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?