Foliations I (Graduate Studies in Mathematics)

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. Part 1. The Foundations -- Chapter 1. Foliated Manifolds 5 -- 1.1. What Is A Foliation? 5 -- 1.2. Foliated Atlases 23 -- 1.3. The Frobenius Theorem 34 -- Chapter 2. Holonomy 45 -- 2.1. Foliated Bundles 46 -- 2.2. The Holonomy Pseudogroup 55 -- 2.3. Germinal Holonomy 59 -- 2.4. Reeb Stability 67 -- Chapter 3. Basic Constructions 71 -- 3.1. Suspension 71 -- 3.2. Pullbacks 79 -- 3.3. Transverse Modifications 81 -- 3.4. Tangential Gluing 90 -- 3.5. Orientation Covers 94 -- 3.6. Deformations 95 -- Chapter 4. Asymptotic Properties 103 -- 4.1. Minimal Sets 103 -- 4.2. Ends Of Manifolds 110 -- 4.3. Ends Of Leaves And Limit Sets 115 -- Part 2. Codimension One -- Chapter 5. Basic Structures 123 -- 5.1. Biregular Covers 123 -- 5.2. Open, Saturated Sets 126 -- 5.3. Proper And Semiproper Leaves 133 -- Chapter 6. Compact Leaves 137 -- 6.1. The Set Of Compact Leaves 137 -- 6.2. The Thurston Stability Theorem 142 -- 6.3. Compact Leaves And Closed Transversals 145 -- Chapter 7. General Position 151 -- 7.1. The Morse Lemma And Homotopies 151 -- 7.2. Poincare -- Bendixson Theory 156 -- 7.3. Analytic Foliations 162 -- Chapter 8. Generalized Poincare -- Bendixson Theory 165 -- 8.1. Foliations And Freshman Calculus 166 -- 8.2. Sacksteder's Theorem 183 -- 8.3. The Theory Of Levels 187 -- 8.4. Leaves Of Finite Depth 197 -- Chapter 9. Foliations Without Holonomy 205 -- 9.1. Open, Saturated Sets Without Holonomy 205 -- 9.2. A Leaf-preserving Flow 208 -- 9.3. A Closed, Nonsingular 1 -- Form 217 -- 9.4. Tischler's Theorem 220 -- 9.5. Isotopy 222 -- Part 3. Arbitrary Codimension -- Chapter 10. Foliation Cycles 231 -- 10.1. Basic Notions 231 -- 10.2. Foliation Currents 239 -- 10.3. Foliations Of Dimension One 253 -- 10.4. Taut Foliations Of Codimension One 257 -- 10.5. Taut Foliations Of Arbitrary Codimension 261 -- Chapter 11. Foliated Spaces 273 -- 11.2. Foliated Atlases 275 -- 11.3. Some Constructions Of Foliated Spaces 281 -- 11.4. Geometry On Foliated Spaces 298 -- 11.5. Holonomy -- Invariant Measures 301 -- Chapter 12. Growth, Invariant Measures And Geometry Of Leaves 309 -- 12.1. Quasi-riemannian Geometry Of Leaves 309 -- 12.2. Growth 311 -- 12.3. Growth And Holonomy -- Invariant Measures 325 -- 12.4. Leaves Closed At Infinity 330 -- 12.5. Dimension Two 331 -- 12.6. Uniformization 341 -- Chapter 13. Entropy Of Foliations 347 -- 13.1. Entropy Of Maps 348 -- 13.2. Entropy Of Groups, Pseudogroups And Foliations 351 -- 13.3. Geometric Entropy 358 -- 13.4. Invariant Measures 373 -- 13.5. Resilient Leaves 379. Alberto Candel, Lawrence Conlon. Includes Bibliographical References And Index. The Book Assumes Only The Material Of A Standard Graduate Course In Algebra. It Is Suitable As A Text For A Year-long Graduate Course. The Subject Is Of Interest To Students Of Algebra, Number Theory And Algebraic Geometry. The Systematic Treatment Presented Here Makes The Book Also Valuable As A Reference.--book Jacket. Ch. 1. Introduction -- Ch. 2. Semisimple Rings And Modules -- 2.1. Basic Notions -- 2.2. Structure Theorems -- 2.3. Idempotents And Blocks -- 2.4. Behavior Under Field Extensions -- 2.5. Theorems Of Burnside And Frobenius-schur -- Ch. 3. Semisimple Group Representations -- 3.1. Examples And General Results -- 3.2. Representations Of Abelian Groups -- 3.3. Decomposition Of The Regular Representation -- 3.4. Applications Of Frobenius's Theorem -- 3.5. Characters -- 3.6. Idempotents And Their Uses -- 3.7. Subfields Of The Complex Numbers -- 3.8. Fields Of Positive Characteristic -- Ch. 4. Induced Representations And Applications -- 4.1. Induced Representations -- 4.2. Mackey's Theorem -- 4.3. Permutation Representations -- 4.4. M-groups -- 4.5. Theorems Of Artin And Brauer -- 4.6. Degrees Of Irreducible Representations -- Ch. 5. Introduction To Modular Representations -- Ch. 6. General Rings And Modules -- 6.1. Jordan-holder And Krull-schmidt Theorems -- 6.2. The Jacobson Radical -- 6.3. Rings Of Finite Length -- 6.4. Finite-dimensional Algebras -- Ch. 7. Modular Group Representations -- 7.1. General Results -- 7.2. Characters And Brauer Characters -- 7.3. Examples -- App. Some Useful Results. Steven H. Weintraub. Includes Bibliographical References (p. 209) And Index. This book is an introduction to the representation theory of finite groups from an algebraic point of view, regarding representations as modules over the group algebra. The approach is to develop the requisite algebra in reasonable generality and then to specialize it to the case of group representations. Methods and results particular to group representations, such as characters and induced representations, are developed in depth. Arithmetic comes into play when considering the field of definition of a representation, especially for subfields of the complex numbers. The book has an extensive development of the semisimple case, where the characteristic of the field is zero or is prime to the order of the group, and builds the foundations of the modular case, where the characteristic of the field divides the order of the group. The book assumes only the material of a standard graduate course in algebra. It is suitable as a text for a year-long graduate course. The subject is of interest to students of algebra, number theory and algebraic geometry. The systematic treatment presented here makes the book also valuable as a reference. A graduate textbook introducing group representation theory to students who have no prior knowledge of it, and may not even know what a representation is. They are expected to have a sound knowledge of linear algebra and a good familiarity with basic module theory. Annotation (c) Book News, Inc., Portland, OR (booknews.com) A guide to the qualitative theory of foliations. It features topics including: analysis on foliated spaces, characteristic classes of foliations and foliated manifolds. It is suitable as a supplementary text for a topics course at the advanced graduate level. We shall begin by giving some basic examples of the objects and questions we will be studying.