A class of left ideals of the Steenrod algebra

Abstract

We study the nested collection of left ideals of $\mathcal{A}$, the mod 2 Steenrod algebra, $L(k) : = \mathcal{A}\{Sq^{2^0}, Sq^{2^1}, Sq^{2^2}, ... , Sq^{2^k}$. We determine the smallest k such that $Sq^n \in l (k)$. We discuss an application which improves upon the results of F. R. Cohen and the first author in their paper comparing the loop of the degree 2 map on a sphere and the H-space squaring map on the loop of a sphere.