the year 1961, and thus gave their support to our various activities. The present list includes not only the National Research Council and five provinces but also almost all of the Canadian life assurance companies and chartered banks, and a large number of industrial and engineering companies.

Sustaining Members Province of Nova Scotia Province of Ontario Province of Quebec

ACKNOWLEDGMENT

Stanton, R. G. St. Pierre, Jacques Sumner, D. B. Sussman, David Sussman, Mrs. David Svilokos, Nikola Thierrin, G. Thierrin, Mrs. G. Thomas, G. H. M. Tingley, Arnold J. Tomiuk, Daniel Trotter, Hale Turgeon, Rev. Jean, s.j. Tutte, W. T. Underhill, Anne B. Van Der Merwe, D. F. Venne, Marc Wallace, P. R. Wallace, Mrs. P. R. Wasan, M. T. \Vatson, R. E. Watson, Mrs. R. E. Williams, W. L. G. Williams, Mrs. W. L. G. Wittenberg, A. Wonenburger, Maria J. Woonton, G. A. Wyman, Max Wyman, Mrs. Max Zassenhaus, Hans

19

University of Waterloo, Ontario University of Montreal McMaster University, Hamilton McGill University, Montreal Long Island University, Brooklyn University of Montreal University of Saskatchewan, Saskatoon Dalhousie University, Halifax University of Ottawa Queen's University, Kingston Loyola College, Montreal University of Toronto Dominion Astrophysical Observatory, Victoria McGill University, Montreal University of Montreal McGill University, Montreal Queen's University, Kingston Southern Illinois University McGill University, Montreal (retired) Laval University, Quebec Queen's University, Kingston McGill University, Montreal University of Alberta, Edmonton Notre Dame University, South Bend, Indiana ACKNOWLEDGMENT

As Executive Director of the Canadian Mathematical Congress, I take pleasure in here recording, on behalf of the Congress, the names of the public bodies and companies which contributed to our general fund during the year 1961, and thus gave their support to our various activities. The present list includes not only the National Research Council and five provinces but also almost all of the Canadian life assurance companies and chartered banks, and a large number of industrial and engineering companies.

Sustaining Members Province of Nova Scotia Province of Ontario Province of Quebec

20

ACKNOWLEDGMENT

Bank of Montreal British Columbia Electric Company Limited Canada Life Assurance Company Canadian Industries, Limited Consolidated Mining and Smelting Company of Canada, Limited Great-West Life Assurance Company Hydro-Electric Power Commission of Ontario Imperial Oil Limited International Nickel Company of Canada, Limited London Life Insurance Company Manufacturers Life Insurance Company Northern Electric Company, Limited Royal Bank of Canada Shawinigan Water and Power Company and associated companies Steel Company of Canada Sun Life Assurance Company of Canada Winnipeg Actuaries Club (funds provided by the life insurance companies wtih Head Offices in Winnipeg)

Other Contributors National Research Council Province of New Brunswick Province of Saskatchewan Abitibi Power and Paper Company, Limited Algoma Steel Corporation, Limited Alliance Mutual Life Insurance Company Allied Chemical Canada, Limited Allsopp and Company, Edmonton Aluminum Company of Canada Limited A. E. Ames and Company Limited A. M. F. Atomics, Limited, Port Hope Anthes-Imperial Company Limited, St. Catharines Aronovitch and Leipsic Limited, Winnipeg J. H. Ashdown Hardware Company Limited L' Assurance Vie Desjardins Babcock-Wilcox and Goldie-McCulloch Limited, Galt Bach-Simpson Limited, London Bailey-Selburn Oil and Gas Ltd., Calgary Bank of Nova Scotia Banque Canadienne Nationale B. P. Baragar and Company Ltd., Edmonton

S. Belzberg Enterprises, Edmonton Beneficial Finance Company of Canada Henry Birks and Sons (Montreal) Limited British American Oil Company Limited British Columbia Forest Products Limited, Vancouver Brody's Limited, Edmonton Bruck Mills Limited, Montreal Buckwold's Limited, Saskatoon Honourable C. J. Burchell, Halifax Burns and Company, Limited Calgary Power, Limited California Standard Company, Calgary Canada Cement Company Limited Canada Iron Foundries Limited Canada Packers Foundation Canada Safeway, Limited Canada Starch Company Limited Canada Steamship Lines Limited Canada Wire and Cable Company Limited Canadair Limited Canadian Acme Screw and Gear Limited Canadian Bank of Commerce Canadian Bank Note Company Limited

ACKNOWLEDGMENT Canadian Bedding Company Limited, Edmonton Canadian Blower and Forge Company Limited, Kitchener Canadian Forest Products Limited, Vancouver Canadian General Electric Company Limited Canadian Ingersoll-Rand Company Limited Canadian Kodak Company Limited Canadian Marconi Company Canadian Oil Companies Limited Canadian Pacific Railway Company Canadian Pratt and Whitney Aircraft Company Limited Canadian Premier Life Insurance Company Canadian Refractories Limited Canadian Salt Company Limited Canadian Sportswear Limited, Winnipeg Canadian Superior Oil of California, Limited, Calgary Canadian Utilities Limited, Edmonton Canadian Wallpaper Manufacturing Limited Canadian Western Natural Gas Company Limited, Edmonton Canadian Westinghouse Company Limited E.G. M. Cape and Company Limited Cartier, Cote and Piette, Montreal Century Geophysical Corporation of Canada, Calgary Christie Grant's Limited, Edmonton Confederation Life Association Consumers' Gas Company, Toronto Continental Can Company of Canada Limited Continental Life Insurance Company Coopt'rative Life Insurance Company, Regina Crane Canada Limited H. E. Crowell, Esq., Halifax Crown Life Insurance Company Cyanamid of Canada Limited M. M. Dillon and Company, Limited Dominion Bridge Company Limited Dominion Chain Company Limited

21

Dominion Electrohome Industries Limited Dominion Engineering Works Limited Dominion Forge Limited Dominion Foundries and Steel Dominion Life Assurance Company Dominion Rubber Company Limited Dominion Stores Limited Dow Chemical of Canada Limited Dower Brothers Limited, Edmonton Dufresne Engineering Company Limited, Montreal Dupont Company of Canada Limited John East Iron Works Limited, Saskatoon Eaton Automotive Products Limited The T. Eaton Company Limited Eckler and Company Limited Edmonton Furniture Limited Edmonton Motors Limited Edmonton Photo Supply Limited Electroline Manufacturing Company Limited, Windsor Empire Life Insurance Company Equitable Life Insurance Company of Canada H. M. E. Evans and Company, Edmonton Excelsior Life Insurance Company Falconbridge Nickel Mines Limited Fashion Dress Shoppe Limited, Edmonton Federal Grain Limited, Winnipeg Federated Cooperatives Limited, Saskatoon Fiberglass Canada Limited Firestone Tire and Rubber Company of Canada, Limited Fischer Bearings Manufacturing Limited A. J. Freiman Limited, Ottawa General Motors of Canada Limited G. M. Gest, Contractors Limited B. F. Goodrich Canada Limited Goodyear Tire and Rubber Company of Canada, Limited Great Atlantic and Pacific Tea Company Limited G. L. Griffith and Sons Limited, Stratford

22

ACKNOWLEDGMENT

Helliwell, MacLachlan and Company, Vancouver Home Oil Company Limited Honeywell Controls Limited Household Finance Corporation of Canada Hudson's Bay Company Hudson Bay Mining and Smelting Company Limited A. T. Hurter, Esq., Montreal Imperial Life Assurance Company of Canada Imperial Tobacco Company of Canada Limited Industrial Acceptance Corporation Limited Industrial Life Insurance Company International Business Machines Company Limited International Harvester Company of Canada Limited Interprovincial Pipe Line Company Investors Syndicate of Canada Limited Jenkins Brothers Limited, Montreal Johnson Wire Works Limited, Montreal Johnstone Walker Limited, Edmonton K. C. S. Limited, Toronto Harold Kline, Esq., Edmonton Irving Kline Limited, Edmonton Kraft Foods Limited K. V. P. Company Limited R. W. L. Laidlaw, Esq., Toronto H. R. Lawson, Esq., Toronto Lawson and Jones Limited, London Lincoln Electric Company of Canada Limited Loblaw Groceterias Company Limited MacDonald Electric Limited, Kitchener Madison Natural Gas Company Limited, Calgary Maple Leaf Mills Limited Marsland Engineering Limited, Kitchener Massey-Ferguson Limited Mathews Conveyer Company Limited, Port Hope McArthur Ladies Wear, Edmonton McCabe Grain Company Limited, Winnipeg

