Foliations, Volume 1

Part of the ""AMS in the Graduate Studies in Mathematics"" series, this volume focuses on the qualitative theory of foliations. It explains the fundamental definitions and theorems to prepare the reader for further exploration of the topic. It considers foliations of codimension one, and also deals with foliations of higher codimension.

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.