Cyclic homology and equivariant homology

Abstract

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 w 3. 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