William M. Mercer Limited, Toronto Metropolitan Life Insurance Company Metropolitan Stores of Canada Limited Mining Corporation of Canada Limited Minnesota Mining and Manufacturing of Canada Limited W. and A. Moir, Halifax Maison's Brewery Limited Monarch Life Assurance Company Monsanto Canada Limited Montreal City and District Savings Bank Montreal Engineering Company Limited Montreal Life Insurance Company Henry Morgan and Company Limited Mutual Life Assurance Company of Canada National Cash Register Company of Canada Limited National Grain Company Limited, Winnipeg National Life Assurance Company of Canada National Steel Car Corporation A. D. Nesbitt, Esq., Montreal Newhouse Wholesale Limited, Edmonton Niagara Finance Company Limited Carl and Nola Nickle Foundation , North American Life Assurance Company Northern Life Assurance Company of Canada Northwest Wholesale Furniture, Edmonton Northwestern Utilities Limited, Edmonton Norton Company of Canada Limited, Hamilton The Ole Evinrude Foundation (Canada) Inc. Ontario Paper Company Foundation Otis Elevator Company, Hamilton Page-Hersey Tubes, Limited S. Panar, Edmonton Parker Sportswear Company Limited, Edmonton N. M. Paterson and Sons Limited, Winnipeg

ACKNOWLEDGMENT Perini Limited Philips Electronics Industries Limited Pioneer Grain Company Limited, Winnipeg Polymer Corporation Limited K. A. Powell (Canada) Limited, Winnipeg Premier Steel Mills Limited, Edmonton Les Prevoyants du Canada Promislow Brothers Limited, Regina Provincial Bank of Canada Prudential Assurance Company Limited of England Pulp and Paper Mill Accessories Limited, Montreal Quebec North Shore Paper Company RCA Victor Company Limited Remington Rand Limited James Richardson and Sons Limited Rolls-Royce of Canada Limited, Montreal Mrs. Doris E. Rooney, Calgary Royal Insurance Company Limited Royal George Hotel, Edmonton Royal Trust Company Rule, Wynn and Rule, Edmonton Sandwell and Company Limited, Vancouver Sangamo Company Limited, Toronto Savage Shoes Limited, Preston Searle Grain Company Limited, Winnipeg L. E. Shaw Limited, Halifax Sheldons Engineering Limited, Galt Shell Oil Company of Canada Limited Shirriff-Horsey Corporation Limited Siemens Edison Swan (Canada) Limited, Winnipeg Silverwood Dairies Limited Simpson-Sears Limited and The Robert Simpson Company Limited Henry Singer Limited, Edmonton Howard Smith Paper Mills Limited N. Smith Belting Works Limited, Toronto Southam Company Limited

23

Southern Canada Power Company Limited Sovereign Life Assurance Company of Canada Sparton of Canada Limited, London Sperry Gyroscope Company of Canada Limited Spruce Falls Power and Paper Company Limited Standard Life Assurance Company Standard Oil Company of British Columbia Limited Sterling Printers, Edmonton Stone and Webster Canada Limited Supertest Petroleum Corporation Limited, London Taylor, Pearson and Carson (Canada) Limited Texaco Canada Limited B. B. Torchinsky and Associates Limited, Saskatoon Toronto-Dominion Bank Traders Finance Corporation Limited Trans Mountain Oil Pipe Line Company Trudeau's Limited, Edmonton United-Carr Fastener Company of Canada Limited, Hamilton United Geophysical Company of Canada, Calgary Vancouver Sun Ventures Limited, Toronto Waterloo Motors, Edmonton Weisler's Limited, Edmonton W. C. Wells, Construction Company Limited, Saskatoon Westclox Canada Limited, Peterborough Western Life Assurance Company Winnipeg Free Press Company Limited Winspear, Hamilton and Anderson, Edmonton The Honourable Robert H. Winters, Toronto Woodward Stores Limited, Vancouver Max Wyman, Esq., University of Alberta

24

ACKNOWLEDGMENT

This list might have contained also almost all of the universities of Canada; they have supported the Congress by sending or helping to send members of their staffs to our meetings and in many cases by contributing directly to the support of the Canadian Journal of Mathematics rather than contributing directly to our treasury. Their contribution is of the greatest importance and is acknowledged with gratitude. Gratitude is also expressed to many individuals who contributed to the success of the Congress and Seminar of 1961, • either financially or otherwise. The National Research Council gave not only generous financial support but also the lively interest of its members and particularly of its late president, Dr. E. W. R. Steacie. We record here our thanks to the Canada Council not only for $2,000 towards the cost of publication of these Proceedings and $2,000 towards the expenses of the research lecturers, but also for the interest shown in the development of mathematics. In particular, we are grateful for the interest shown by its director, Dr. A. W. Trueman, in the efforts of the Congress to strengthen the contribution of mathematics to the cultural life of Canada. This was the second meeting of the Congress in Montreal. The first one was held in 1945 at McGill University when the organization was founded. We are grateful to the rector of the University of Montreal, Monseigneur Lussier, for his co-operation in extending to the Congress the use of the facilities of the University and to the staff for their untiring assistance. Many were involved in the advance preparations but special mention should be made of the members of the Department of Mathematics who did so much to make the Seminar and Congress of 1961 another outstanding success.

L. F. S. RITCEY Executive Director

PRESIDENTIAL ADDRESS

DERIVATIVES AND INTEGRALS WITH RESPECT TO A BASE FUNCTION R. L.

JEFFERY

1. Introduction. This study arose out of work started by Lusin (1) which led to the following theorems: Let F(x) and G(x) be two functions on [a, b] which are continuous and generalized absolutely continuous, (ACG). If F'(x) = G'(x), except for a set of measure zero, then F(x) and G(x) differ by a constant. If F(x) is continuous and ACGon [a, b] and if F'(x) = f(x) except for a set of zero measure then F(x) - F(a)

=

f

"J(t)dt

a

where the integration is in the generalized Denjoy sense. In this address corresponding results are presented when the measure function and integration are in terms of a base function w which is non-decreasing. The study was carried out during the 1961 session of the Summer Research Institute of the Canadian Mathematical Congress and the 1962 session of the Summer Research Institute of the Australian Mathematical Society. The outcome was a constructive approach to a generalized integral for which the base function is first of bounded variation and then of generalized bounded variation in the restricted sense. The latter results will be published elsewhere. Fundamental results in the same direction have been obtained by Ward (2) and by Henstock (3 ). Their work is based on the descriptive approach of Perron. In both studies there is involved what might be called a point by point examination of the functions which enter. For this reason it was found desirable to give, in terms suitable for the purpose 25

26

R. L.

JEFFERY

in hand, the definition of a measure function and to state its properties which will be used. 1.2. Functions in class \j. A function Fin class \j has the following properties. F is defined on ( - co, co) = 3 except at the points of a denumerable set N. For x E 3 F(x+) and F(x-) both exist and are finite where the limits are taken over S - N. For x E S- N, F(x+) = F(x-) = F(x). 1.3. An outer measure function relative to a function w E \j which is non-decreasing and bounded. Let w (x) E \j be bounded on S - N and be such that if X1, X2 E S' - N, X1 > X2, then w (x1) :,;;. w(x2). For an open interval (a, b) let µ*(a, b) = w(b-) - w(a+ ). For any set A E 3 let µ*(A) = inf

,I: µ*(xi, x';)

where (x ;, x /) is a set of open intervals covering A. If 0 is the empty set µ* (0) = 0. µ*(A) is then an outer measure function in the sense of Monroe (2, chapter II). In what follows lower case Greek letters denote open sets and for any set A, A denotes the complement of A, A = S - A. If A is any set and a :) A then a = Ua;, a; n a 1 = 0, i ~ j, µ*(A) < µ*(a;). Results I-VI follow from (4, chapter 11, 5).

I. µ*(A)
0 is given there is then a mutually exclusive finite set of closed intervals Vi, v2, •.. , Vv, for which

+

µ('Jv,)

< µ*(E) + e,

µ*('J En v,)

>

µ*(E) -

E.

