Table Of ContentCesar Camacho
Alcides Lins Neto
Geometric Theory
of
Foliations
Translated by Sue E. Goodman
Springer Science+
Business Media, LLC
Library of Congress Cataloging in Publication Data
Camacho, Cesar, 1943-
Geometric theory of foliations.
Translation of: Teoria geometrica das folheacoes.
Bibliography: p.
Includes index.
1. Foliations (Mathematics) 2. Geometry, Differential.
I. Lins Neto, Alcides, 1947- . II. Title.
QA613.62.C3613 1985 514 73 84-10978
I.
CIP-Kurztitelaufn,all,me der Deutschen Bibliothek
Camacho, C~sar:
Geometric theory of foliations/~sar Camacho;
Alcides Lins Neto. Transl. by Sue E. Goodman -
Boston; Basel; Stuttgart: Birkhiuser, 1985.
Einheitssacht.: Theoria geom~trica das
folhea¢es <d t. >
NE: Lins Neto, Alcides: 27
All rights reserved. No part of this publication may be reproduced, stored
in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without prior
permission of the copyright owner.
c Springer Science+Business Media New York 1985
Originally published by Birkhlluser Boston, Inc. in 1985
Softcover reprint of the hardcover 1st edition 1985
ISBN 978-1-4684-7149-6 ISBN 978-1-4612-5292-4 (eBook)
DOI 10.1007/978-1-4612-5292-4
CONTENTS
Introduction 1
Chapter I - Differentiable Manifolds
§ 1. Differentiable manifolds 5
§2. The derivative 13
§3. Immersions and submersions 14
§4. Submanifolds 16
§5. Regular values 17
§6. Transversality 17
§7. Partitions of unity 18
Chapter II - Foliations
§1. Foliations 21
§2. The leaves 31
§3. Distinguished maps 32
§4. Plane fields and foliations 35
§5. Orientation 36
§6. Orientable double coverings 37
§7. Orientable and transversely orientable foliations 38
Notes to Chapter II 41
Chapter III - The Topology of the Leaves
§1 . The space of leaves 47
§2. Transverse uniformity 48
§3. Closed leaves 51
§4. Minimal sets of foliations 52
Notes to Chapter III 53
Chapter IV - Holonomy and the Stability Theorems
§ 1. Holonomy of a leaf 62
§2. Determination of the germ of a foliation in a 67
neighborhood of a leaf by the the holonomy
of the leaf
§3. Global trivialization lemma 69
§4. The local stability theorem 70
§5. Global stability theorem. Transversely orientable case 72
§6. Global stability theorem. General case 78
Notes to Chapter IV 80
Chapter V - Fiber Bundles and Foliations
§ 1. Fiber bundles 87
§2. Foliations transverse to the fibers of a fiber bundle 91
§3. The holonomy of 5' 93
§4. Suspension of a representation 93
§5. Existence of germs of foliations 100
§6. Sacksteder's Example 102
Notes to Chapter V 106
Chapter VI - Analytic Foliations of Codimension One
§1. An outline of the proof of Theorem 1 116
§2. Singularities of maps f: 1Rn - 1R 118
§3. Haefliger's construction 121
§4. Foliations with singularities on D2 123
§5. The proof of Haefliger's theorem 127
Chapter VII - Novikov's Theorem
§1. Sketch of the proof 131
§2. Vanishing cycles 133
§3. Simple vanishing cycles 138
§4. Existence of a compact leaf 140
§5. Existence of a Reeb component 149
§6. Other results of Novikov 152
§7. The non-orientable case 157
Chapter VIII - Topological Aspects of the
Theory of Group Actions
§1. Elementary properties 159
§2. The theorem on the rank of S3 163
§3. Generalization of the rank theorem 165
§4. The Poincare-Bendixson theorem for actions of R2 168
§5. Actions of the group of affine transformations 171
of the line
Appendix - Frobenius' Theorem
§ 1. Vector fields and the Lie bracket 175
§2. Frobenius' theorem 182
§3. Plane fields defined by differential forms 184
Exercises 189
Bibliography 199
Index 203
INTRODUCTION
Intuitively, a foliation corresponds to a decomposition of a manifold into a
union of connected, disjoint submanifolds of the same dimension, called leaves,
which pile up locally like pages of a book.
