Foliated bundles and characteristic classes.

*(English)*Zbl 0308.57011
Lecture Notes in Mathematics. 493. Berlin-Heidelberg-New York: Springer-Verlag. xiv, 207 p. DM 23.00; $ 9.50 (1975).

The foliated bundles studied in this book generalize the bundle of transverse frames of a foliation on a differentiable manifold. Namely, following the authors’ definition, a foliated bundle is a \(G\)-principal bundle \(P\to M\) where \(M\) is a foliated manifold and \(P\) is endowed with a “flat partial connection”, i.e. an involutive distribution which projects onto the foliation of \(M\), it is transverse to the fibers and is invariant by right translations. The notion of a foliated bundle covers a wide range of geometrically important situations. The authors’ major contribution to the theory of foliated bundles consists in the construction of the theory of the characteristic classes of these bundles, which is a deep and interesting generalization of the classical Chern-Weil theory. In the case of the transverse frame bundle of a foliation, a theory of the characteristic classes of foliations is obtained which, in the most important cases, gives the same results as the other known theories, especially the theory of R. Bott and A. Haefliger [Bull. Am. Math. Soc. 78, 1039–1044 (1972; Zbl 0262.57010)]. It is important to point out that by using local connections, the authors were able to work also in the complex analytic and algebraic categories, but the present book refers especially to the \(C^\infty\) category. The authors’ theory is exposed in several publications which are mentioned in the list of the references of this book. As for the book itself, it consists of lectures given by the authors at the universities of Heidelberg and Illinois and, besides giving a readable introduction to the subject, it also contains, in the last chapters still unpublished results.

The book contains eight chapters. The first chapter presents classical problems on involutive bundles and discusses partial connections on principal bundles. The second chapter studies foliated bundles. Several examples are given and adapted and basic (projectable) connections are considered. The third chapter is intended to give an introduction to the theory of the characteristic classes and it deals with the particular and more simple case of the flat bundles.

The core of the book is chapter 4: “Characteristic classes of foliated bundles”, where the general construction of the theory is developed. Namely, the classical Chern-Weil homomorphism is considered and it is shown that, in the foliated case, it has the important property of preserving some suitably defined filtrations, whence it induces homomorphisms for the corresponding cohomology and spectral sequences. Next, if \(H\subset G\) and an \(H\)-reduction of \(P\) are given, a generalized characteristic homomorphism with values on \(H^*(M)\) may be defined, which gives the announced theory.

The remaining three chapters contain new developments and applications. Namely, chapter 5 is concerned with the computation of the cohomology of the relative Weil algebras on which the characteristic homomorphism is defined. In same cases, especially under the hypothesis that the Lie algebras of \(G\) and \(H\) form a reductive pair, these computations can be completed. Chapters 6 and 7 study non-trivial examples of characteristic classes for flat and foliated bundles. Particularly, in chapter 7, the case of homogeneous foliated bundles is solved by reducing the necessary calculations to some algebraic problems.

Finally, the last chapter 8 gives the method of extending the theory to the complex analytic and algebraic categories. Here, the main construction is that of a semisimplicial Weil algebra, which is next used in the construction of a characteristic homomorphism.

The book contains eight chapters. The first chapter presents classical problems on involutive bundles and discusses partial connections on principal bundles. The second chapter studies foliated bundles. Several examples are given and adapted and basic (projectable) connections are considered. The third chapter is intended to give an introduction to the theory of the characteristic classes and it deals with the particular and more simple case of the flat bundles.

The core of the book is chapter 4: “Characteristic classes of foliated bundles”, where the general construction of the theory is developed. Namely, the classical Chern-Weil homomorphism is considered and it is shown that, in the foliated case, it has the important property of preserving some suitably defined filtrations, whence it induces homomorphisms for the corresponding cohomology and spectral sequences. Next, if \(H\subset G\) and an \(H\)-reduction of \(P\) are given, a generalized characteristic homomorphism with values on \(H^*(M)\) may be defined, which gives the announced theory.

The remaining three chapters contain new developments and applications. Namely, chapter 5 is concerned with the computation of the cohomology of the relative Weil algebras on which the characteristic homomorphism is defined. In same cases, especially under the hypothesis that the Lie algebras of \(G\) and \(H\) form a reductive pair, these computations can be completed. Chapters 6 and 7 study non-trivial examples of characteristic classes for flat and foliated bundles. Particularly, in chapter 7, the case of homogeneous foliated bundles is solved by reducing the necessary calculations to some algebraic problems.

Finally, the last chapter 8 gives the method of extending the theory to the complex analytic and algebraic categories. Here, the main construction is that of a semisimplicial Weil algebra, which is next used in the construction of a characteristic homomorphism.

Reviewer: Izu Vaisman

##### MSC:

57R30 | Foliations in differential topology; geometric theory |

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

55R10 | Fiber bundles in algebraic topology |

57N65 | Algebraic topology of manifolds |

57R20 | Characteristic classes and numbers in differential topology |

53C10 | \(G\)-structures |

55R40 | Homology of classifying spaces and characteristic classes in algebraic topology |