Introduction to the geometry of foliations. Part B: Foliations of codimension one.

*(English)*Zbl 0552.57001
Aspects of Mathematics, Vol. E3. Braunschweig - Wiesbaden: Friedr. Vieweg & Sohn. X, 298 p. DM 52.00; $ 21.50; £14.00 (1983).

This book gives a very extensive and detailed account of many important topics in the theory of foliations of codimension one. As Part B of a two volume work [for Part A (1983) see Zbl 0486.57002)], it begins with Chapter 4, a treatment of basic constructions and examples. The transverse one-dimensional foliation plays a major role here and throughout the book. An unusual feature of this book is that the transverse foliation is constructed even in the delicate case in which the codimension-one foliation is only of class \(C^ 0.\)

Chapter 5 treats basic structure theory, featuring the structure of open, saturated sets in compact, foliated manifolds of codimension one. Also treated are holonomy, linear holonomy, stability theorems for compact leaves, and minimal sets.

In Chapter 6, the study of exceptional minimal sets, begun in the previous chapter, is continued. A complete proof is given of Sacksteder’s important result that, arbitrarily near any semiproper, exceptional leaf, there is a leaf with nontrivial linear holonomy (provided the foliation has \(C^ 2\)-smoothness). In particular, local minimal sets of exceptional type always have resilient leaves.

Chapter 7 treats one-sided holonomoy, vanishing cycles, and closed transversals. The main result proved here is that the presence of a nullhomotopic, closed transversal implies the existence of a leaf with one-sided holonomy and of a leaf with a nontrivial vanishing cycle.

The detailed structure theory of closed, foliated manifolds without holonomy is treated in Chapter 8. The important theorem of Tischler, Joubert, and Moussu, for foliations defined by a closed, nonsingular 1- form is proved. The \(C^ 0\) case is also treated and it is proved that the lifted foliation on the universal cover is a fibration over the real line. Using this, they prove the theorem of Sacksteder-Imanishi that, in the \(C^ 2\) case, such a folitation is homeomorphic to one defined by a closed, nonsingular 1-form. Even for the \(C^ 0\) case, they obtain the Tischler-like theorem that M fibers over the circle.

In Chapter 9 the theory of growth (of groups, homogeneous spaces, Riemannian manifolds, orbits of pseudogroups, and leaves of foliations) is treated. A number of theorems, due to various authors, are proved that relate growth properties to other geometric features of codimension-one foliations. A direct and elementary proof is given that, in the presence of a resilient leaf, there is an open, saturated set of leaves with exponential growth. This is an important observation vis à vis a recent result of Duminy on the vanishing of the Godbillon-Vey class whenever no leaf is resilient.

Chapter 10 gives an extensive treatment of holonomy-invariant measures and the relations with growth. A major part of these results is due to J. F. Plante.

Although, as the authors admit, a number of important topics are omitted, some selection was necessary and the topics chosen make this book a goldmine of information for mathematicians interested in foliation theory.

Chapter 5 treats basic structure theory, featuring the structure of open, saturated sets in compact, foliated manifolds of codimension one. Also treated are holonomy, linear holonomy, stability theorems for compact leaves, and minimal sets.

In Chapter 6, the study of exceptional minimal sets, begun in the previous chapter, is continued. A complete proof is given of Sacksteder’s important result that, arbitrarily near any semiproper, exceptional leaf, there is a leaf with nontrivial linear holonomy (provided the foliation has \(C^ 2\)-smoothness). In particular, local minimal sets of exceptional type always have resilient leaves.

Chapter 7 treats one-sided holonomoy, vanishing cycles, and closed transversals. The main result proved here is that the presence of a nullhomotopic, closed transversal implies the existence of a leaf with one-sided holonomy and of a leaf with a nontrivial vanishing cycle.

The detailed structure theory of closed, foliated manifolds without holonomy is treated in Chapter 8. The important theorem of Tischler, Joubert, and Moussu, for foliations defined by a closed, nonsingular 1- form is proved. The \(C^ 0\) case is also treated and it is proved that the lifted foliation on the universal cover is a fibration over the real line. Using this, they prove the theorem of Sacksteder-Imanishi that, in the \(C^ 2\) case, such a folitation is homeomorphic to one defined by a closed, nonsingular 1-form. Even for the \(C^ 0\) case, they obtain the Tischler-like theorem that M fibers over the circle.

In Chapter 9 the theory of growth (of groups, homogeneous spaces, Riemannian manifolds, orbits of pseudogroups, and leaves of foliations) is treated. A number of theorems, due to various authors, are proved that relate growth properties to other geometric features of codimension-one foliations. A direct and elementary proof is given that, in the presence of a resilient leaf, there is an open, saturated set of leaves with exponential growth. This is an important observation vis à vis a recent result of Duminy on the vanishing of the Godbillon-Vey class whenever no leaf is resilient.

Chapter 10 gives an extensive treatment of holonomy-invariant measures and the relations with growth. A major part of these results is due to J. F. Plante.

Although, as the authors admit, a number of important topics are omitted, some selection was necessary and the topics chosen make this book a goldmine of information for mathematicians interested in foliation theory.

Reviewer: L.Conlon

##### MSC:

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

57R30 | Foliations in differential topology; geometric theory |