Geometry of characteristic classes. Transl. from the Japanese by the author.

*(English)*Zbl 0976.57026
Translations of Mathematical Monographs. Iwanami Series in Modern Mathematics. 199. Providence, RI: American Mathematical Society (AMS). xiii, 185 p. (2001).

The present book is devoted to characteristic classes, which were introduced and studied during the last three decades. It deals with characteristic classes of flat bundles, of foliations and of surface bundles.

The first chapter contains an exposition of de Rham homotopy theory, and describes the main results of D. Sullivan for simplicial complexes which are nilpotent and of finite type. The contents are not directly related to the following chapters on characteristic classes.

The second chapter deals with characteristic classes of flat bundles. It contains an exposition of the theory of flat bundles and their classifying spaces, and a short introduction to Chern-Weil theory, cohomology of Lie algebras, and Gelfand-Fuks cohomology. For a principal bundle over a \(C^\infty\)-manifold \(M\) with structure group a Lie group \(G\), a characteristic homomorphism is defined from the cohomology of the Lie algebra of \(G\) with respect to a maximal compact Lie subgroup \(K\) to the real cohomology group of the base \(M\). The elements of the image of this homomorphism are called characteristic classes of the flat bundle. The characteristic homomorphism is defined by using the connection form of the flat bundle. Results of Borel and Harishandra are used to show that these characteristic classes of flat bundles are non-trivial. There is a brief description of Chern-Simons theory. Gelfand-Fuks cohomology is used to define characteristic classes for flat \(F\)-product bundles, where \(F\) is a \(C^\infty\)-manifold.

The third chapter is devoted to characteristic classes of foliations. After a short introduction to the theory of foliations, the author defines the Godbillon-Vey class of codimension one foliations, and describes the example of Thurston on continuous variation of the Godbillon-Vey class. Canonical forms on frame bundles of higher order are studied, and the Bott vanishing theorem is proved. Then discontinuous invariants are discussed, which were first introduced by the author.

Chapter 4 deals with characteristic classes of surface bundles. It starts with sections on the mapping class group of closed surfaces, and on the classification of surface bundles. For an orientable surface bundle \(\pi:E\to M\) let \(e\in H^2(E;Z)\) denote the Euler class of the tangent bundle along the fibres, and \(e_i= \pi_* e^{i+1}\in H^{2i} (M;Z)\), where \(\pi_*\) is the Gysin homomorphism induced by \(\pi\). \(e_i\) is called the \(i\)-th characteristic class of the surface bundle. The construction of ramified coverings is discussed, and used to show that the classes \(e_i\) are non-trivial. There is a discussion on the algebraic independence of characteristic classes of surface bundles and on the Nielsen realisation problem.

The book concludes with a section on directions and problems for future research.

The first chapter contains an exposition of de Rham homotopy theory, and describes the main results of D. Sullivan for simplicial complexes which are nilpotent and of finite type. The contents are not directly related to the following chapters on characteristic classes.

The second chapter deals with characteristic classes of flat bundles. It contains an exposition of the theory of flat bundles and their classifying spaces, and a short introduction to Chern-Weil theory, cohomology of Lie algebras, and Gelfand-Fuks cohomology. For a principal bundle over a \(C^\infty\)-manifold \(M\) with structure group a Lie group \(G\), a characteristic homomorphism is defined from the cohomology of the Lie algebra of \(G\) with respect to a maximal compact Lie subgroup \(K\) to the real cohomology group of the base \(M\). The elements of the image of this homomorphism are called characteristic classes of the flat bundle. The characteristic homomorphism is defined by using the connection form of the flat bundle. Results of Borel and Harishandra are used to show that these characteristic classes of flat bundles are non-trivial. There is a brief description of Chern-Simons theory. Gelfand-Fuks cohomology is used to define characteristic classes for flat \(F\)-product bundles, where \(F\) is a \(C^\infty\)-manifold.

The third chapter is devoted to characteristic classes of foliations. After a short introduction to the theory of foliations, the author defines the Godbillon-Vey class of codimension one foliations, and describes the example of Thurston on continuous variation of the Godbillon-Vey class. Canonical forms on frame bundles of higher order are studied, and the Bott vanishing theorem is proved. Then discontinuous invariants are discussed, which were first introduced by the author.

Chapter 4 deals with characteristic classes of surface bundles. It starts with sections on the mapping class group of closed surfaces, and on the classification of surface bundles. For an orientable surface bundle \(\pi:E\to M\) let \(e\in H^2(E;Z)\) denote the Euler class of the tangent bundle along the fibres, and \(e_i= \pi_* e^{i+1}\in H^{2i} (M;Z)\), where \(\pi_*\) is the Gysin homomorphism induced by \(\pi\). \(e_i\) is called the \(i\)-th characteristic class of the surface bundle. The construction of ramified coverings is discussed, and used to show that the classes \(e_i\) are non-trivial. There is a discussion on the algebraic independence of characteristic classes of surface bundles and on the Nielsen realisation problem.

The book concludes with a section on directions and problems for future research.

Reviewer: K.H.Mayer (Dortmund)

##### MSC:

57R20 | Characteristic classes and numbers in differential topology |

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

57R32 | Classifying spaces for foliations; Gelfand-Fuks cohomology |

55P62 | Rational homotopy theory |

20F38 | Other groups related to topology or analysis |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

57R30 | Foliations in differential topology; geometric theory |

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

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |