On the Action of Steenrod Squares on Polynomial Algebras

Abstract

Let ${P_s}$ be the $\bmod - 2$ cohomology of the elementary abelian group $(Z/2Z) \times \cdots \times (Z/2Z)$ ($s$ factors). The $\bmod - 2$ Steenrod algebra $A$ acts on ${P_s}$ according to well-known rules. If ${\mathbf {A}} \subset A$ denotes the augmentation ideal, then we are interested in determining the image of the action ${\mathbf {A}} \otimes {P_s} \to {P_s}$: the space of elements in ${P_s}$ that are hit by positive dimensional Steenrod squares. The problem is motivated by applications to cobordism theory [P1] and the homology of the Steenrod algebra [S]. Our main result, which generalizes work of Wood [W], identifies a new class of hit monomials.