Cyclic Homology

This book is a comprehensive study of cyclic homology theory. The first part deals 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 S1-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 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. 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.

In this 2nd edition a new chapter 13 on MacLane (co)homology is added. It is compared with a variant of algebraic K-theory called stable K-theory. It turns out that these two theories are isomorphic. The main tool for this comparison is a third theory constructed from derived functors over the category of polynomial functors. This chapter is a transition from the content of the first 12 ch.

Subject at the forefront of research A very much needed book Loday is well-known both as one of the leading researchers in the field and also as a very clear and precise expositor A diversity of approaches A comprehensive monograph from a unified point of view There is no competitive work available

Inhalt

1. Hochschild Homology.- 2. Cyclic Homology of Algebras.- 3. Smooth Algebras and Other Examples.- 4. Operations on Hochschild and Cyclic Homology.- 5. Variations on Cyclic Homology.- 6. The Cyclic Category, Tor and Ext Interpretation.- 7. Cyclic Spaces and Sl-Equivariant Homology.- 8. Chern Character.- 9. Classical Invariant Theory.- 10. Homology of Lie Algebras of Matrices.- 11. Algebraic K-Theory.- 12. Non-commutative Differential Geometry.- 13. Mac Lane (co)homology.- Appendices.- A. Hopf Algebras.- B. Simplicial.- C. Homology of Discrete Groups and Small Categories.- D. Spectral Sequences.- E. Smooth Algebras.- References.- References 19921996.- Symbols.