Given a pointed space X, let $J(X)$ denote its James construction. There is a natural inclusion $X\rightarrow J(X)$ which can equivalently be described as the unit map $\eta_X:X\rightarrow \Omega \Sigma X$. The homotopy fiber of the counit map $\epsilon_X:\Sigma \Omega X\rightarrow X$ is known to be $\Omega X * \Omega X$. Is there an analogous result for the homotopy fiber of $\eta_X$ in terms of $X$?

$\begingroup$ I have the feeling that an analogous result would involve the homotopy cofiber of $\eta$, not the homotopy fiber. By the way, is $\ast$ the wedge product? $\endgroup$– Konrad VoelkelApr 17, 2013 at 17:09

$\begingroup$ * is the join, ie the suspension of the smash product. $\endgroup$– Fernando MuroApr 17, 2013 at 18:19

$\begingroup$ I believe for $X=S^{n}$ then one can get some answers plocally by looping the EHP sequence. In general I am not sure what can be said. $\endgroup$– Callan McGillApr 17, 2013 at 20:38

5$\begingroup$ There is not even a known identification of the homotopy cofiber $C$ of $e_X$ in general. There are some partial results: its suspension $\Sigma C$ is weak equivalent to $\Sigma (X\wedge \Omega \Sigma X)$. In the metastable range (approximately 3 times the connectivity of $X$), the cofiber coincides with the "cojoin," i.e., the holim of the diagram $\Sigma X \to \Sigma X \vee \Sigma X \leftarrow \Sigma$ given by the two inclusions. $\endgroup$– John KleinApr 17, 2013 at 22:57

$\begingroup$ Thanks for your comments. It seems there is not a known answer for this question, as John implicitly claimed above. $\endgroup$– MatanPApr 19, 2013 at 13:53
1 Answer
This might not be a satisfactory answer, but let me point out that S.c. Wong constructed a "little cube" model for the homotopy fiber of $\Omega^{k1}E^n : \Omega^{k1}\Sigma^{k}X \to \Omega^{k1+n}\Sigma^{k+n} X$ in this paper in 1994.
Note that May's little cube model for $\Omega\Sigma X$ is essentially the James construction.