A similar lemma is given in (7, p. 618), where the intervals are nonoverlapping rather than mutually exclusive. Let a ::) E with µ (a) < µ* (E) + e. Delete from V all intervals not on a. Then a = 'Ja;, a; n aj = 0, i ~ j. Let n

an= U

CX;

i

with n sufficiently great to insure that for e arbitrary, but fixed, µ*(En an)

> µ*(E)

-

E.

Let (a;, b 1) be an interval of an. Let 01 > 02 > ... be a sequence of positive numbers with ok - 0 and let Ek C E n (a,, b;) be such

28

R. L.

JEFFERY

that if x E Ek C En (a 1, bi) there is at least one interval [x, x+hk] of V which is on (a;, bi) with hk > ok and with x + hk E (a 1, b 1). By VI, § 1.3, µ*(Ek) - µ*En (a;, b;) as ok - 0. Each point x E En (a 1, b;) is in Ek for ok sufficiently near zero or fork sufficiently great. Now take k so great that µ*( Ek )

>

µ*[ E ('i (a i, b 1)]

-

E , 2n

n the index of an. Let Xo = inf Ek. If Xo E Ek, let V1

If Xo

~

Ek take X1

+ ho], ho

= [xo, Xo

>

= h,. 0

> ok.

Xo, X1 E Ek, and such that

(1)

where Et is the first of a sequence Ei, e2, ••• , for which LEt In this case let v1 = [x1, X1 + h1], h1 > ok. Take x2 E Ek, X2 > X1 + h1 and so that


ok. This procedure can be continued, and because ok > 0 is fixed and because x + h,, E (a;, b1) for x E Ek, it follows that an integer p is reached for which Uv,, i = 1, ... , p, contains all of Ek on [a, b] except the set U Ek n (x, + h,, xt+ 1). It then follows from relations such as (1) and (2) that µ*[u (Ek {J V1)] > µ*[Ek {J (a1, b;)] Et

L

(3)

µ*[u E {J

Vj]

> µ*[Ek

{J (at, b1)] - !._.

Then, because Ek C E, it follows that µ*(u E {J V1)

>

µ*[E {J (a;, b;)] -

n

!._.

n The closed intervals v1 are mutually exclusive. If this construction is carried out for each of the intervals of the set an then the total set of intervals v 1 thus obtained will be finite in number, will be mutually exclusive, and, because v 1 E an C a, it follows that µ(Uv 1)

< µ*(E)

+ E.

29

PRESIDENTIAL ADDRESS

Also, because of (3), the total set

µ*[UE n

Vt

is such that

vtl > µ*(E) -

E.

THEOREM 2.1. If F is non-decreasing, FE except for a set of w-measure zero.

15,

then D.,F exists

Let Ea be the set for which DF+ - !2_F+ > o and suppose that for osufficiently near zero µ*(Ea) > 0. Let Yi be a dense denumerable set on - ro < y < ro and let

Ei = [x: x E Ea, Yi -

¼o
0, it follows that µ*(E 1) > 0 for at least one value io of i. Let E; 0 = Eo, Yio = Yo• Then for x E Eo (l) d = Yo - ¼o < DF+, !2_F+ < Yo - ¾o = c, d - c > ½o. Also for x E Eo we have F(x w(x

(2)

+ ht) + h 1)

- F(x-) w(x-) -

0, hi - 0, x + h 1 E 3 - N. Lemma 2.1 can now be used to select from the intervals associated with E 0 in (2) a mutually exclusive set of closed intervals v1°, v2°, ... , vn°, v;0 = [x 1, x/], X; E Eo, x;' E 3 - N, and F(x/) - F(xi-) w(x 1 ) - w(x 1 - )

(3) and for which, if (4)

µ( U vi°)

E




µ*Eo -

E.

Because F(x) and w(x) are continuous at x/ and because F(xi-) and w(xi-) exist there is an open interval (~;, ~/) with ~; < x 1 < xt' < U, with ~11 ~ / E 3 - N and with

(5)

F(~/) - F(~i)

( .

W ~1) -

( ) < c.

W ~i

Furthermore, because {vi°} is a mutually exclusive set of closed intervals, ~i, U can be so chosen that the intervals (~., ~/) are mutually exclusive, and so that

30

R. L.

(6)

µ[ U (~t, ~/)]

JEFFERY


0, h; - 0, x + h; E 3 - N, (x, x + ht) on an interval (~Jt U), and with F(x w(x

(8 )

+ h;)

+ ht)

- F(x- )> d - w(x-) ·

Lemma 2.1 can now be used to select from the set of intervals associated with E 0 by (8) a mutually exclusive set vi', v2', . .. , v/, v/ = [xt, x/], for which F(x/) - F(x;-) w(x;) - w(x;-)

(9) (10)

µ( U v/)


d

'

n (u

v;')]

>

µ*E0

-

E.

Then relations (9) and (10), with the last part of (4), give (11)

}:{F(x/) - F(xt-)1

>

d(µ*(Eo) - 2Ej.

But because each (xt, x/) is on an interval (~ 1, is non-decreasing we have, using (7), (12)

}:(F(x/) - F(x;)}

< }:(F(U)

~/)

and because F

< c(µ*(Eo) + 2Ej. independently of o, it

- F(~;)l

Then, because d - c > ½o and E is arbitrary follows that (11) and (12) are contradictory. We conclude, therefore, that DF+ = !2_F+ = DF+ except for a set of w-measure zero. The proof of Lemma 2.1 can obviously be altered to conform to the case in which each point of a set E is the right end-point of a sequence of intervals [x + h, x], h < 0. With Lemma 2.1 altered in this way it can be used, as in the foregoing, to show that the left derivative of F with respect to w exists except for a set of w-measure zero. The derivative exists at points of discontinuity of w, and it can then be shown, as in (5, p. 123) that the left derivative and the right derivative are equal except for a denumerable set at each of which w is continuous, hence a set

31

PRESIDENTIAL ADDRESS

which has w-measure zero. Obviously the left and right derivatives are equal at a point of discontinuity of w. This completes the proof of Theorem 2.1. 2.2. If D.,F exists on an w-measurable set E with 0 then D.,F is w-measurable on E.

THEOREM

µ (E)

>

Let a be a real number and let


a].

If D.,F is not measurable on E then for some a the sets Ea, Ea are not w-separated. If such is the case, for c > 0, let Eca

If

Ct


0 so that

+

(8) 0




li-i

39

PRESIDENTIAL ADDRESS

Let (a 11 b 1) be the intervals complementary to ~, on [l', m'J. Then because a 1, b, E ~, it follows from (2) that I:IF(b 1 - ) - F (a1+) I converges. There is then an integer N such that (9)

f IF(b1-) -

F(a,+)

n+l

I
N.

For n fixed, n > N, let [rm, Sm] be the closed intervals complementary to the set (a 1 , b1), ••• , (an, bn). Then U[rm, Sm] :) ~'. The method used to show that Iii is Lebesgue-Stieltjes integrable with respect to w over ~, can be used in conjunction with (8) to determine a set of mutually exclusive intervals corresponding to [xk, ~/] in (7), each interval on an interval of the set [rm, sm] and with, if xk

= h, h, ~/ E ~',

(10)

I: Fm,+)

- F(h-)

>

L

l;-1µ(e1) -

> J~, f fdw and such that (11)

µ{U ~,

n

E

- 2E

[~1, ~~,]} > µ(~') - E',

where €' is sufficiently small to insure that if [17;, 11/] is a set of non-overlapping intervals with 17;, 11/ E ~, and µ[U~' n[11;, 11/ll < E1 then (12) ~k is for some m also a point Sm and a point of discontinuity then h' = hN ow consider the open intervals (~k, ~k') which are complementary to the set [~ 1, Ul on the set [rm, sm]. Then from (11) µ[U~' a1, U)] < E1• If an end-point ~k of such an interval is not a limit point on the right of points of~, then an interval (~k, '17k), 11k E ~', of the set (a;, b;), i > n, can be dropped from the left end of ak, ~k') where ~I n ak, 11k) = 0. If ~k is a limit point on the right of points of~', there is '17k E ~', ~k < '17k n, can be dropped from the right end of (11k, h'), 11k' E ~', ~, n (11k, ~k') = 0. If ~k, is a limit point

If in (10)

of

w

n

40

R. L.

JEFFERY

on the left of points of ~I there is '1'/k 1 EI~, '1'/k < '1'/k', IFa/-) - F ('1'/k'+)I < Ek• Denote the open intervals (~k, '1'/k), (,,.,k', h') by (11p, 11/). Then, since Ek is arbitrary, (13) Furthermore, it follows from (11) and (12) that (14)

We now have the interval [l', m'] covered by the mutually exclusive set of closed intervals U[~i> ~/] and U['l'/k, ,,.,/] some of which may be points, and the open intervals U (11p, 11/) and (a1, v1) U ... U (an, bn) complementary to this mutually exclusive set of closed intervals. We then have F(m') - F(l') =

+

L

L

IF(~~+) - F(~i-)) +

IF(,,.,~+) - F(,,.,k-)\ +

n

L

L

IF(bi-)-F(a;+))

1

IF(11;-)

+ F(11p+)).

