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Cyclic Homology in Non-Commutative Geometry

  • Joachim Cuntz
  • Georges Skandalis
  • Boris Tsygan

Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 121)

Table of contents

About this book

Introduction

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.

Keywords

K-theory algebra cyclic homology homology non-commutative geometry

Authors and affiliations

  • Joachim Cuntz
    • 1
  • Georges Skandalis
    • 2
  • Boris Tsygan
    • 3
  1. 1.Institute of MathematicsUniversity of MünsterMünsterGermany
  2. 2.Centre de Mathématiques de JussieuUniversité Paris 7 Denis DiderotParisFrance
  3. 3.Department of MathematicsNorthwestern UniversityEvanstonUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-06444-3
  • Copyright Information Springer-Verlag Berlin Heidelberg 2004
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-07337-3
  • Online ISBN 978-3-662-06444-3
  • Series Print ISSN 0938-0396
  • Buy this book on publisher's site