Intuitively, a foliation corresponds to a decomposition of a manifold into a union of connected, disjoint submanifolds of the same dimension, called leaves, which pile up locally like pages of a book.
The theory of foliations, as it is known, began with the work of C. Ehresmann and G. Reeb, in the 1940's; however, as Reeb has himself observed, already in the last century P. Painlevé saw the necessity of creating a geometric theory (of foliations) in order to better understand the problems in the study of solutions of holomorphic differential equations in the complex field.
The development of the theory of foliations was however provoked by the following question about the topology of manifolds proposed by H. Hopf in the 1930's: "Does there exist on the Euclidean sphere S3 a completely integrable vector field, that is, a field X such that X - curl X -= O?" By Frobenius' theorem, this question is equivalent to the following: "Does there exist on the sphere S3 a two-dimensional foliation?"
This question was answered affirmatively by Reeb in his thesis, where he presents an example of a foliation of S 3 with the following characteristics: There exists one compact leaf homeomorphic to the two-dimensional torus, while the other leaves are homeomorphic to two-dimensional planes which accumulate asymptotically on the compact leaf. Further, the foliation is C. Also in the work are proved the stability theorems, one of which, valid for any dimension, states that if a leaf is compact and has finite fundamental group then it has a neighborhood consisting of compact leaves with finite fundamental group. Reeb's thesis motivated the research of other mathematicians, among whom was A. Haefliger, who proved in his thesis in 1958, that there exist no analytic two-dimensional foliations on S3 . In fact Haefliger's theorem is true
in higher dimensions.
The example of Reeb and others, which were constructed later, posed the following question, folkloric in the midst of mathematics: "Is it true that every foliation of dimension two on S3 has a compact leaf?" This question was answered affirmatively by S. P. Novikov in 1965, using in part the methods introduced by Haefliger in his thesis. In fact Novikov's theorem is much stronger. It says that on any three-dimensional, compact simply connected manifold, there exists a compact leaf homeomorphic to the two-dimensional torus, bounding a solid torus, where the leaves are homeomorphic to two-dimensional planes which accumulate on the compact leaf, in the same way as in the Reeb foliation of S3 .
One presumes that the question initially proposed by Hopf, was motivated by the intuition that there must exist nonhomotopic invariants which would serve to classify three-dimensional manifolds. In fact this question did not succeed in this objective, since any three-dimensional manifold does admit a two-dimensional foliation. However, a refinement proposed by J. Milnor with the same motivation, had better results. In effect, Milnor defined the rank of a manifold as the maximum number of pairwise commutative vector fields, linearly independent at each point, which it is possible to construct on the manifold. This concept translates naturally in terms of foliations associated to actions of the group Et'. The problem proposed by Milnor was to calculate the rank of S. This problem was solved by E. Lima in 1963 by showing that the rank of a compact, simply connected, three-dimensional manifold is one. Later H. Rosenberg, R. Roussarie and D. Weil classified the compact three-dimensional manifolds of rank two.
In this book we intend to present to the reader, in a systematic manner, the sequence of results mentioned above. The later development of the theory of foliations, has accelerated, especially in the last ten years. We hope that this book motivates the reading of works not treated here. Some of these are listed in the bibliography.
César Camacho
Alcides Lins Neto