It then follows from (9), (10), (13), (14) that F(m') - F(l') Then, since

E

> J~, f fdw

+ L I F(b;-)

- F(a;+)} - 4E.

is arbitrary, it follows that

F(m') - F(l')

> J~, f fdw + L

IF(b;-) - F(ai+ )}.

The foregoing arguments can be used, in conjunction with the sets ei where e; = Elx: x E ~',

l;-1

< f < l;j

to show that F(m') - F(l')

< J~, f fdw + L

IF(b;-) - F(a;+)\.

L

I F(b;-) - F(ai+)},

It then follows that F(m') - F(l') =

f fdw J~,

+

and because the points l', m' can be taken arbitrarily near l, m respectively and because F is AC - w ovtr ~ it follows that

41

PRESIDENTIAL ADDRESS

(15)

F(m-) - F(l+) =

n

f- tel que Z = Z' = Z", f = f' = f", S" < S et S" < S'. Soit 3 (.p-) la classe quotient de .s;>-;m par la relation r" : hA ,,..._, hA' si, et seulement si, ii existe h E hA et h' E hA' tels que h ,,..._, h' modulo r'. Cette classe est une classe sous-locale pour l'ordre quotient de celui de .s;>-;m par r" et I' application ,i' : m(S,f, Z) modulo r" - Z

l'etale sur 3o- Une classe h- E J(.p-) sera appeleefeuilletage elementaire de seconde espece sur ,i' (h-). Soit C une classe complete de feuilletages elementaires de seconde espece, c'est-a-dire une sous-classe de 3(-S;>-) saturee par induction et agregation et admettant pour base une sous-classe C' telle que, pour tout c E C' et tout c' E C', c I\ c' soit defini et ,i' (c) I\ 11' (c') = ,i'(c' I\ c). Alors C s'identifie a un feuilletage de seconde espece (m, .p) elargie sur Z = V,i'(C). Remarque. La categorie .p- peut Hre remplacee par une categorie locale ~- au-dessus de ~- contenant met 3 comme sous-groupoides satures par induction; ii faut alors remplacer .s;>- par la classe ~des elements h E ~- tels que a(h) E 3o et (3(h) E mo. On pourrait aussi considerer la classe 3>-(.s;>-) des jets locaux atomiques des elements de .s;>- au-dessus de ~- (voir 5). De la relation d'equivalence r, on deduit une relation d'equivalence clans 3>-(.s;>-) dont les classes d'equivalence sont appelees germes de feuilletages (voir 4 et 16); la classe des germes de feuilletages est un faisceau sur l'espace (meta-) topologique des germes atomiques de structures de l'espece 3 0 , dont les sections au-dessus de Z E 3o sont Jes feuilletages de seconde espece (m, .p-) elargie. Exemples (1) Soit ~' la categorie de tous Jes homomorphismes r fois differentiables, metant le groupoide A,/ et 3 le groupoide des automorphismes locaux d'ordre r d'une variete differentiable Vn de dimension n. Un atlas complet A compatible avec A,/ forme d'homomorphismes r fois differentiables d'ouverts de Vn clans RP sera

128

CHARLES EHRESMANN

appelefeuilletage r fois differentiable de seconde espece. En particulier, si A est forme d'homomorphismes r fois differentiables de rang constant, A est appele feuilletage r fois differentiable regulier A; est sous-jacent a une structure de variete feuilletee r fois differentiable bien determinee (voir ci-dessous). Les singularites d'un feuilletage A r fois differentiable de seconde espece correspondent aux germes de A provenant d'un homomorphisme h de rang non constant au voisinage d'un point. (2) Soit ,8 = 2(58 # \j) (voir paragraphe 4); soit .S:, la classe des triplets (s, prJ- 1, u), ou u = (f, s X s')>S # \jet pr 1 est la projection canonique de p(s) X p' (s') sur p(s). Alors .S:, verifie les conditions 1, 2, 3 du debut du paragraphe et on peut Jui associer la classe .s:,-(>S # \j) des homomorphismes de 2(58 # \j) vers 58. Aune structure feuilletee :Z de l'espece 58 # \j elargie correspond !'atlas complet A de .S:,-(,S # \j) engendre par les cartes (s, prJ- 1, u), ou u = (f, s X s')>S # \j E B; cet atlas complet definit un feuilletage de seconde espece (58, .S:, (58 # \j)) elargie sous-jacent a :z. Soit A E ~ (58, .S:,) o un feuilletage de seconde espece (58, .S:,) elargie sur Z E Bo- Soit r (Z) la topologie sur P1 (Z) sous-jacente a Z. Les traces sur !es ouverts de r(Z) des ensembles J- 1 (x), ou (s, f, z) E A et x E p (s), forment une base d'une topologie T' sur p 1 (Z). Le couple des topologies (r(Z), T') definit sur Pt(Z) un feuilletage topologique sous-jacent au feuilletage de seconde espece A. Si T' est localement connexe, une feuille de (r(Z), T') sera appelee feuille de A. Si A est le feuilletage de seconde espece sous-jacent a la structure feuilletee :Z E 2 (58 # \j) 0, les feuilletages topologiques sous-jacents a :Z et a A sont identiques et ce sont des feuilletages localement simples. Soit (T, T') un feuilletage topologique localement simple sur E; pour tout ouvert simple Ude E, soit Uw son espace transverse et u !'application canonique de U sur Uw. Soit '.t: le groupoide des homeomorphismes d'un espace topologique sur un autre et soit B la classe des ouverts de E relativement a T. Soit .s:,- la classe de toutes Jes applications continues d'un element de B clans un espace topologique. La sous-classe de .s:,- formee des triplets ( Uw, u, U) engendre un atlas complet A compatible avec '.t: qui definit sur E un feuilletage de seconde espece ('.t:, .s:,-) elargie sous-jacent au feuilletage

STRUCTURES FEUILLETEES

129

topologique (T, T'). Le feuilletage topologique sous-jacent a A est aussi le feuilletage ( T, T').

II. Structure transversale d'un feuilletage 1. P seudogroupe d' holonomie Soient (T, T') un feuilletage topologique localement simple sur E et U un ouvert simple. Soit v un relevement d'un sous-espace B de l'espace transverse u· clans U, c'est-a-dire une application continue de B clans U (muni de la topologie induite par T) telle qui v(x") appartienne a la plaque de U ayant x· pour image canonique clans U. Le sousespace v(B) de U sera appele espace quasi-transversal elementaire. Si B = u·, v(B) sera appele espace transversal elementaire. Les germes des espaces quasi-transversaux elementaires forment un ouvert E/ de l'espace EA de tous les germes de sous-espaces de l'espace E muni de la topologie T. Muni de la topologie induite par celle de EA, l'espace E/ sera appele espace quasi-transversal total. Le sous-espace de E/ forme des germes d'espaces transversaux elementaires sera appele espace transversal total de (T, T') et note Et. Un ouvert suppose relativement separe de E,A (respectivement de Et) sera appele espace quasi-transversal (respectivement transversal). En vertu du theoreme de Zorn, tout espace (quasi-)transversal est contenu clans un espace (quasi-)transversal maximal. La classe 8 des couples (h, N), ou N est un espace transversal elementaire et h un homeomorphisme de N sur un espace transversal elementaire N', est un sous-groupoide local de ~ et un groupoide local au-dessus de (t pour la projection 1r: (h, N) - (h, 1r(N)), ou 1r(N) est !'ensemble sous-jacent au sous-espace N. Un element de 8 sera appele isomorphisme transversal elementaire et on appellera isomorphisme localement transversal une classe complete C de 8 compatible avec 1r. Si C est forme d'unites (c'est-a-dire d'espaces transversaux elementaires), C definit un espace transversal sans point double. 8 contient les sous-groupoYdes suivantes : (1) Le groupoYde des couples (h, N) tels que, pour tout x E 1r(N), X et h(x) appartiennent a une meme feuille de (T, T'). Un element de ce sous-groupoi"de 8 8 sera appele isomorphisme transversal strict elementaire.

