Spherical classes and the algebraic transfer
We study a weak form of the classical conjecture which predicts that there are no spherical classes in QoS0 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 Pk = F2[x1,... ,Xk] with |xi| = 1. The general linear group CLk = 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 jk : F2 ⊗A(PkGL →(F2 ⊗APk)GLk induced by the identity map on Pk. is zero in positive dimensions for k > 2. In other words, every Dickson invariant (i.e. element of PkGLk) of positive dimension belongs to A+· Pk 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 ⊗GLkPk), the range of jk and to show it commuting through jk with Kameko's Sq0 on F2 ⊗GLk P(Pk), the domain of jk. We compute explicitly the action of Sq0 on P(F2 ⊗GLkPk) for k≤4. ©1997 American Mathematical Society.