Hopf invariants for mapping cones

The relative James construction provides a functorial Hopf invariant defined on the fiber of a pinch map. Here we give a lifting of this Hopf invariant from the fiber of one pinch map to the fiber of another. An application is made to the study of the Cohen Moore Neisendorfer splitting of the loops of a Moore space in order to understand the obstructions that come in the study of the double suspension conjecture.