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**Homologie cyclique et K-théorie.**
*(French)*
Zbl 0648.18008

Astérisque, 149. Publié avec le concours du Centre National de la Recherche Scientifique. Paris: Société Mathématique de France. 147 p.; FF 100.00; $ 17.00 (1987).

In the introduction, the author refers any reader who wants a resume to six notes in the C. R. Acad. Sci., Paris, Sér. I 297, 381-384, 447-450, 515-516, and 557-560 (1983; Zbl 0528.18008, Zbl 0528.18009, Zbl 0528.18010, and Zbl 0532.18009), ibid. 302, 321-324 (1986; Zbl 0593.55004), and ibid. 303, 507-510 (1986; Zbl 0601.18007), and one note in Current trends in algebraic topology, Semin. London/Ont. 1981, CMS Conf. Proc. 2, 19-27 (1982; Zbl 0553.18006).

The following abstract is given at the end of the book: “This book is an expanded version of some ideas related to the general problem of characteristic classes in the framework of Chern-Weil theory. These ideas took their origins independently from the work of Alain Connes and the author. They were motivated by considerations of operator algebras and algebraic K-theory. The good framework to develop these considerations is cyclic homology (or non-commutative De Rham homology). This enables us to extend this classical Chern-Weil theory far beyond its original scope (at least for the general linear group). Cyclic homology is the natural target for characteristic classes and its computation is a matter of homological algebra as was shown by A. Connes. On the other hand, the objects we are taking the characteristic classes of are elements of the K-theory of a ring. This K-theory (algebraic or topological) is difficult to compute in general. Cyclic homology (also called “additive K-theory” by Feigin and Tsygan) appears therefore as a first step to compute the K- groups for general algebras. We have tried to make this book as self- contained as possible. Together with the motivations provided by A. Connes, the reader should not find any special difficulty to read it. In particular, the first chapters can be easily integrated in a graduate course on the subject.”

The following abstract is given at the end of the book: “This book is an expanded version of some ideas related to the general problem of characteristic classes in the framework of Chern-Weil theory. These ideas took their origins independently from the work of Alain Connes and the author. They were motivated by considerations of operator algebras and algebraic K-theory. The good framework to develop these considerations is cyclic homology (or non-commutative De Rham homology). This enables us to extend this classical Chern-Weil theory far beyond its original scope (at least for the general linear group). Cyclic homology is the natural target for characteristic classes and its computation is a matter of homological algebra as was shown by A. Connes. On the other hand, the objects we are taking the characteristic classes of are elements of the K-theory of a ring. This K-theory (algebraic or topological) is difficult to compute in general. Cyclic homology (also called “additive K-theory” by Feigin and Tsygan) appears therefore as a first step to compute the K- groups for general algebras. We have tried to make this book as self- contained as possible. Together with the motivations provided by A. Connes, the reader should not find any special difficulty to read it. In particular, the first chapters can be easily integrated in a graduate course on the subject.”

Reviewer: H.J.Munkholm

### MSC:

18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |

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

55R50 | Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory |

18-02 | Research exposition (monographs, survey articles) pertaining to category theory |

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

53C05 | Connections (general theory) |