130

CHARLES EHRESMANN

(2) Le groupoi'de H engendre par la classe Hs des (h, N) tels que N = v(Uv),h(1r(N)) C Uetuhv = Identitede Uv,ouuestl'application canonique de U sur UV. Le pseudogroupe H sera appele

pseudogroupe d' holonomie. (3) Pour tout x E E, le groupoi:de Hx des couples (h, N) EH, tels que 1r(N) I\ h(1r(N)) contienne le point x. Soit 8A le groupoi:de des jets locaux j/(h, N), ou (h, N) E 8 et x E 1r(N); 8A admet Et pour classe d'objets. Soit hA !'ensemble des jets locauxj/(h, N), ou x decrit 1r(N); Jes ensembles hA forment une base d'une topologie sur 8A, induisant sur Et la topologie consideree ci-dessus. Pour cette topologie, 8A est un groupoide topologique (voir 7). Soient 8/ (respectivement HA, H/) Jes sous-groupoides de 8A formes des jets j/(h, N), ou (h, N) E 8 8 (respectivement E H, E Hx); ces sous-groupoides sont ouverts clans 8A. Un ouvert Q de 8A, tel que la correspondance definie par !es couples (a(X), {3(X)), ou X E n, soit une application biunivoque, sera appele isomorphisme transversal; Q s'identifie a un homeomorphisme de l'espace transversal a(Q) sur l'espace transversal /3(!2). Les elements n forment un pseudogroupe 8'; H s'identifie a un sous-pseudogroupe faible de 8 1 ; le sous-pseudogroupe H de 8 1 ayant pour base H a pour elements Jes isomorphismes transversaux Q contenus clans HA. Le groupoide i1 quotient de HA par le groupoide reunion des groupoides H/ est appele groupoide d'holonomie de (T, T') et un element de 11, jet d'holonomie. La classe d'equivalence X de T/ = j/(h, N) E HA est formee des jets locaux S1TJS, ou s E HxA, s' E Hx'A et x' = h(x); Jes unites a droite et a gauche de X sont respectivement H/ et Hx'A; de plus, on posera: aA(X) = x et {3A (X) = x'. On supposera generalement que par tout point de E passe au moins un espace transversal. Soit F une feuille de (T, T'); soit x la classe des jets d'holonomie X tels que aA(X) = x, ou x E F; alors on a : {3A ( x) = F.

2. Groupoide transverse d' holonomie Soit (T, T') un feuilletage topologique localement simple sur

STRUCTURES FEUILLETEES

131

E; soit 3 le groupoide des couples (h, U), ou U est un ouvert simple de (T, T') et h un isomorphisme de (T, T')u sur (T, T')u,, Alors 3 s'identifie a un sous-pseudogroupe de '.l: et 3 contient les sous-pseudogroupes suivants : (1) Le groupoi'de 3., des couples (h, U) tels que, pour tout x E U, x et h(x) appartiennent a une meme feuille de (T, T'). (2) Le groupoide Sc engendre par !es couples (h, U), ou U et h(U) sont des ouverts distingues d'un meme ouvert simple U 1 et ou x et h(x) appartiennent a une meme plaque de U 1 pour tout

XE u.

A (h, U) E 3 correspond une application h" de u· sur U"', ou U"' sont !es espaces transverses de (T, T') u et de (T, T') u' respectivement et U' = h ( U). Soit 3/p le groupoide ordonne quotient (voir 3) de 3 par la relation d'equivalence : (h, U) '"" (h 1, U 1) si, et seulement si, U = U 1, h(U) = h 1(U1) et h1" = h". La classe d'equivalence de (h, U) E 3 sera identifiee au triplet (U', h", U), ou U' = h(U), et 3/ p s'identifie a un sous-groupoide du groupoide 3' suivant : 3' est le groupoide des triplets ( U', f, U), ou fest un homeomorphisme de l'espace transverse U" sur l'espace transverse U"'. On munit 3' de la relation d'ordre: (U',f, U) < (Ui',f1, U 1) si, et seulement si, U est distingue clans U1, U' est distingue clans Ui' et u'f = Jiu, ou u et u' sont les injections canoniques de U" et U"' clans U 1" et U 1"' respectivement. Un element de 3' sera appele isomorphisme transverse. 3' admet pour sous-groupoide le groupoide H' des triplets ( U', f, U) tels que f soit l'isomorphisme canonique associe a une chaine pure (voir § I. 1); H' est engendre par les triplets ( U', f, U) tels que U soit distingue clans U' et que f soit l'injection de U" clans U"', supposee etre une surjection. H', muni de la structure d'ordre induite par 3', est appele groupoYde des isomorphismes d' holonomie. Remarque. On pourrait aussi munir 3' de la relation d'ordre : (U',f, U) < (Ui',Ji, U1) si, et seulement si, UC U1, U' C Ui', u'f = j 1u; pour cette relation, 3' est une categorie locale (clans laquelle un element induit par une unite peut ne pas etre une unite) dont 3/ p est une sous-categorie locale et H' une classe prelocale (3 ).

u· et

132

CHARLES EHRESMANN

Soient U et U' deux ouverts simples de E. Soit H' ( U', U) la sous-classe de H' formee des triplets (Ui',Ji, U 1), ou Ui' et U1 sont des ouverts satures de U' et U respectivement; alors H'(U', U) est une classe locale appelee classe transverse d' holonomie relative a ( U', U). Soient U' et U"' les espaces transverses de U et de U'; !'application: (U1',fi, U1) - (U1'",fi, Ui') identifie H(U', U) a une classe 'P(U', U) d'homeomorphismes d'un sous-espace de U' sur un sous-espace de U"', appeles homeomorphismes d'holonomie de U' vers U"'. En particulier, si U = U', on peut identifier H'(U, U) et 'P(U, U); on obtient ainsi un pseudogroupe que l'on designera par 'P ( U) et appellera pseudogroupe transverse d' holonomie relatif a U. Le groupoi:de des jets locaux des applications de 'P( U) sera appele groupoide transverse d' holonomie relatif a U. Soient x E E et x' E E; designons par x'Sx' la classe des elements (U',f, U) E 3' ou x"' = f(x-), x E U et x' E U'; soit x'Px la relation d'equivalence clans x'Sx': (U',J, U) ,..._, (U1',Ji, U1) si, et seulement si, il existe (U2',J2, U2) E x'Sx' tel que (U2',h U2) < (U',f, U) et (U2',h, U2) < (Ui',ii, U1). La classe (U',J, U) modulo x'Px sera designee par ,,,j,/ ( U', j, U) et appelee jet local transverse. Par passage au quotient relativement a la relation d'equivalence engendree par les relations x'Px on obtient le groupoi:de P.(3') des jets locaux transverses. Le sous-groupoi:de P(H') de P.(3') forme des jets x'j/(U',J, U), ou (U',j, U) E H', sera appele groupoide d' holonomie et note fl'. PROPOSITION. Si par tout point x de E il passe au moins un espace transversal, H-' est identique au groupoide d'holonomie H- (§ 1).

Si X = (U',J, U) EH', soit W(X) !'ensemble des elements x,'jx/(U',f, U) EH-', ou x1 E U et xi' E U'. Les ensembles W(X) forment une base d'une topologie sur H-'. PROPOSITION. Muni de la topologie precedente, H-' est un groupoide topologique (voir 7).

Pour tout x E E, le sous-groupe Hx-' de H-' forme des jets locaux xjl( U', f, U) sera appele grouped' holonomie en x de la feuille F passant par x. L'unite de Hx-', appelee germe d'espace transverse en x, s'identifie a !'ensemble des couples ( U, Pu), ou U est un ouvert simple contenant x et Pu la relation d'equivalence sous-jacente a

STRUCTURES FEUILLETEES

133

(T, T') u (c'est-a-dire le germe en x de la partition locale sousjacente).

