The a -decomposability of the Singer Construction

Let RsM denote the Singer construction on an unstable module M over the Steenrod algebra A at the prime two; RsM is canonically a subobject of Ps⊗M , where Ps = F2[x1, . . . , xs] with generators of degree one. Passage to A -indecomposables gives the natural transformation RsM → F2⊗A (Ps⊗M), which identifies with the dual of the composition of the Singer transfer and the Lannes-Zarati homomorphism. The main result of the paper proves the weak generalized algebraic spherical class conjecture, which was proposed by the first named author. Namely, this morphism is trivial on elements of positive degree when s > 2. The condition s > 2 is necessary, as exhibited by the spherical classes of Hopf invariant one and those of Kervaire invariant one.