##
**Cyclic homology.**
*(English)*
Zbl 0780.18009

Grundlehren der Mathematischen Wissenschaften. 301. Berlin: Springer-Verlag. xvii, 454 p. (1992).

Cyclic homology appeared in algebraic topology at the beginning of the eighties. It was dealt with, independently, by A. Connes, B. Tsygan, D. Quillen and the author of this book, J. L. Loday. A great variety of sources of this theory (index theorems for non-commutative Banach algebras, Lie algebra homology of matrices, algebraic \(K\)-theory, homology of \(S^ 1\)-spaces) and the appearance (lately) of applications in differential geometry and global analysis around the index formulae (and other fundamental topics of global analysis) justify the author’s opinion “…cyclic homology theory illuminates a great many interactions between algebra, topology, geometry, and analysis”.

The contents of the book can be divided into three main themes: Chapters 1–5: cyclic homology of algebras; Chapters 6–8: cyclic sets and \(S^1\)-spaces; Chapters 9–11: Lie algebras and Abelian \(K\)-theory. The last chapter contains (without proofs) some important applications to global analysis on foliated manifolds and other domains.

Chapter 1 deals with classical Hochschild homology (whose “variant” is just cyclic homology). The definition of the functor \(HC_n\) of cyclic homology appears in Chapter 2 together with the basic properties, such as, for example, Connes periodicity. The third chapter is devoted to the computation of Hochschild and cyclic homologies of some particular types of algebras such as tensor and symmetric algebras, universal enveloping algebras of Lie algebras and smooth algebras. A relationship with the de Rham cohomology is given. Chapter 4 concerns some important operations on cyclic homology such as conjugation, derivation, product, coproduct and others. The object of this chapter is to explain the behaviour of Hochschild and cyclic homologies with respect to these operations. Some ways of modifying cyclic homology are studied in Chapter 5.

The second part of this book starts with Chapter 6. Here, the relationship between the finite cyclic groups and the simplicial category is described in detail, which allows to interpret cyclic homology as a Tor-functor and cyclic cohomology as an Ext-functor. An application to the equivariant homology of \(S^1\)-spaces is given in Chapter 7. In Chapter 8 the author studies the Chern-Connes character in the non- commutative case (the classical construction is given in the first sections). This chapter ends with applications to the Bass trace conjecture (due to B. Eckmann) and to the idempotent conjecture (as done by A. Connes).

The third part of the book is devoted to the explanation of the relationship between cyclic theory and (1) homology of matrices, (2) Lie algebra homology, (3) algebraic \(K\)-theory. The classical invariant theory (needed in the sequel) is presented in Chapter 9. The Loday- Quillen-Tsygan theorem computing the homology of the Lie algebra of matrices in terms of cyclic homology is one of the main results of Chapter 10. The relationship between algebraic \(K\)-theory and cyclic homology is studied in Chapter 11, whose aim is the Goodwillie isomorphism between relative algebraic \(K\)-theory and relative cyclic homology of a nilpotent ideal in characteristic zero. In the rest of the chapter, the results of Chapter 8 are used to construct Chern character maps from algebraic \(K\)-theory to cyclic homology and, next, to secondary characteristic classes (due to Karoubi).

The aim of the last chapter is to draw one’s attention to some applications of cyclic theory to the Godbillon-Vey invariant, to the index theorem for Fredholm modules and to the Novikov conjecture on higher signatures and its \(K\)-theoretic analogue.

The contents of the book can be divided into three main themes: Chapters 1–5: cyclic homology of algebras; Chapters 6–8: cyclic sets and \(S^1\)-spaces; Chapters 9–11: Lie algebras and Abelian \(K\)-theory. The last chapter contains (without proofs) some important applications to global analysis on foliated manifolds and other domains.

Chapter 1 deals with classical Hochschild homology (whose “variant” is just cyclic homology). The definition of the functor \(HC_n\) of cyclic homology appears in Chapter 2 together with the basic properties, such as, for example, Connes periodicity. The third chapter is devoted to the computation of Hochschild and cyclic homologies of some particular types of algebras such as tensor and symmetric algebras, universal enveloping algebras of Lie algebras and smooth algebras. A relationship with the de Rham cohomology is given. Chapter 4 concerns some important operations on cyclic homology such as conjugation, derivation, product, coproduct and others. The object of this chapter is to explain the behaviour of Hochschild and cyclic homologies with respect to these operations. Some ways of modifying cyclic homology are studied in Chapter 5.

The second part of this book starts with Chapter 6. Here, the relationship between the finite cyclic groups and the simplicial category is described in detail, which allows to interpret cyclic homology as a Tor-functor and cyclic cohomology as an Ext-functor. An application to the equivariant homology of \(S^1\)-spaces is given in Chapter 7. In Chapter 8 the author studies the Chern-Connes character in the non- commutative case (the classical construction is given in the first sections). This chapter ends with applications to the Bass trace conjecture (due to B. Eckmann) and to the idempotent conjecture (as done by A. Connes).

The third part of the book is devoted to the explanation of the relationship between cyclic theory and (1) homology of matrices, (2) Lie algebra homology, (3) algebraic \(K\)-theory. The classical invariant theory (needed in the sequel) is presented in Chapter 9. The Loday- Quillen-Tsygan theorem computing the homology of the Lie algebra of matrices in terms of cyclic homology is one of the main results of Chapter 10. The relationship between algebraic \(K\)-theory and cyclic homology is studied in Chapter 11, whose aim is the Goodwillie isomorphism between relative algebraic \(K\)-theory and relative cyclic homology of a nilpotent ideal in characteristic zero. In the rest of the chapter, the results of Chapter 8 are used to construct Chern character maps from algebraic \(K\)-theory to cyclic homology and, next, to secondary characteristic classes (due to Karoubi).

The aim of the last chapter is to draw one’s attention to some applications of cyclic theory to the Godbillon-Vey invariant, to the index theorem for Fredholm modules and to the Novikov conjecture on higher signatures and its \(K\)-theoretic analogue.

Reviewer: Jan Kubarski (Łódź)

### MSC:

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

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

18G90 | Other (co)homology theories (category-theoretic aspects) |

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 |