Relations among the kernels and images of {S}teenrod squares acting on right $\mathcal{A}$-modules

TL;DR

This paper explores the relationships between kernels and images of Steenrod squares acting on homology groups, introducing new tools to understand when certain inclusions hold or fail, with implications for algebraic topology.

Contribution

It establishes a general relationship between kernels and images of Steenrod squares on right $ ext{A}$-modules and develops new machinery to analyze when reverse inclusions occur.

Findings

Identifies conditions under which the reverse inclusion of kernels and images holds.

Provides counterexamples showing the failure of reverse inclusion for small cases.

Introduces homotopy systems and null subspaces as tools for analyzing Steenrod algebra actions.

Abstract

In this note, we examine the right action of the Steenrod algebra $\mathcal{A}$ on the homology groups $H_*(BV_s, \F_2)$, where $V_s = \F_2^s$. We find a relationship between the intersection of kernels of $Sq^{2^i}$ and the intersection of images of $Sq^{2^{i+1}-1}$, which can be generalized to arbitrary right $\mathcal{A}$-modules. While it is easy to show that $\bigcap_{i=0}^{k} \mathrm{im}\,Sq^{2^{i+1}-1} \subseteq \bigcap_{i = 0}^k \mathrm{ker}\,Sq^{2^i}$ for any given $k \geq 0$, the reverse inclusion need not be true. We develop the machinery of homotopy systems and null subspaces in order to address the natural question of when the reverse inclusion can be expected. In the second half of the paper, we discuss some counter-examples to the reverse inclusion, for small values of $k$, that exist in $H_*(BV_s, \F_2)$.