ON COMPUTATION HOCHSCHILD AND CYCLIC HOMOLOGY OF HOMOGENEOUS SPACES

Abstract

Cyclic and Hochschild homology HH*(X), HC*(X) of a topological space X has appeared to be of wide interest since the publication of the papers by Burghelea [5], Burghelea and Fiedorowicz [6], and Goodwillie [12]. As far as HH*(X) can be identified to $H_* (X^{S^1})$ and HC*(X) to $H_* (ES^1\times {}_{S^1} X^{S^1})$ (the homology of a free loop space $X^{S^1}$ and the homology of the associated bundle $ES^1\times {}_{S^1} X^{S^1}$ [12]), cyclic and Hochschild homology provide the powerful technique of studying a free loop space. The computation of Hochschild homology is closely connected with the investigation of closed geodesics on manifolds and related dynamical systems. The computation of cyclic homology is connected with the groups of diffeomorphism of manifolds. It should be mentioned that the investigation of various topological invariants of $X^{S^1}$ is very important in view of their role in mathematical physics [27].