3. Deroulements d'un espace feuillete Soit (T, T') un feuilletage localement simple sur E. Soient w un ouvert simple de E et ww son espace transverse. Soit X = (U',f, U) E H', ou U' est un ouvert sature clans w pour la relation d'equivalence Pw sous-jacente a (T, T')w- Soit xw' une plaque de U'; soit ~ = xv,j/'(U',f, U) la classe des triplets (U/,f1, U,) EH' verifiant les conditions suivantes : Ui' est sature pour Pw; il existe (U2',J2, U2) EH' tel que U2' est sature pour Pw, (U2',J2, U2) < (U',f, U), (U2',J2, U2) < (Ui',Ji, U1), x E U2, xw' est la plaque de U2' contenant hu2(x), ou u2 est !'application canonique de U2 sur U 2w. On peut identifier ~ avec j/'(iju), jet local de E clans ww, OU u et i sont les applications canoniques de u sur uw et de uw/ clans ww. Soit E*(w) la classe des jets ~ ainsi definis. Soit W(X) !'ensemble des jets locaux x1v'jx/'( U', f, U), OU X1 1 decrit U' et X1 decrit U. Les ensembles W(X) forment une base d'une topologie T* sur E*(w). Designons par Q la reunion des feuilles F de (T, T') rencontrant w. Soit a* !'application : ~ - x de E*(w) clans Q et /3* !'application ~ - xw/ de E*(w) clans WV. PROPOSITION. Avec les notations precedentes, E* (w) est etale au-dessus de Q par la projection a* et le feuilletage (T*, T*') determine sur E*(w) para* est simple.Lafeuille F* = 13*- 1 (xw'),ou x' E w,est un revetement normal de F 3 x' pour la projection a*; le groupe des automorphismes de ce revetement s' identifie au groupe d' holonomie f1 x' ', en associant a s E H/ l'automorphisme l - st, oil l E F*.

On appelle E*(w) le deroulement faible de (T, T') au-dessus de

w. La topologie de E* (w) peut ne pas etre separee meme si T est

separee. Soit ~ la classe des chaines pures munie de Ia loi de composition definie par : (W1, • • • t Wn; W1.2, • • • t Wn-1.n) (Wf, • • • t W~,; WL2, • • • , Wn'-1,n')

= (W1, • • • t Wn, w;, • • • , W~,; W1,2, • • • t Wn-1,n, WL2, • • • t w~'-1.n') si, et seulement si, lVi' = Wn. ~ est une categorie libre dont les unites sont les chaines pures reduites a un seul element (W).

134

CHARLES EHRESMANN

Soit r la plus petite relation d'equivalence sur (£ compatible avec la loi de composition et avec les relations elementaires : (W)""' (W, W; W);

(W1, W2; W1,2) (W2, Wi. W1,2) ,..._,. (W1).

La classe d'equivalence r modulo r sera notee [r]. La classe (£/r est un groupoide pour la loi de composition deduite de celle de (£ par passage au quotient. (£/r s'identifie au groupoide libre associe au graphe dont les sommets sont les chaines pures (W) et les aretes les elements (W1, W2; W1,2). La categorie (£ est munie de la relation d'ordre : (W1, ... , Wn; W1,2, ... , Wn-1,n) < (Wi', .... ,Wn'; W1.2', .... , Wn'-1,n,') si, et seulement si, n = n', W 1 est distingue clans Wt' et la plaque de TVi,1+1 determinee par x est contenue clans la plaque de TV/,;+1 determinee par x. Soit (£" la classe des chaines pures pointees; (£" est une categorie pour la loi de composition : (x, r, x') (xi', r', x") = (x, r r', x") si, et seulement si, x' = X1 1 et le compose r r' est defini clans (£. Soit (£"/r le groupoide quotient de(£" par la relation d'equivalence r : (x, r, x') "-' (x 1, r 1, x 1') si, et seulement si, x = x 1, x' = xi' et r""' r 1 modulo r. La classe (x, r, x') modulo r sera notee [x, r, x']. (£" est munie de la relation d'ordre : (x, r, x') < (xi, r 1, xi') si, et seulement si, x = x 1 , x' = xi' et r < r 1 ; par passage au quotient, (£'jr est aussi muni d'une relation de preordre. Soit r' la plus petite relation d'equivalence sur (£"/r qui soit compatible avec la loi de composition et avec la relation : C "-' C' si, et seulement si, C < C' ou si C' < C. Nous designerons par (£le groupoide quotient de (£' /r par r'. Soit m un recouvrement de E par des ouverts simples U. Soit C£m· la sous-categorie de (£" formee des chaines pures pointees (x, W1, ... , Wn; W1,2, ... , Wn-1.n, x') telles que W1 soit un ouvert sature clans un ouvert U E m. Soit (£m· / r le sous-groupoide de (£'jr quotient de (£m· par la relation d'equivalence rm induite par r sur (fa"; cette relation d'equivalence est aussi la plus petite relation d'equivalence compatible avec la loi de composition et avec les relations elementaires entre elements de C£m·. Soit rm' la plus petite relation d'equivalence sur (£m'jr compatible avec la loi de composition et avec la relation : C ,..._,. C' si, et seulement si, ii existe C" E (£"/r tel que C" < C et C" < C'. Pour tout c E f£m·,

135

STRUCTURES FEUILLETEES

la classe (c modulo rm) modulo rm' sera notee [c] modulo nJt 1 • Soit C£m- le groupoide quotient de f£m'/ r par r,;n'. L'application : [c] modulo rm' - [c]modulo r' est une application de C£m- sur (£-, qui n'est generalement pas biunivoque. Pour tout r = (W1, ... , Wn; W1.2, ... , Wn-1,n) E f£m, soit U(r) l'ensemble des elements C = [x, r, x'] modulo 11R' E C£m-, ou x E W 1 et x' E Wn, L'ensemble U(r) sera muni de la topologie T(r) ayant pour base les ensembles U(r, Wi', Wn') des elements [x, r, x'] modulo nJt 1 ou x E W 1', x' E Wn', W 1' et Wn' etant des ouverts contenus clans W 1 et Wn, On munit aussi U(r) de la topologie T'(r) ayant pour base les ensembles U(r, Vi', Vn') des classes [x, r, x'] modulo rm', ou x E Vi', x' E Vn', Vi' et Vn' etant des plaques de W 1' et Wn'• Le couple (T(r), T'(r)) definit un feuilletage topologique sur U(r). PROPOSITION. Avec les notations precedentes, C£m- est un groupoide topologique pour la topologie T'iR (respectivement Tm') ayant pour base les ensembles U(r, Wi', Wn') (respectivement U(r, V 1', Vn')) et le couple (Tm, Tm') est un feuilletage topologique.

Le sous-espace de ~m- forme des unites s'identifie la topologie T (respectivement T'). Nous designerons par er et les projections :

aE

muni de

rr

er: [x, 13- : [x,

r, x'] modulo rffl' - x' r, x'] modulo rm' - x

de C£m- sur E. Soit 91 un es pace transversal; designons par .Em (91) le souses pace de C£m- forme des elements C tels que 13-(C) E 91. Alors la projection a- etale .Em(91) sur le sous-espace de E reunion des feuilles de (T, T') qui rencontrent 91. On appellera .Em(91) le deroulement relatif a fR au-dessus de 91. Le feuilletage (TIJ"t, T1J1') induit sur .Em(91) par (Tm, Tm') s'identifie au feuilletage defini par l'etalement a-. PROPOSITION. Avec les notations precedentes, le feuilletage (TIJ"t, T1J1') de EIJ"t (91) est simple et son espace transverse est identifie a 91 par !'application 13-.

