Consider a fiber sequence of spaces

$$F \overset{i}{\to} E \to B$$

The cofiber $C(i)$ of the inclusion of the fiber comes with a canonical map $C(i) \to B$. Its possible to show (using some point set topology) that the fiber of that map is homotopy equivalent to $\Omega B \ast F$ hence we obtain a new fiber sequence

$$\Omega B \ast F \to C(i) \to B$$

To demonstrate the power of this statement consider the case of the path space fibration where $B= X , E = PX \cong \ast , F = \Omega X$. The new fiber sequence we obtain would be

$$\Sigma (\Omega X \wedge \Omega X) \cong \Omega X \ast \Omega X \to \Sigma \Omega X \to X$$

This implies immediately that whenever $X$ is $n$-connective for $n\ge 1$ then the canonical map $\Sigma \Omega X \to X$ is ($2n-1$)-connective (which is sort of the Eckman Hilton dual of Freudenthal suspension theorem).

Question: Is there an Eckman-Hilton dual to this statement?

To be precise, let $X \to Y \overset{\pi}{\to} Z$ be a cofiber sequence and let $F := fib(\pi)$ be the fiber $\pi$. Again there's a canonical (upto homotopy) map $f: X \to F$ and we may take its cofiber to obtain a new cofiber sequence

$$X \to F \to C(f)$$

Is there a "closed formula" for $C(f)$ (and for the maps involved) in terms of $X,Y$ and $Z$ (and the maps) similar to the one from above?

  • $\begingroup$ IS there a reason to expect it to be a loop space?! What is the case with the cofibre sequences $X\to *\to \Sigma X$. So, I thought something like $\Omega Map_*(Z,\Sigma X)$ might be a dual object to expect to appear as the cofibre of $X\to F$ ?!? $\endgroup$ – user51223 May 4 '19 at 14:49

This sort of question was studied a lot by Ganea (see here and here). The first piece of bad news is that duality is not perfect: there is no formula for that cofiber that depends only on $\Sigma X$ and $Z$. A counterexample was supplied by Barratt (as Remark 3.5 in the first linked paper above).

Even worse, the homotopy type of that cofiber does not have a formula just in terms of $X$ and $Z$! A counterexample is supplied in the second linked paper as 1.4.

The best we can do is that the suspension of the cofiber is given by $\Omega Z \ast X$.

This observation leads to a nice proof of the James splitting, incidentally- with no calculation.

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    $\begingroup$ Thanks! What a bummer though... :( $\endgroup$ – Saal Hardali May 7 '19 at 11:58

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