On the action of Steenrod squares on polynomial algebras

Abstract

Let P s {P_s} be the mod − 2 \bmod - 2 cohomology of the elementary abelian group ( Z / 2 Z ) × ⋯ × ( Z / 2 Z ) (Z/2Z) \times \cdots \times (Z/2Z) ( s s factors). The mod − 2 \bmod - 2 Steenrod algebra A A acts on P s {P_s} according to well-known rules. If A ⊂ A {\mathbf {A}} \subset A denotes the augmentation ideal, then we are interested in determining the image of the action A ⊗ P s → P s {\mathbf {A}} \otimes {P_s} \to {P_s} : the space of elements in P s {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.