Suspension splittings and James–Hopf invariants

Abstract

James constructed a functorial homotopy decomposition for path-connected, p ointed CW-complexes X. We generalize this to a p-local functorial decomposition of ΣA, where A is any functorial retract of a looped co-H-space. This is used to construct Hopf invariants in a more general context. In addition, when A = ΩY is the loops space of a co-H-space, we show that the wedge summands of ΣΩY further functorially decompose by using an action of an appropriate symmetric group. As a valuable example, we give an application to the theory of quasi-symmetric functions.