Subject: Hilton-Eckmann dual to James construction Date: Mon, 29 Jul 2002 14:32:33 -0400 From: John Klein Organization: Wayne State University Here's a question for the list: Let X be a based space. Then there is a tower of homotopy functors ... -> J^3(X) -> J^2(X) -> J^1(X) where, J^k(X) is, roughly, the Hilton-Eckmann dual to the k-th filtration of the James construction (= free monoid on X). Specifically J^1(X) := X, and J^k(X) is constructed from J^{k-1}(X) as a homotopy pullback J^k(X) := holim(J^{k-1}(X) --> X_k <-- X^{v k}) , where X^{v k} = k-fold wedge of copies of X, and X_k = the homotopy limit of the punctured k-cubical diagram S |--> X^{v S} (S ranges over all subsets of {1,2,3,...,k}, and X^{v S} means the functions S --> X which are supported on a single element.) The map J^{k-1}(X) --> X_k given by thinking of {1,2,...,k-1} as a subset of {1,2,...,k} in k-different ways. My question: what's the homotopy type of the homotopy inverse limit J^infty(X) := lim_k J^k(X) ? note: In constrast with the James construction, at cofiltration two we get J^2(X) = Sigma Omega X, i.e., the suspension of the loop space of X (this is classical; I learned about it in one of Ganea's papers, but it probably is much older). So I don't expect that J^infty(X) to coincide with Sigma Omega X, but seems to me that the latter is a retract of the former. Has anyone ever looked at anything like this?