Cyclic Homology in Non-Commutative Geometry

Cyclic homology was introduced in the early eighties independently by Connes and Tsygan. They came from different directions. Connes wanted to associate homological invariants to K-homology classes and to describe the index pair ing with K-theory in that way, while Tsygan was motivated by algebraic K-theory and Lie algebra cohomology. At the same time Karoubi had done work on characteristic classes that led him to study related structures, without however arriving at cyclic homology properly speaking. Many of the principal properties of cyclic homology were already developed in the fundamental article of Connes and in the long paper by Feigin-Tsygan. In the sequel, cyclic homology was recognized quickly by many specialists as a new intriguing structure in homological algebra, with unusual features. In a first phase it was tried to treat this structure as well as possible within the traditional framework of homological algebra. The cyclic homology groups were computed in many examples and new important properties such as prod uct structures, excision for H-unital ideals, or connections with cyclic objects and simplicial topology, were established. An excellent account of the state of the theory after that phase is given in the book of Loday.

Operator algebras and non-commutative geometry is one of the most exciting and active research areas in mathematics Contributing authors are top experts in the field, making this subseries unique Comprehensive coverage of the field Includes supplementary material: sn.pub/extras

Inhalt
I. Cyclic Theory, Bivariant K-Theory and the Bivariant Chern-Connes Character by J. Cuntz: 1. Cyclic Theory; 2. Cyclic Theory for Locally Convex Algebras; 3. Bivariant K-Theory; 4. Infinite-Dimensional Cyclic Theories; A. Locally Convex Algebras; B. Standard Extensions.- II. Noncommutative Geometry, the Transverse Signature Operator, and Hopf Algebras (after A. Connes and H. Moscovici) by G. Skandalis: 1. Preliminaries; 2. The Local Index Formula; 3. The Diff-Invariant Signature Operator; 4. The 'Transverse' Hopf Algebra.- III. Cyclic Homology by B. Tsygan: 1. Introduction; 2. Hochschild and Cyclic Homology of Algebras; 3. The Cyclic Complex C^{lambda}_{bullet}; 4. Non-Commutative Differential Calculus; 5. Cyclic Objects; 6. Examples; 7. Index Theorems; 8. Riemann-Roch Theorem for D-Modules.