The theory of foliations, as it is known, began with the work of C. Ehresmann
and G. Reeb, in the 1940's; however, as Reeb has himself observed, already
in the last century P. Painleve saw the necessity of creating a geometric theory
(of foliations) in order to better understand the problems in the study of solutions
of holomorphic differential equations in the complex field.
The development of the theory of foliations was however provoked by the
following question about the topology of manifolds proposed by H. Hopf in the
1930's: "Does there exist on the Euclidean sphere S3 a completely integrable
vector field, that is, a field X such that X· curl X • 0?" By Frobenius'
theorem, this question is equivalent to the following: "Does there exist on the
sphere S3 a two-dimensional foliation?"
This question was answered affirmatively by Reeb in his thesis, where he
presents an example of a foliation of S3 with the following characteristics:
There exists one compact leaf homeomorphic to the two-dimensional torus, while
the other leaves are homeomorphic to two-dimensional planes which accu
mulate asymptotically on the compact leaf. Further, the foliation is C"". Also
in the work are proved the stability theorems, one of which, valid for any dimen
sion, states that if a leaf is compact and has finite fundamental group then
it has a neighborhood consisting of compact leaves with finite fundamental
group. Reeb's thesis motivated the research of other mathematicians, among
whom was A. Haefliger, who proved in his thesis in 1958, that there exist no
analytic two-dimensional foliations on S3• In fact Haefliger's theorem is true
in higher dimensions.
The example of Reeb and others, which were constructed later, posed the
following question, folkloric in the midst of mathematics: "Is it true that every
2 Geometric Theory of Foliations
foliation of dimension two on S3 has a compact leaf?" This question was
answered affirmatively by S. P. Novikov in 1965, using in part the methods
introduced by Haefliger in his thesis. In fact Novikov's theorem is much
stronger. It says that on any three-dimensional, compact simply connected
manifold, there exists a compact leaf homeomorphic to the two-dimensional
torus, bounding a solid torus, where the leaves are homeomorphic to two
dimensional planes which accumulate on the compact leaf, in the same way
as in the Reeb foliation of S3•
One presumes that the question initially proposed by Hopf, was motivated
by the intuition that there must exist nonhomotopic invariants which would
serve to classify three-dimensional manifolds. In fact this question did not suc
ceed .in this objective, since any three-dimensional manifold does admit a two
dimensional foliation. However, a refinement proposed by J. Milnor with the
same motivation, had better results. In effect, Milnor defined the rank of a
manifold as the maximum number of pairwise commutative vector fields,
linearly independent at each point, which it is possible to construct on the mani
fold. This concept translates naturally in terms of foliations associated to actions
of the group Rn. The problem proposed by Milnor was to calculate the rank
of S3• This problem was solved by E. Lima in 1963 by showing that the rank
of a compact, simply connected, three-dimensional manifold is one. Later H.
Rosenberg, R. Roussarie and D. Weil classified the compact three-dimensional
manifolds of rank two.
In this book we intend to present to the reader, in a systematic manner, the
sequence of results mentioned above. The later development of the theory of
foliations, has accelerated, especially in the last ten years. We hope that this
book motivates the reading of works not treated here. Some of these are listed
in the bibliography.
We wish to express here our appreciation to Airton Medeiros and Roberto
Mendes for various suggestions and especially to Paulo Sad for his collaboration
in the reading and criticism of the text.
Rio de Janeiro, May 1979
C~sar Camacho
Alcides Lins Neto
Addendum to the English edition
This book is a translation of TEO RIA GEOMETRICA DAS FOLHEAQOES,
no. 12 of the Series Projeto Euclides, published by IMP A - CNPq (BRAZIL).
In this translation the arguments of some theorems were improved and an ap-
Introduction 3
pendix about elementary properties of the fundamental group was suppressed.
We wish to acknowledge Sue Goodman for the excellent work of translation.
Rio de Janeiro, March 1984
Cesar Camacho
Alcides Lins Neto