Soit F une feuille de (T, T'), fR(F) le recouvrement de F par

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CHARLES EHRESMANN

les plaques V des ouverts U E m. Soit ~9l(Fl. !'ensemble des chaines de plaques pointees (x, 'Y, x') = (x, V1, ... , Vn; V1,2, ... , Vn-1,n,x') OU VE m(F) pour tout i < n. On definit comme precedemment (en considerant sur F le feuilletage trivial (T', T')) une loi de composition clans ~9l(F). et des relations nR(F) et nRrF/. Soit ~9l(F)- = (~91(F\·;nR(F))/rm(F) 1 • Pour tout element (x, 'Y, x') de ~9l(F) • ii existe des chaines pures pointees appartenant a ~m· dont (x, 'Y, x') soit la chaine de plaques associee reliant x a x'; soit C(x, 'Y, x') une de ces chaines. Soit 'P !'application : [x, 'Y, x'] modulo ra,F/--+ [C(x, 'Y, x')] modulo rm'. PROPOSITION. Avec les notations precedentes, 'Pest un isomorphisme de ~9l(F)- sur le sous-groupoide de ~91- forme des elements C tels C) = X, OU X decrit F. que

rr (

Supposons x fixe clans F. Soit Em (x) le sous-espace topologique de ~9l(F)- forme des elements [x, 'Y, x'] modulo rm1F>' et ~m-(x) le sous-groupe de ~9l(F)- forme des elements [x, 'Y, x] modulo ra 1Fi'; ~m-(x) s'identifie aussi a un sous-groupe de ~m-. PROPOSITION. Avec les notations precedentes, Em(F) (x) est un revetement normal de F pour ['application : [x, 'Y, x'] modulo nRcF>'--+ x'; ~m-(x) est antiisomorphe au groupe des automorphismes de ce revetement, Si m(F) est formed' ouverts simplement connexes, Em(F) (x) s'identijie au revetement universe! de F et ~m-(x) au groupe fondamental de F.

L'antiisomorphisme est !'application qui associe a s E Q:m-(x) l'automorphisme du revetement y--+ sy, ou y E Em-(C) le jet local z,,jl(Wn, h, W1) E H'. Soit C = [C] modulo ra' = [C1 ] modulo ra' ou C1 E ~m·; alors on a j>-(C) = j>-(C1). Soit y; !'application : C--+ j>-(C). v

PROPOSITION. Avec les notations precedentes, y; est un foncteur contravariant de Q:m- sur H'. En particulier, on a une antirepresentation de ~m-(x) sur le groupe d'holonomie Hx', ou x E F.

Soit WE m. Soit ~m-(W) !'ensemble des elements [x, r, x'] modulo ra' tels que le premier element W 1 de r soit un ouvert

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sature de W; cet ensemble est muni de la relation d'equivalence rot": [x, r, x']modulo m' ,..._, [xi, r 1, xi'] modulo rot' si, et seulement si, x' = Xi', r = r1 et Xt appartient a la plaque xw de W contenant x. La classe d'equivalence ([x, r, x']modulo rm')modulo rm" sera notee [xw, r, x'],,cm). Soit Em(W) !'ensemble des classes d'equivalence modulo rm". Soit r une chaine pure (Wi, . .. , Wn; W1,2, ... 1 Wn-1,n), OU W1 est sature clans W, et soit U(r, Wi', Wn') !'ensemble des elements [xw, r, x'],,(m), OU xw est une plaque de Wi', x' E Wn, Wi' est un ouvert sature clans W 1 et Wn' un ouvert contenu clans Wn; soit U(r, xw, Vn') !'ensemble des elements [xw, r, x'],,(m), OU x' E Vn', Vn' etant une plaque de Wn'. Soit Tm (W) (respectivement Tm' (W)) la topologie sur Em(W) ayant pour base les ensemble U(r, W 1', Wn') (respectivement U(r, xW, Vn')). Le couple (Tm(W), Tm'(W))est un feuilletage topologique simple dont l'espace transverse s'identifie a l'espace transverse WW de W par }'application : [xw, r, x'],,cm)

---t

xw.

De plus !'application a-: [xw, r, x'],, ---t x' est un etalement de Em(W) sur la reunion des feuilles de (T, T') rencontrant W. On appellera .Em(W) le deroulement relatif a mau-dessus de W. Si N est un espace transversal elementaire contenu clans W et applique canoniquement sur Ww, !'application : [x, r, x'] modulo r'm

---t

[xw, r, x'],,

ou x E N, est un isomorphisme des espaces feuilletes .Em(N) sur Em(W). Soit x !'application : [xw,

r, x'],, ---t z"j;,(W1,J1. Wn)

de Em(W) sur E*(W), ou f designe l'isomorphisme d'holonomie associe a la chaine pure r. PROPOSITION. Avec les notations precedentes, x est un etalement de Em(W) sur E*(W).

Remarquons que la topologie de Em(W) peut ne pas etre separee meme si T est une topologie separee. En particulier, on pourra considerer pour m le recouvrement

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forme par tous les ouverts simples; m(F), ou Fest une feuille de (T, T'), est alors le recouvrement de F forme par toutes les plaques de F. L'espace Em(W) sera note dans ce cas E(W); !'application :

[xy, r, x'J,,rm) -

[xy, r, x'],,

est un etalement de Em(W) sur E(W), pour tout recouvrement m. Nous supposerons dans la fin de ce paragraphe que tout point de E admet un voisinage qui soit un ouvert simple dont toutes Jes plaques soient simplement connexes. PROPOSITION. Si m' est un recouvrement de E forme d'ouverts simples dont toutes les plaques soient simplement connexes, toute Jeuille F de Em' ( W) s' ident~tie au revetement universe! de la J euille or (F'). PROPOSITION. Soit m' un recouvrement de E f orme d' ouverts simples dont toutes les plaques soient simplement connexes; pour tout recouvrement m de E Jorme d' ouverts simples, l' es pace Em' ( W) muni de la topologie Tm' ( W) est etale au-dessus de l' es pace Em ( W) muni de la topologie Tm(W).

L'etalement est !'application : [xy,

r, x'],,..+1) E A' pour tout X < m, et dont la loi de composition est definie par : si, et seulement si, a 1m = a 1, 1 • Soit ~(~)/r le groupoide quotient de ~(~) par la plus petite relation d'equivalence r compatible avec la loi de composition et avec les relations elementaires :

Soit

P>..

une relation elementaire sur ~(~) :

a

une famille A de telles relations P>.. est associe le groupoide (~, A')/A quotient de~(~) par la plus petite relation d'equivalence p compatible avec la Joi de composition et avec la relation r et les relations P>.. E A; ce groupoide est un groupoide quotient du groupoide libre ~(~)/r (cette construction est un cas particulier de la construction d'un groupoide par la donnee de generateurs et de relations, les generateurs etant ici Jes couples (aJt ak), ou (j, k) E A'). Soit Sf (K(w)/G') le groupoide 5ll)/Aw suivant : ~w est la famille des ouverts (i, s), ou (i, s) E IX G; 5ill est !'ensemble des couples ((i, s), (j, s')) tels que (i,j) E Mou i = j, et que (j, s') soit contenu clans (i, s;j); Aw est la famille des relations :

mw,

((i1, s1), ... , (im, Sm)) ,..._., (i1, s1)

si

(ii, S1)

= (im,

Sm)

et s'il existe un element de m' qui contienne (ikt sk), OU k < m. Soit Sr(r /G') le groupoide (~v, $ill)/Av ou ~vest la famille des plaques VA(i, G's), ou (i, s) E IX G, et Av la famille des relations : ( VA (ii, G' s1), ... , VA Cim, G' Sm)) ,..._., VA (i1, G' s1) si, pour une relation

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appartenant cl Aw, on a ((i1, s1), ... , (im, Sm)) ,..._, (ii, s1). Soit de meme St(r' /G') le groupoi'de obtenu en rempla!;ant clans St(r/G') les plaques VA(i, G's) par les plaques VA'(i, G's). Les groupoi'des Jr(g(w)/G'), St(r/G'), et St(r'/G') sont isomorphes. En vertu du lemme qui va suivre, ii en resulte que les groupes fondamentaux de FA/G', de FA'/G' et de F' sont isomorphes. Soit B un espace topologique dont chaque point admette un voisinage ouvert simplement connexe. Supposons donnes les elements suivants : (1) Un recouvrement m: de 513 par des ouverts connexes a 1, ou j E J. (2) Une famille d'ouverts connexes aJI" ou (j, k) decrit un sousensemble A' de J X J, telle que a 1 V ak C a 1k; de plus, si a 1 n ak r5- 0, on a (j, k) EA'; enfin, aJJ = a 1. (3) Un ensemble S d'ouverts simplement connexes tel que, pour tout (j, k) E A', ii existe un ouvert appartenant a S qui contienne la reunion des ouverts a 1'k' pour lesquels a 1k n a 1'k' r5- 0. Soit p, Ia relation sur ~(m:) (voir plus haut) : (a,11 • • • , a,,,.) ,..._, a,1 si v 1 = vm et s'il existe un ouvert appartenant a S contenant la reunion des ouverts a.,..,.+it ou µ < m. Soit A une famille de telles relations p., contenant en particulier toutes Jes relations Px clans lesquelles on a : ax,. n ax,.+1 r5- 0 et ax,. n ax,,._ 1 x1 r5- 0 pour tout µ < m-1. Soit St(B) le groupoi'de (m:, A')/A, quotient de ~(m:) par la relation d'equivalence p engendree par Ia relation r et Jes relations p, E A. Les unites de St(B) s'identifient aux elements a1, ou j E J. LEMME. Avec les hypotheses precedentes, le groupe fondamental de B est isomorphe au sous-groupe de St(B) d'unite a 1.

