Cyclic Homology (grundlehren Der Mathematischen Wissenschaften ; 301)

This is a comprehensive study of cyclic homology theory. It opens with details of Hochschild and cyclic homology of associative algebras, their variations (periodic theory, dihedral theory) and the comparison with de Rham comology theory. The second part deals with cyclic sets, cyclic spaces, their relationships with S1-equivariant homology and the Chern character of Connes. The third section is devoted to the homology of the Lie algebra of matrices (the Loday-Quillen-Tsygan theorem) and its variations (namely non-commutative Lie homology). This is followed by an account of algebraic K-theory and its relationship to cyclic homology. The book concludes with an overview of some applications to non-commutative differential geometry (foliations, Novikov conjecture, idempotent conjecture) as devised by Alain Connes. Most of the results treated in this book have already appeared in research articles. However, some are new (non-commutative Lie homology for instance) and many proofs are either more explicit or simpler than the existing ones. This book is a comprehensive study of cyclic homology theory. The first partdeals with Hochschild and cyclic homology of associative algebras, their variations (periodic theory, dihedral theory) and the comparison with de Rham comology theory. The second part deals with cyclic sets, cyclic spaces, their relationships with S 1-equivariant homology and the Chern character of Connes. The third part is devoted to the homology of the Lie algebra of matrices (the Loday-Quillen-Tsygan theorem) and its variations (namely non-commutative Lie homology). The fourth part is an account of algebraic K-theory and its relationship to cyclic homology. The last chapter is an overview of some applications tonon-commutative differential geometry (foliations, Novikov conjecture, idempotent conjecture) as devised by Alain Connes. Most of the results treated in this book have already appeared in research articles. However some are new (non-commutative Lie homology for instance) and many proofs are either more explicit or simpler than the existing ones. Though this book was thought of a basic reference for researchers, several part of it are accessible to graduate students, since the material is almost self contained. It also contains a comprehensive list of references on the subject. From the reviews: "This is a very interesting book containing material for a comprehensive study of the cyclid homological theory of algebras, cyclic sets and S1-spaces. Lie algebras and algebraic K-theory and an introduction to Connes'work and recent results on the Novikov conjecture. The book requires a knowledge of homological algebra and Lie algebra theory as well as basic technics coming from algebraic topology. The bibliographic comments at the end of each chapter offer good suggestions for further reading and research. The book can be strongly recommended to anybody interested in noncommutative geometry, contemporary algebraic topology and related topics." European Mathematical Society Newsletter In this second edition the authors have added a chapter 13 on MacLane (co)homology. Front Matter....Pages I-XVII Hochschild Homology....Pages 1-49 Cyclic Homology of Algebras....Pages 50-87 Smooth Algebras and Other Examples....Pages 88-113 Operations on Hochschild and Cyclic Homology....Pages 114-154 Variations on Cyclic Homology....Pages 155-197 The Cyclic Category, Tor and Ext Interpretation....Pages 198-222 Cyclic Spaces and S 1 -Equivariant Homology....Pages 223-252 Chern Character....Pages 253-276 Classical Invariant Theory....Pages 277-294 Homology of Lie Algebras of Matrices....Pages 295-336 Algebraic K -Theory....Pages 337-376 Non-commutative Differential Geometry....Pages 377-394 Back Matter....Pages 395-454 This book is a comprehensive study of cyclic homology theory together with its relationship with Hochschild homology, de Rham cohomology, S1 equivariant homology, the Chern character, Lie algebra homology, algebraic K-theory and non-commutative differential geometry. Though conceived as a basic reference on the subject, many parts of this book are accessible to graduate students. A comparative study of cyclic homology theory, dealing with such topics as Hochschild and cyclic homology of associated algebras, their variations and their comparison with de Rham cosmology theory. There is also discussion of cyclic sets and spaces, and the Chern character of Connes. Jean-louis Loday. Includes Bibliographical References And Index.