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

Description

The Geometric Theory of Foliations is one of the fields in Mathematics that gathers several distinct domains: Topology, Dynamical Systems, Differential Topology and Geometry, among others. Its great development has allowed a better comprehension of several phenomena of mathematical and physical nature. Our book contains material dating from the origins of the theory of foliations, from the original works of C Ehresmann and G Reeb, up till modern developments. In a suitable choice of topics we are able to cover material in a coherent way bringing the reader to the heart of recent results in the field. A number of theorems, nowadays considered to be classical, like the Reeb Stability Theorem, Haefliger's Theorem, and Novikov Compact leaf Theorem, are proved in the text. The stability theorem of Thurston and the compact leaf theorem of Plante are also thoroughly proved. Nevertheless, these notes are introductory and cover only a minor part of the basic aspects of the rich theory of foliations. Preliminaries -- Plane fields and foliations -- Topology of the leaves -- Holonomy and stability -- Haefliger's theorem -- Novikov's compact leaf -- Rank of 3-manifolds -- Tischler's theorem -- Plante's compact leaf theorem -- Currents, distributions, foliation cycles and transverse measures -- Foliation cycles: a homological proof of Novikov's compact leaf theorem.