The hit problem for the dickson algebra

Abstract

Let the mod 2 Steenrod algebra, .4, and the general linear group, GL(fc,F2), act on Pk := F2[x1 ,...,xk] with |x1| = 1 in the usual manner. We prove the conjecture of the first-named author in Spherical classes and the algebraic transfer, (Trans. Amer. Math Soc. 349 (1997), 3893-3910) stating that every element of positive degree in the Dickson algebra Dk := (Pk)GL(k'F2) is A-decomposable in Pfc for arbitrary k > 2. This conjecture was shown to be equivalent to a weak algebraic version of the classical conjecture on spherical classes, which states that the only spherical classes in Q0S0 are the elements of Hopf invariant one and those of Kervaire invariant one. ??2001 American Mathematical Society.