Cyclic Homology

A comparative study of cyclic homology theory, dealing with such topics as Hochschild and cyclic homology of associated algebras, their variations and their comparison with de Rham cosmology theory. There is also discussion of cyclic sets and spaces, and the Chern character of Connes.

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This is a comprehensive study of cyclic homology theory. It opens with details of Hochschild and cyclic homology of associative algebras, their variations (periodic theory, dihedral theory) and the comparison with de Rham comology theory. The second part deals with cyclic sets, cyclic spaces, their relationships with S1-equivariant homology and the Chern character of Connes. The third section is devoted to the homology of the Lie algebra of matrices (the Loday-Quillen-Tsygan theorem) and its variations (namely non-commutative Lie homology). This is followed by an account of algebraic K-theory and its relationship to cyclic homology. The book concludes with an overview of some applications to non-commutative differential geometry (foliations, Novikov conjecture, idempotent conjecture) as devised by Alain Connes. Most of the results treated in this book have already appeared in research articles. However, some are new (non-commutative Lie homology for instance) and many proofs are either more explicit or simpler than the existing ones.