BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 3, May 1974

FOLIATIONS BY H. BLAINE LAWSON, JR.1 TABLE OF CONTENTS

1. 2. 3. 4. 5. 6. 7. 8.

Definitions and general examples. Foliations of dimension-one. Higher dimensional foliations; integrability criteria. Foliations of codimension-one; existence theorems. Notions of equivalence; foliated cobordism groups. The general theory; classifying spaces and characteristic classes for foliations. Results on open manifolds; the classification theory of Gromov-Haefliger-Phillips. Results on closed manifolds; questions of compact leaves and stability.

Introduction. The study of foliations on manifolds has a long history in mathematics, even though it did not emerge as a distinct field until the appearance in the 1940's of the work of Ehresmann and Reeb. Since that time, the subject has enjoyed a rapid development, and, at the moment, it is the focus of a great deal of research activity. The purpose of this article is to provide an introduction to the subject and present a picture of the field as it is currently evolving. The treatment will by no means be exhaustive. My original objective was merely to summarize some recent developments in the specialized study of codimension-one foliations on compact manifolds. However, somewhere in the writing I succumbed to the temptation to continue on to interesting, related topics. The end product is essentially a general survey of new results in the field with, of course, the customary bias for areas of personal interest to the author. Since such articles are not written for the specialist, I have spent some time in introducing and motivating the subject. However, this article need not be read linearly. §§ 1, 2, 3 and 5 fall into the category of "basic material." §§ 4, 8 and the combination 6-7 are essentially independent of each other. I would like to thank Bill Thurston and André Haefliger for making several valuable suggestions for improving the manuscript. An address delivered before the 78th Annual Meeting of the Society in Las Vegas, Nevada, on January 17, 1972 by invitation of the Committee to Select Hour Speakers for the Summer and Annual Meetings, under the title Foliations of compact manifolds', received by the editors May 7, 1973. AMS (MOS) subject classifications (1970). Primary 57D30. 1 Work partially supported by NSF grant GP29697. Copyright ® American Mathematical Society 1974

369

370

H. B. LAWSON, JR.

[May

1. Definitions and general examples. A manifold is, roughly speaking, a space locally modelled on affine space; and a submanifold is a subset locally modelled on an affine subspace. In this spirit, a foliated manifold is a manifold modelled locally on an affine space decomposed into parallel affine subspaces. r DEFINITION 1. By a p-dimensional, class C foliation of an m-dimensional manifold M we mean a decomposition of M into a union of disjoint connected subsets {U(XXQ, and transition functions gafi : Ua n Up->Diff(Q) such that ha o h~^\x, y)=(x, ga/j(x)(y)). If the transition functions are locally constant, the bundle is said to have discrete structure group. Note that under this assumption, the codimensionq (Q fit together to give a foliation of M. Every such bundle can be constructed in the following way. Let 9?:7r1(jP)-^Diff(ô) be a homomorphism and denote by P the universal covering space of P. Then TT^P) acts jointly on the product PxQ, and we define M=PxQl7T1(P). The action preserves the product structure, and so the product foliation of PxQ (arising from P x Q-+Q) projects to a foliation of M. This is the foliation we described above. Note that each leaf looks like a many-valued cross-section of the bundle M-^>?P. In fact, 7T restricted to any leaf is a covering map. To see this note that if J5? is the leaf corresponding to Px{x}

FOLIATIONS BY H. BLAINE LAWSON, JR.1 TABLE OF CONTENTS

1. 2. 3. 4. 5. 6. 7. 8.

Definitions and general examples. Foliations of dimension-one. Higher dimensional foliations; integrability criteria. Foliations of codimension-one; existence theorems. Notions of equivalence; foliated cobordism groups. The general theory; classifying spaces and characteristic classes for foliations. Results on open manifolds; the classification theory of Gromov-Haefliger-Phillips. Results on closed manifolds; questions of compact leaves and stability.

Introduction. The study of foliations on manifolds has a long history in mathematics, even though it did not emerge as a distinct field until the appearance in the 1940's of the work of Ehresmann and Reeb. Since that time, the subject has enjoyed a rapid development, and, at the moment, it is the focus of a great deal of research activity. The purpose of this article is to provide an introduction to the subject and present a picture of the field as it is currently evolving. The treatment will by no means be exhaustive. My original objective was merely to summarize some recent developments in the specialized study of codimension-one foliations on compact manifolds. However, somewhere in the writing I succumbed to the temptation to continue on to interesting, related topics. The end product is essentially a general survey of new results in the field with, of course, the customary bias for areas of personal interest to the author. Since such articles are not written for the specialist, I have spent some time in introducing and motivating the subject. However, this article need not be read linearly. §§ 1, 2, 3 and 5 fall into the category of "basic material." §§ 4, 8 and the combination 6-7 are essentially independent of each other. I would like to thank Bill Thurston and André Haefliger for making several valuable suggestions for improving the manuscript. An address delivered before the 78th Annual Meeting of the Society in Las Vegas, Nevada, on January 17, 1972 by invitation of the Committee to Select Hour Speakers for the Summer and Annual Meetings, under the title Foliations of compact manifolds', received by the editors May 7, 1973. AMS (MOS) subject classifications (1970). Primary 57D30. 1 Work partially supported by NSF grant GP29697. Copyright ® American Mathematical Society 1974

369

370

H. B. LAWSON, JR.

[May

1. Definitions and general examples. A manifold is, roughly speaking, a space locally modelled on affine space; and a submanifold is a subset locally modelled on an affine subspace. In this spirit, a foliated manifold is a manifold modelled locally on an affine space decomposed into parallel affine subspaces. r DEFINITION 1. By a p-dimensional, class C foliation of an m-dimensional manifold M we mean a decomposition of M into a union of disjoint connected subsets {U(XXQ, and transition functions gafi : Ua n Up->Diff(Q) such that ha o h~^\x, y)=(x, ga/j(x)(y)). If the transition functions are locally constant, the bundle is said to have discrete structure group. Note that under this assumption, the codimensionq (Q fit together to give a foliation of M. Every such bundle can be constructed in the following way. Let 9?:7r1(jP)-^Diff(ô) be a homomorphism and denote by P the universal covering space of P. Then TT^P) acts jointly on the product PxQ, and we define M=PxQl7T1(P). The action preserves the product structure, and so the product foliation of PxQ (arising from P x Q-+Q) projects to a foliation of M. This is the foliation we described above. Note that each leaf looks like a many-valued cross-section of the bundle M-^>?P. In fact, 7T restricted to any leaf is a covering map. To see this note that if J5? is the leaf corresponding to Px{x}