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Cyclic Homology in Non-Commutative Geometry

  • Book
  • © 2004

Overview

Authors:
  1. Joachim Cuntz

    1. Institute of Mathematics, University of Münster, Münster, Germany

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  2. Georges Skandalis

    1. Centre de Mathématiques de Jussieu, Université Paris 7 Denis Diderot, Paris, France

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  3. Boris Tsygan

    1. Department of Mathematics, Northwestern University, Evanston, USA

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  • 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

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

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Table of contents (3 chapters)

  1. Front Matter

    Pages I-XIII
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  2. Cyclic Theory, Bivariant K-Theory and the Bivariant Chern-Connes Character

    • Joachim Cuntz
    Pages 1-71

  3. Cyclic Homology

    • Boris Tsygan
    Pages 73-113

  4. Noncommutative Geometry, the Transverse Signature Operator, and Hopf Algebras [after A. Connes and H. Moscovici]

    • Georges Skandalis
    Pages 115-134

  5. Back Matter

    Pages 135-137
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Keywords

About this book

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.

Reviews

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)

Authors and Affiliations

  • Institute of Mathematics, University of Münster, Münster, Germany

    Joachim Cuntz

  • Centre de Mathématiques de Jussieu, Université Paris 7 Denis Diderot, Paris, France

    Georges Skandalis

  • Department of Mathematics, Northwestern University, Evanston, USA

    Boris Tsygan

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