The cyclic groups and the free loop space
type: Item
instance of: scholarly article
title: The cyclic groups and the free loop space
zbMATH Open document ID: 0632.57028
DOI: 10.1007/BF02564455
author: Gunnar Carlsson
author: Ralph L. Cohen
published in: Commentarii Mathematici Helvetici
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140092
review text: The authors study \(\tilde T(Y)=T(Y)/T(point)\), where \(T(Y)=ES^ 1\times_{S^ 1}F(Y)\) \((F(Y)=space\) of free loops on Y, with the natural \(S^ 1\)-action). The main result establishes a homology isomorphism \(Z(X)\to \tilde T(\Sigma X)\), for a connected complex X, where \(Z(X)\) is a combinatorial functor of X, obtained from the Dyer- Lashof construction by replacing the permutation groups by cyclic subgroups. The homology of the right-hand side is evaluated by combining results of Burghelea-Fiedorowicz and Goodwillie (giving an equality between \(\bar H_*\tilde T(Y)\) and Connes' cyclic homology \(\overline{HC}_*(Y))\) and Loday-Quillen (computation of \(\overline{HC}_*(\Sigma X))\). The homology of the left-hand side is handled by observing that \(Z(X)\) has a natural filtration, whose manageable quotients stably split \(Z(X)\) as their wedge. The authors also show how the Dyer-Lashof construction is obtainable from their Z- construction by stabilization.
Mathematics Subject Classification ID: 57T25
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 55P47
Mathematics Subject Classification ID: 55N25
zbMATH DE Number: 4028443
zbMATH Keywords: suspension
zbMATH Keywords: equivariant free loop-space homology
zbMATH Keywords: free loops
zbMATH Keywords: Dyer-Lashof construction
zbMATH Keywords: cyclic homology
zbMATH Keywords: Z-construction
reviewed by: Ştefan Papadima
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4028443