Generalized Steenrod-Hopf invariants for stable homotopy theory

Abstract

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.

Peterson-Stein Formulas in the Adams Spectral Sequence

“…We also note that this lemma corrects Proposition 5 of[3] from which Propositions 1 and 2 now follow except the minus signs in their statements disappear.) PROOF OF LEMMA.…”

Hopf invariants and Browder’s work on the Kervaire invariant problem

“…More precisely in case q + 1 = 2k, if (M2q, F) is a framed manifold whose Thorn invariant is/ E irlq(S°), then Browder showed that the Kervaire invariant, K(M2q, F), is one if and only if/projects to h\ in £¿'2*+'(5°). On the other hand, the present author [3] introduced homotopy invariants Ix and 72 which are defined on certain subgroups of the homotopy groups trJ(X) of a spectrum X and which take their values in the E2 term of an Adams spectral sequence for X. For the mod 2 Adams spectral sequence of S° and an element/ E ir2q(S°), I2(f) is always defined, and if q + 1 = 2k, A2 survives if and only if there is/ G w2*+<_2(S°) so that 72(/) = A2.…”

“…Occasionally we shall regard £ as a functor on the stable category and denote induced maps on Ex by a subscript "*" and induced maps on (sub) quotients of £, by a subscript "*" (which convention is consistent with the above usage of sharped morphism letters for induced homomorphisms on Ext). In [3], it was shown that Proof. Let x(q + 1) G H2q+x(C(d)) be the element in the basis dual to the basis {Sq*x} for //2?+I(C(w)) corresponding to Sq9+1x, where x G //?…”