Constructing product fibrations by means of a generalization of a theorem of Ganea

A theorem of Ganea shows that for the principal homotopy fibration $\Omega B\to F\to E$ induced from a fibration $F\to E\to B$, there is a product decomposition $\Omega (E/F)\approx \Omega B\times \Omega (F*\Omega B)$. We will determine the conditions for a fibration $X\to Y\to Z$ to yield a product decomposition $\Omega (Z/Y)\approx X\times \Omega (X*Y)$ and generalize it to pushouts. Using this approach we recover some decompositions originally proved by very computational methods. The results are then applied to produce, after localization at an odd prime $p$, homotopy decompositions for $\Omega {J_{k}\left (S^{2n}\right )}$ for some $k$ which include the cases $k=p^{t}$. The factors of $\Omega {J_{p^{t}}\left (S^{2n}\right )}$ consist of the homotopy fibre of the attaching map $S^{2np^{t}-1}\to {J_{p^{t}-1}\left (S^{2n}\right )}$ for ${J_{p^{t}}\left (S^{2n}\right )}$ and combinations of spaces occurring in the Snaith stable decomposition of $\Omega ^{2} S^{2n+1}$.