Demonstration. Soit x un point donne de B. Soit 'Y = (a 1u . . . , a 1,,.) un element de ~(m:), OU XE ail; designons par 'YA !'ensemble des suites pointees (x, 'Y, x'), ou x' decrit a 1,,.; cet ensemble est muni de la topologie ayant pour base d'ouverts Jes ensembles -yA(A) formes des suites (x, 'Y, x') telles que x' E A et A est un ouvert contenu clans a 1,,.. Soit BA l'espace quotient de l'espace somme topologique des espaces 'YA par la relation d'equivalence r" : (x, 'Y, x') ,...., (x, ar,, ... , ar,,.', x") si, et seulement si, x' = x" et

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(aiu ... , a 1m, ai'm', ... , ai' 1 ) ,..._, ail modulo p. L'espace BA est connexe: tout element de BA peut se representer sous la forme (ai, ... , aim• x'), ou a 1 est un ouvert choisi contenant x. Soit Bun revetement universe! de B pointe en x au-dessus de x; soit b la projection canonique de B sur B. La suite (x, 'Y, x') se releve en une suite (x, a111 • •• , aim• x') telle que a1,,. et a1"+ 1 soient les composantes connexes de b- 1(a 1") et b- 1(aiP+J qui appartiennent a une meme composante connexe de b- 1 (a 1µJp+ 1 ). L'application o: (x, 'Y, x') - x' est compatible avec la relation r"; il en resulte une application BA de BA sur B, qui definit BA comme revetement de B. Puisque BA est connexe, on a BA = B. Par suite, le groupe fondamental de B s'identifie au sous-groupe de Sr(B) forme des classes (ai, a 111 • •• , a 1m, a 1) modulo r", c'est-a-dire au sous-groupe de Sr(B) d'unite a1. Un raisonnement analogue au precedent permet plus generalement de demontrer la proposition.

Soit (T, T') un feuilletage localement simple tel que T soit une topologie separee. Soit F une feuille compacte telle que tout x E F admette un voisinage ouvert simple dont l' espace transverse est separe et dont toutes les plaques sont simplement connexes. Si tout voisinage de F contient un tube t(w) d'fime F, il existe un tube t(w') d'fime F tel que le groupe fondamental de toute jeuille F' contenue dans t (w') soit isomorphe au groupe f ondamental d' un revetement de F. PROPOSITION.

Remarque. En utilisant la structure uniforme de F, on pourrait construire les recouvrements (U 1)1,r et (U 11 ),1,J ,M de F de sorte que les conditions plus strictes indiquees clans Reeb (13) soient verifiees. Dans ce cas, la consideration des groupoides Sr ( r /G') et Sr(r' /G') serait remplacee par celle des nerfs des recouvrements (VA(i, G's)) et (VA'(i, G's)), ou (i, s) E IX G; ces nerfs sont isomorphes. L'isomorphisme des groupes fondamentaux de F' et de FA/G' resulte alors du lemme suivant, qui se demontre d'une maniere analogue a celle utilisee clans le lemme precedent. LEMME. Soit B un es pace topologique; soit m: un recouvrement de B par des ouverts connexes a 1, ou j E J, verifiant la condition : si a 1 n a~ ~ 0, il existe un ouvert simplement connexe contenant la reunion de a 1 et ak. Alors le groupe fondamental de B est isomorphe au groupe f ondamental du nerf du recouvrement m:.

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THEOREME. Soil (T, T') un feuilletage localement simple et t(w) un tube parfait d' ame F, tel que l' espace transverse ww soil une variete topologique. Alors le groupe structural G de t(w) est l'ensemble des elements de (w) de source ww, ou (w) est le pseudogroupe transverse d'holonomie relatif a w. De plus, les deroulements E*(w) et EA(w) sont isomorphes.

Demonstration. Soit s l'isomorphisme d'holonomie associe a une chaine pure fermee admettant w pour premier et dernier element. Soit xw' E ww; la feuille F' con tenant £' rencontre w en p (x') plaques. Puisque p(x') est majore par le nombre d'elements de G, la transformation s est periodique. Soit s' E G tel que ix-'s = ixJs'; ii existe un voisinage de xw clans ww dont tous les points sont fixes par l'isomorphisme d'holonomie s- 1s'. Par consequent s- 1s' est une transformation periodique dont !'ensemble des points fixes admet un point interieur. En utilisant un theoreme de Newman-Smith {voir 17 et 18), on en deduit que s- 1s' est l'identite de ww, done que s = s' E G. CoROLLAIRE. Tout element de (w) dont la source est connexe est une restriction d'une transformation de G.

En effet, deux elements distincts de G n'ayant pas de restriction commune, un agregat de restrictions d'elements de G dont la source est connexe est une restriction d'un element de G. PROPOSITION. Avec les hypotheses du theoreme precedent, l'ensemble u des points xw' de WW tels que le groupe d'holonomie fix,' soit reduit a son unite est un ouvert partout dense de ww. Si F' est une feuille contenant une plaque xw' E u, alors t(w) est un tube parfait d'ame F'.

Demonstration. Puisque deux elements de G qui ont le meme jet local en un point sont identiques, pour tout x' E w le groupe fix,' est isomorphe au sous-groupe G' de G laissant fixe la plaque xw' qui contient x'; soit q(x') le nombre d'elements de G'. Soit ww' un voisinage de xw' ne rencontrant pas s(ww'), pour touts E G, s ~ G'; pour tout xw" E ww', on a q(x") < q(x'). Si q(x") = q(x') pour tout xw" E ww', toute transformation de G' laisse fixes tousles points de ww', done est l'identite de ww; par suite q(x') = 1. II en resulte que, si q(x') > 1, il existe au moins un point xw" E ww' tel que

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q(x") < q(x'). L'ensemble u des points xv' tels que q(x') = 1 est un ouvert de wv; si le complementaire de u contenait un ouvert u', ii existerait x 1 E u' tel que q(x 1) soit le minimum de q sur u'; d'apres ce qui precede, q(x 1) = 1, ce qui est contraire a l'hypothese. Done u est un ouvert partout dense clans wv. Remarque. Le theoreme et la proposition precedents sont encore vrais clans les cas plus generaux auxquels le theoreme de Smith (18) s'applique. Soit (T, T') un feuilletage localement simple et F une feuille propre de (T, T'). On ( U', U) tels que : (1) a(l) est base d'un filtre convergent vers la plaque xw et, pour tout i; E l, on a xw qa(i;). (2) La plaque £' est adherente au filtre engendre par {3(l). Remarquons que la propriete est independante du choix de x et x'. Elle entraine l'instabilite de (T, T') autour de F et de F'. Si l'espace transverse Uw est separe, cette definition entraine qu'il existe i; E l tel que xw' 1(3 (I;).

5. Complements relatifs a certaines structures feuilletees Gas de deuxfeuilletages supplementaires. Soient (T, T') et (T, Ti') deux feuilletages topologiques supplementaires sur E (c'est-a-dire sous-jacents a une structure de produit local de l'espece 't X 't elargie). Alors tout point x de E admet un voisinage U qui est un ouvert simple relativement aux deux feuilletages et tel que toute plaque de U pour l'un des feuilletages soit un espace transversal elementaire pour l'autre; U sera appele voisinage bi-simple de x. Supposons qu'il existe un tube t(w) d'ame F relativement a (T, T') tel que les ouverts w 1 soient bi-simples. Soit G le groupe structural de t(w). Soit Y; une plaque quelconque de w 1 relative a (T, Ti'); a tout element (w 1, J, w;) du groupoide structural r de t(w) correspond l'homeomorphismef> 1. To prove this result, first note that there exists 0 that

0