Generalized Steenrod-Hopf invariants for stable homotopy theory

Abstract:

In his paper On the groups $J(X)$. IV, Adams suggested that one might try to continue his $d$ and $e$ invariants to a sequence of higher homotopy invariants, each defined upon the vanishing of its predecessors and each taking its value in a certain Ext group. Recently he pointed out the efficacy of relocating his $d$ and $e$ invariants in Ext groups formed over a certain abelian category of comodules. It is the purpose of this note to carry out the program suggested above in a setting of the sort just mentioned. More specifically, for each homology theory which is representable by a comutative ring spectrum and whose ring of cooperations is flat over the coefficient ring, a sequence of higher homotopy invariants is constructed whose first term is Adams’ $e$ invariant for this theory.