Introduction to Foliations and Lie Groupoids
This is just a small book nicely covering principal notions of the theory of foliations and its relations to recently introduced notions of Lie groupoids and Lie algebroids. After the first chapter, containing a definition of a foliation and main examples and constructions, the authors introduce the key notion of holonomy of a leaf, a definition of an orbifold and they prove the Reeb and the Thurston stability theorems. Chapter 3 contains the Haefliger theorem (there are no analytic foliations of codimension 1 on S3) and the Novikov theorem (concerning existence of compact leaves in a codimension 1 transversely oriented foliation of a compact three-dimensional manifold). The Molino structure theorem for foliations defined by nonsingular Maurer-Cartan forms is treated in Chapter 4. A holonomy groupoid of a foliation is a basic example of so called Lie groupoids. The last two chapters describe properties of Lie groupoids, a notion of weak equivalence between Lie groupoids, a special class of étale groupoids, and a Lie algebroid as an infinitesimal version of a Lie groupoid. The book is based on course lecture notes and it still keeps its qualities and nice presentation.