Cyclic Homology in Non- Commutative Geometry Series: Encyclopaedia of Mathematical Sciences

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Cyclic Homology in Non- Commutative Geometry Series: Encyclopaedia of Mathematical Sciences

springer.com Joachim Cuntz, Georges Skandalis, Boris Tsygan Cyclic Homology in Non- Commutative Geometry Series: Encyclopaedia of Mathematical Sciences 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 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 2004, XIII, 137 p. 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 Printed book first phase it was tried to treat this structure as well as possible within the traditional Hardcover framework of homological algebra. The cyclic homology groups were computed in many 109,99 € | £99.99 | $139.99 examples and new important properties such as prod• uct structures, excision for H-unital [1]117,69 € (D) | 120,99 € (A) | CHF ideals, or connections with cyclic objects and simplicial topology, were established. An excellent 130,00 account of the state of the theory after that phase is given in the book of Loday. Softcover 109,99 € | £99.99 | $139.99 [1]117,69 € (D) | 120,99 € (A) | CHF 130,00 eBook 93,08 € | £79.50 | $109.00 [2]93,08 € (D) | 93,08 € (A) | CHF 104,00 Available from your library or springer.com/shop MyCopy [3] Printed eBook for just € | $ 24.99 springer.com/mycopy Order online at springer.com / or for the Americas call (toll free) 1-800-SPRINGER / or email us at: [email protected]. / For outside the Americas call +49 (0) 6221-345-4301 / or email us at: [email protected]. The first € price and the £ and $ price are net prices, subject to local VAT. Prices indicated with [1] include VAT for books; the €(D) includes 7% for Germany, the €(A) includes 10% for Austria. Prices indicated with [2] include VAT for electronic products; 19% for Germany, 20% for Austria. All prices exclusive of carriage charges. Prices and other details are subject to change without notice. All errors and omissions excepted. [3] No discount for MyCopy. Part of .

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Cyclic Homology and Equivariant Homology
Invent.math. 87, 403-423 (1987) [nffelltiolle$ mathematicae Springer-Verlag1987 Cyclic homology and equivariant homology John D.S. Jones Mathematics Institute,University of Warwick, Coventry CV4 7AL, UK Introduction The purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of Connes [9-11], see also Loday and Quillen [20], and "IF equivariant homology and cohomology theories. Here II" is the circle group. The most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic spaces so precise statements will be postponed until w3. In this introduction we explain some of the formal similarities between the cyclic theory and the equivariant theory and give two examples where the general results apply. Let A be an associative algebra over a commutative ring K. Then one can form the cyclic homology HC.(A) and cohomology HC*(A) of A. These groups have periodicity operators HC,(A)-.~HC,,_z(A), HC"(A)-.HC"+Z(A). Connes [10] has defined products in cyclic cohomology and using this product structure HC*(K) becomes a polynomial ring K[u] where u has degree 2. The groups HC*(A) now become modules over this polynomial ring and the action of u corresponds to the periodicity operator. Thus it seems reasonable to regard K[u] as the natural coefficients for cyclic cohomology and then to make HC.(A) into a module over K[u] by using the periodicity operator. However it is clear that every element of HC.(A) is u-torsion so HC*(A) and HC.(A) cannot be dual over K[u].
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