On the action of Steenrod squares on polynomial algebras II

Abstract

Let A (2) be the mod-2 Steenrod algebra, and let P s = F 2[x 1, …, x s ] be the mod-2 cohomology of the s-fold product of RP x with itself, with its usual structure as an A (2)-module. A polynomial P ε P s is said to be hit if it is in the image of the action A(2) bo P s → P s , wher A(2) is the augmentation ideal of A (2). In this paper we state two equivalent forms of a conjecture that a certain family of monomials is hit, and prove the conjecture in a special case. In the process, we use information about the canonical antiautomorphism χ of A (2) to show that a hit polynomial P remains hit when multiplied by any polynomial raised to a sufficiently high 2-power. The relevant 2-power depends only on the Milnor basis elements required to hit P.