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**Cyclic homology.
2nd ed.**
*(English)*
Zbl 0885.18007

Grundlehren der Mathematischen Wissenschaften. 301. Berlin: Springer. xviii, 513 p. (1998).

Cyclic homology appeared some 15 years ago from several areas in mathematics. A. Connes developed his variant as a noncommutative counterpart of De Rham cohomology, and in 1983 B. Tsygan as well as D. Quillen and J.-L. Loday found cyclic homology to be the primitive part of the Lie algebra homology of matrices. This established cyclic homology as a Lie analogue of algebraic \(K\)-theory, which is of particular importance in the homology theory of \(S^1\)-spaces.

In 1992 J.-L. Loday published the first comprehensive treatise on this new framework in algebra and geometry (1992; Zbl 0780.18009), which, due to its consequent completeness, up-to-dateness, and encyclopaedic versality, very soon became the acknowledged standard text on the subject.

In the present second edition of this excellent monograph, the author has left the well-proved text intact. However, apart from the correction of some misprints, errors and inaccuracies, another improvement has been taken up. Namely, the author has added a new chapter, at the end of the book, which is entitled “MacLane (co-)homology”. This new Chapter 13, written jointly with T. Pirashvili, relates the material on Hochschild homology, algebraic \(K\)-theory, and cohomology of small categories (as treated in the previous chapters) to the classical, occasionally almost forgotten MacLane cohomology, as well as to the stable (co-)homology of Eilenberg-MacLane spaces and to stable \(K\)-theory in the sense of F. Waldhausen.

Among the appendices, Appendix C on the homology of discrete groups and small categories has been enhanced accordingly, whereas a new Appendix E has been added. This new appendix, written by M. O. Ronco, is to help the reader to understand the relationship between the various definitions of “smooth” algebras in the literature.

Finally, for the convenience of the reader, and for the sake of actuality, a second list of references has been appended. This additional bibliography refers to the recent research articles on periodic cyclic homology and on topological cyclic homology theory, which appeared during the period 1992-1996.

All in all, one can state that this second edition of an important standard text is presented in a profitably enhanced and up-dated version, which certainly increases its lasting significance and its outstanding character in the relevant literature.

In 1992 J.-L. Loday published the first comprehensive treatise on this new framework in algebra and geometry (1992; Zbl 0780.18009), which, due to its consequent completeness, up-to-dateness, and encyclopaedic versality, very soon became the acknowledged standard text on the subject.

In the present second edition of this excellent monograph, the author has left the well-proved text intact. However, apart from the correction of some misprints, errors and inaccuracies, another improvement has been taken up. Namely, the author has added a new chapter, at the end of the book, which is entitled “MacLane (co-)homology”. This new Chapter 13, written jointly with T. Pirashvili, relates the material on Hochschild homology, algebraic \(K\)-theory, and cohomology of small categories (as treated in the previous chapters) to the classical, occasionally almost forgotten MacLane cohomology, as well as to the stable (co-)homology of Eilenberg-MacLane spaces and to stable \(K\)-theory in the sense of F. Waldhausen.

Among the appendices, Appendix C on the homology of discrete groups and small categories has been enhanced accordingly, whereas a new Appendix E has been added. This new appendix, written by M. O. Ronco, is to help the reader to understand the relationship between the various definitions of “smooth” algebras in the literature.

Finally, for the convenience of the reader, and for the sake of actuality, a second list of references has been appended. This additional bibliography refers to the recent research articles on periodic cyclic homology and on topological cyclic homology theory, which appeared during the period 1992-1996.

All in all, one can state that this second edition of an important standard text is presented in a profitably enhanced and up-dated version, which certainly increases its lasting significance and its outstanding character in the relevant literature.

Reviewer: W.Kleinert (Berlin)

### MSC:

18G60 | Other (co)homology theories (MSC2010) |

19D55 | \(K\)-theory and homology; cyclic homology and cohomology |

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

19-02 | Research exposition (monographs, survey articles) pertaining to \(K\)-theory |

16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |

17B55 | Homological methods in Lie (super)algebras |

55P35 | Loop spaces |

58J20 | Index theory and related fixed-point theorems on manifolds |

58J22 | Exotic index theories on manifolds |

14Fxx | (Co)homology theory in algebraic geometry |