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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.
Editorial remark: For more information concerning this article see [M. Ungheretti, Bull. Lond. Math. Soc. 49, No. 1, 95–101 (2017; Zbl 1378.55007)].
Reviewer: R.Vogt

MSC:
55N35 Other homology theories in algebraic topology
55P35 Loop spaces
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
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