We study a weak form of the classical conjecture which predicts that there are no spherical classes in Q_oS⁰ except the elements of Hopf invariant one and those of Kervaire invariant one. The weak conjecture is obtained by restricting the Hurewicz homomorphism to the homotopy classes which are detected by the algebraic transfer. Let P_k = F₂[x₁,... ,X_k] with |x_i| = 1. The general linear group CL_k = GL(k,F₂) and the (mod 2) Steenrod algebra A act on Pfc in the usual manner. We prove that the weak conjecture is equivalent to the following one: The canonical homomorphism j_k : F₂ ⊗_A(P_kGL →(F₂ ⊗_AP_k)GL_k induced by the identity map on P_k. is zero in positive dimensions for k > 2. In other words, every Dickson invariant (i.e. element of P_k^GLk) of positive dimension belongs to A⁺· P_k for k > 2, where A⁺ denotes the augmentation ideal of v4. This conjecture is proved for k = 3 in two different ways. One of these two ways is to study the squaring operation Sq⁰ on P(F₂ ⊗_GLkP_k), the range of j_k and to show it commuting through j_k with Kameko's Sq⁰ on F₂ ⊗_GLk P(P_k), the domain of j_k. We compute explicitly the action of Sq⁰ on P(F₂ ⊗_GLkP_k) for k≤4. ©1997 American Mathematical Society.

Hu'Ng N.H.V.

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