Let ???? = ?? ??k?? be Singer's invariant-theoretic model of the dual of the lambda algebra with H
k(????) ?? Tork
A(2,2), where A denotes the mod 2 Steenrod algebra. We prove that the inclusion of the
Dickson algebra, Dk, into ??k?? is a chain-level representation of the Lannes-Zarati dual homomorphism *k
: 2??ADk?? Tork
A(2,2)??H k(????). The Lannes-Zarati homomorphisms themselves, k, correspond to an
associated graded of the Hurewicz map H : ??*s(S0) ?? - H(Q0S0). Based on this result, we discuss some
algebraic versions of the classical conjecture on spherical classes, which states that Only Hopf invariant one
and Ke. rva. ire. invariant one classes are detected by the Hurewicz homomorphism. One of these algebraic
conjectures predicts that every Dickson element, i. e. element in D/t, of positive degree represents the
homology class 0 in Torj4(F2,F2) for k2. We also show that factors through F2Kerdk, where dkdenotes the
differential of PA. Therefore, the problem of determining F2Ker?kshould be of interest. ?? 2001 American
Mathematical Society.
