  Let $R_s M$ denote the Singer construction on an unstable module $M$ over the
Steenrod algebra $A$ at the prime two; $R_s M$ is canonically a subobject of
$P_s\otimes M$, where $P_s$ is the polynomial algebra on s generators of degree
one. Passage to $A$-indecomposables gives the natural transformation $R_s M
\rightarrow F \otimes_A (P_s \otimes M)$, 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.
