The cyclic groups and the free loop space
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.
scientific article; zbMATH DE number 4028443