Cyclic Homology in Non-Commutative Geometry: 121 (Encyclopaedia of Mathematical Sciences, 121) - Hardcover

Cuntz, Joachim; Skandalis, Georges; Tsygan, Boris

Hardcover

ISBN 10: 3540404694 ISBN 13: 9783540404699

Publisher: Springer, 2003

Synopsis

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.

About this title

Review

From the reviews:

"This volume of the ‘Encyclopedia of Mathematical Sciences’ is a very important and useful contribution to the literature on cyclic homology and noncommutative geometry. ... This book contains three expository articles, covering very important recent results." (Alexander Gorokhovsky, Mathematical Reviews, 2005 k)

Synopsis

This volume contains contributions by three authors and treats aspects of noncommutative geometry that are related to cyclic homology. The authors give rather complete accounts of cyclic theory from different and complementary points of view. The connections between topological (bivariant) K-theory and cyclic theory via generalized Chern-characters are discussed in detail. This includes an outline of a framework for bivariant K-theory on a category of locally convex algebras. On the other hand, cyclic theory is the natural setting for a variety of general index theorems. A survey of such index theorems (including the abstract index theorems of Connes-Moscovici and of Bressler-Nest-Tsygan) is given and the concepts and ideas involved in the proof of these theorems are explained.

Edition details

• Publisher: Springer
• Publication date: 2003
• Language: English
• ISBN 10: 3540404694
• ISBN 13: 9783540404699
• Binding: Hardcover
• Number of pages: 150