Cyclic homology and equivariant homology
type: Item
instance of: scholarly article
title: Cyclic homology and equivariant homology
zbMATH Open document ID: 0644.55005
DOI: 10.1007/BF01389424
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143427
review text: 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 [\textit{M. Ungheretti}, Bull. Lond. Math. Soc. 49, No. 1, 95--101 (2017; Zbl 1378.55007)].
Mathematics Subject Classification ID: 55N35
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 16E40
zbMATH DE Number: 4050538
zbMATH Keywords: cochain algebra of a topological space
zbMATH Keywords: chain algebra of a topological group
zbMATH Keywords: cyclic homology
zbMATH Keywords: periodic complex
zbMATH Keywords: \(S^1\)-equivariant Borel homology
zbMATH Keywords: free loop space
zbMATH Keywords: classifying space
zbMATH Keywords: cyclic space
zbMATH Keywords: cocyclic space
zbMATH Keywords: cyclic and cocyclic structures
zbMATH Keywords: cyclic and cocyclic objects
author: John D. S. Jones
reviewed by: Rainer Vogt
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4050538