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 $X$?
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$\begingroup$ 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 VoelkelCommented Apr 17, 2013 at 17:09
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$\begingroup$ * is the join, ie the suspension of the smash product. $\endgroup$– Fernando MuroCommented Apr 17, 2013 at 18:19
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$\begingroup$ I believe for $X=S^{n}$ then one can get some answers p-locally by looping the EHP sequence. In general I am not sure what can be said. $\endgroup$– Callan McGillCommented Apr 17, 2013 at 20:38
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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 "co-join," i.e., the holim of the diagram $\Sigma X \to \Sigma X \vee \Sigma X \leftarrow \Sigma$ given by the two inclusions. $\endgroup$– John KleinCommented Apr 17, 2013 at 22:57
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$\begingroup$ Thanks for your comments. It seems there is not a known answer for this question, as John implicitly claimed above. $\endgroup$– MatanPCommented Apr 19, 2013 at 13:53
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1 Answer
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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 $\Omega^{k-1}E^n : \Omega^{k-1}\Sigma^{k}X \to \Omega^{k-1+n}\Sigma^{k+n} X$ in this paper in 1994.
Note that May's little cube model for $\Omega\Sigma X$ is essentially the James construction.
