# zbMATH — the first resource for mathematics

Cyclic homology and equivariant homology. (English) Zbl 0644.55005
Given an associative algebra $$A$$ over a commutative ring, A. Connes introduced its cyclic homology $$HC_*(A)$$. Let $$HP_*(A)$$ and $$HC^- _*(A)$$ be the homologies of the related periodic complex and its negative part. Of particular importance in topology are the cases where $$A$$ is the cochain algebra $$S^*X$$ of a topological space $$X$$ or the chain algebra $$S_*G$$ of a topological group $$G$$. The author studies the relationship of the homology groups $$HC_*$$, $$HC^-_*$$ and $$HP_*$$ for $$S^*X$$ and $$S_*G$$ to the $$S^1$$-equivariant Borel homology and cohomology groups of the free loop space $$LX$$ on $$X$$ respectively of $$LBG$$, where $$BG$$ is the classifying space of $$G$$. In particular, he derives the isomorphism $$HC_*(S_*G)\cong H_*(ES^1\times_{S^1}LBG)$$ obtained independently by Burghelea-Fiedorowicz and by Goodwillie. The approach here is quite different: The author obtains his results as a consequence of more general results relating the cyclic homologies of the cyclic chain complexes $$S_*X$$ of a cyclic space $$X$$ and $$S^*Y$$ of a cocyclic space $$Y$$ to the Borel homology of the realization $$| X|$$ and the Borel cohomology of the realization $$| Y|$$, and in making use of both cyclic and cocyclic structures.
The paper is fairly self contained because the author gives a recollection of Connes cyclic category, cyclic and cocyclic objects, and some elementary properties, and of the various cyclic homology groups in the beginning sections, and includes a section with the definitions of $$S^1$$-equivariant homology and cohomology theories. For the latter he uses an explicit model for the chains on $$ES^1\times_{S^1}Z$$ for an $$S^1$$-space $$Z$$, which is particularly suited for the proofs of his main results.