Geometry, dynamics, and topology of foliations : a first course

194 Pages·2017·2.108 MB·English

by  Rojas, Carlos Arnoldo Morales, Scárdua, Bruno

GEOMETRY,
DYNAMICS AND
TOPOLOGY OF
FOLIATIONS
A First Course
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GEOMETRY,
DYNAMICS AND
TOPOLOGY OF
FOLIATIONS
A First Course
Bruno Scardua
Federal University of Rio de Janeiro, Brazil
Carlos Arnoldo Morales Rojas
Federal University of Rio de Janeiro, Brazil
World Scientific
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Library of Congress Cataloging-in-Publication Data
Names: Scardua, Bruno. | Rojas, Carlos Arnoldo Morales.
Title: Geometry, dynamics, and topology of foliations : a first course / by
Bruno Scardua (University of Rio de Janeiro, Brazil),
Carlos Arnoldo Morales Rojas (Federal University of Rio de Janeiro, Brazil).
Description: New Jersey : World Scientific, 2017. | Includes bibliographical references and index.
Identifiers: LCCN 2016059937 | ISBN 9789813207073 (hardcover : alk. paper)
Subjects: LCSH: Foliations (Mathematics) | Differential topology.
Classification: LCC QA613.62 .S33 2017 | DDC 514/.72--dc23
LC record available at https://lccn.loc.gov/2016059937
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Preface
The Geometrical Theory of Foliations is one of the fields in Mathematics
thatgathersseveraldistinctivedomainssuchas; Topology,DynamicalSys-
tems,DifferentialTopologyandGeometry,etc. Itoriginatedfromtheworks
ofC.EhresmannandG.Reeb([Ehresmann(1947)]),([EhresmannandReeb
(1944)]). Thehugedevelopmenthasallowedabettercomprehensionofsev-
eral phenomena of mathematical and physical nature. Classical theorems,
liketheReebstabilitytheorem,Haefliger’stheorem,andNovikov’scompact
leaf theorem, are now searched for holomorphic foliations. Several authors
havebegantoinvestigatesuchphenomena(e.g. C.Camacho,A.LinsNeto,
E. Ghys, M. Brunella, R. Moussu, S. Novikov and others). The study of
such field presumes knowledge of results, techniques of the real case and
superior familiarity with the classical aspects of Holomorphic Dynamical
Systems.
There is a number of important books dedicated to the study of folia-
tions, specially inthe non-singularsmoothframework. We shall notlist all
of them, but we cannot avoid mentioning the books of Camacho and Lins
Neto ([Camacho and Lins-Neto (1985)]), Candel and Conlon ([Candel and
Conlon (2000)]), C. Godbillon ([Godbillon (1991)]) and Hector and Hirsch
([Hector and Hirsch (1987)]), which are among our favorite. Each of these
has influenced in our text. From the choice of topics, to the path taken
in some demonstrations. In our viewpoint, the book of Camacho and Lins
Netoisimportantforitswisechoiceoftopics,andforaimingatthegeome-
tryoffoliations. ThebookofHectorandHirschhasinfluencedusspecially
in the interplaybetweengeometryanddynamics offoliations. The bookof
C. Godbillon is very interesting for the wide range of topics that are cov-
ered. The proofs are elegant, usually short, but still precise. Finally, the
book of Candel and Conlon, divided into two volumes, is a very detailed
vii
viii Geometry, Dynamics and Topology of Foliations - a first course
introduction to the general theory of foliations. There, one can find a very
complete exposition of the important results in the theory of codimension
onefoliations. Specialattentionisgiventothetheoryofminimalsets. This
materialhasgreatlyinfluencedourexpositionhere. Theirbookistherefore
a natural complement to any introduction to the theory of foliations, and
should used as a reference any further information and courses.
Finally,wewouldliketomentionthatourpersonalinterestforfoliations
has been greatly supported by some amazing works from several authors.
Besides the authors mentioned above or in the text itself we wish to men-
tion P. Schweitzer, D. Calegari, P. Molino, S. Fenley, P. Thondeur and T.
Tsuboi.
These notes are mainly introductory and only cover part of the basic
aspects of the rich theory of foliations. In particular, additional extensive
information in some of the results presented here, may be searched in the
bibliographywegive. Wehavetriedtoclarifythegeometryofsomeclassical
results and provide motivation for further study. Our goal is to highlight
thisgeometricalviewpointdespitesomeloss(?) offormalism. Wehopethat
this text may be useful to those who appreciate Mathematics. Specially,
to the students that are interested in this exquisite and conducive field of
Mathematics.
This text is divided into two basic parts. The first part, which cor-
responds to the first eight chapters, consists of an exposition of classical
results in Geometric Theory of (real) Foliations. Special attention is paid
to the classicalReebStability theorems,Haefliger’stheoremandNovikov’s
compact leaf theorem.
Starting at Chapter 9, the second part contains a robust proof of
Plante’s Theorem on growth and compact leaves. This is followed by the
basic ingredients of the theory of foliation cycles and currents which is de-
veloped in Chapter 10. Then in Chapter 11 this is applied in D. Sullivan’s
homological proof of Novikov’s compact leaf theorem. It is based in a mix
of topological argumentation and invariant measure theory for foliations.
Chapter A is dedicated to some more specialized results concerning the
structure of codimension one foliations on closed manifolds. We present
the results of Dippolito on the structure of codimension one foliations and
semi-stability. AlsowepresentCatwell-Conlon’sresultonthe minimal sets
for such foliations.
We presentin Chapter 11, D. Sullivan’s homological proofof Novikov’s
compact leat theorem. This appears to be an applicable procedure for
complex foliations. We invite the reader to think about it.
Preface ix
In the last part of the book we give an exposition of some important
results on the structure of codimension one foliations. We state results of
Dippolito and Cantwell-Conlonon the their structure.
We hope the reader will enjoy reading this book as much as we have
enjoyed writing it.
Carlos Morales and Bruno Sc´ardua