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A theorem of Ganea shows that for the principal homotopy - bration B ! F ! E induced from a bration F ! E ! B ,t here is a product decomposition (E=F) B (F B). We will determine the conditions for a bration X ! Y ! Z to yield a product decomposition (Z=Y ) X(XY ) and generalize it to pushouts. Using this approach we recover some decompositions originally proved by very computational meth- ods. The results are then applied to produce, after localization at an odd prime p, homotopy decompositions for Jk S2n for some k which include the cases k = pt .T he factors of J p t S 2 n consist of the homotopy bre of the attaching map S2np t 1 ! Jpt 1 S2n for Jpt S2n and combinations of spaces occurring in the Snaith stable decomposition of 2S2n+1.

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