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Foliations I
Alberto Candel, California Institute of Technology, Pasadena, CA, and Lawrence Conlon, Washington University, St. Louis, MO
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Graduate Studies in Mathematics
2000; 402 pp; hardcover
Volume: 23
ISBN-10: 0-8218-0809-5
ISBN-13: 978-0-8218-0809-2
List Price: US$63
Member Price: US$50.40
Order Code: GSM/23
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Foliations II - Alberto Candel and Lawrence Conlon

This is the first of two volumes on the qualitative theory of foliations. This volume is divided into three parts. It is extensively illustrated throughout and provides a large number of examples.

Part 1 is intended as a "primer" in foliation theory. A working knowledge of manifold theory and topology is a prerequisite. Fundamental definitions and theorems are explained to prepare the reader for further exploration of the topic. This section places considerable emphasis on the construction of examples, which are accompanied by many illustrations.

Part 2 considers foliations of codimension one. Using very hands-on geometric methods, the path leads to a complete structure theory (the theory of levels), which was established by Conlon along with Cantwell, Hector, Duminy, Nishimori, Tsuchiya, et al. Presented here is the first and only full treatment of the theory of levels in a textbook.

Part 3 is devoted to foliations of higher codimension, including abstract laminations (foliated spaces). The treatment emphasizes the methods of ergodic theory: holonomy-invariant measures and entropy. Featured are Sullivan's theory of foliation cycles, Plante's theory of growth of leaves, and the Ghys, Langevin, Walczak theory of geometric entropy.

This comprehensive volume has something to offer a broad spectrum of readers: from beginners to advanced students to professional researchers. Packed with a wealth of illustrations and copious examples at varying degrees of difficulty, this highly-accessible text offers the first full treatment in the literature of the theory of levels for foliated manifolds of codimension one. It would make an elegant supplementary text for a topics course at the advanced graduate level. Foliations II is Volume 60 in the AMS in the Graduate Studies in Mathematics series.

Readership

Advanced graduate students and research mathematicians interested in manifold theory and topology and related fields.

Reviews

"Both volumes are written with great care and a deep knowledge of the subject ... provide the most extensive elaboration of several aspects of foliation theory."

-- Mathematical Reviews

"The large number of well-chosen examples is one of the most striking features of the book . [It] contains several beautiful figures which help one to imagine and better understand situations described formally in the text. Therefore, graduate students, young researchers, and in fact, everybody interested in foliations, should profit from this book."

-- Mathematical Reviews

"Overall presentation is first-rate ... diagrams ... are well-crafted and reflect the strongly `graphical' nature of the subject ... A prospective reader who cares to invest the time needed to plough seriously through the book ought to be rewarded with a gratifying mathematical experience ... can also be recommended to more advanced researchers, who would enjoy seeing a compendium of major results."

-- Bulletin of the London Mathematical Society

"The authors pay great attention to examples, and you can find a large number of them in the book ... They are well-chosen and will keep the interest of the reader on a high level ... [The book is] a fundamental source for everybody with a serious interest in foliations."

-- European Mathematical Society Newsletter

Table of Contents

The Foundations
  • Foreword to Part 1
  • Foliated manifolds
  • Holonomy
  • Basic constructions
  • Asymptotic properties
Codimension One
  • Foreword to Part 2
  • Basic structures
  • Compact leaves
  • General position
  • Generalized Poincarè-Bendixson theory
  • Foliations without holonomy
Arbitrary Codimension
  • Foreword to Part 3
  • Foliation cycles
  • Foliated spaces
  • Growth, invariant measures and geometry of leaves
  • Entropy of foliations
  • Bibliography
  • Index
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