Symmetric A actions on $\mathcal{A}(2)$

TL;DR

This paper classifies symmetric left actions of the mod 2 Steenrod algebra on its subalgebra, exploring their structure, variety, and relation to cohomology of self maps, expanding understanding of algebraic actions in topology.

Contribution

It characterizes the variety of symmetric actions of the Steenrod algebra on (2), including their enumeration, structure, and relation to cohomological self maps, extending prior classifications.

Findings

Identified 256 symmetric actions arising from specific cohomological maps.

Described 1600 total  actions on (2) and their algebraic structure.

Analyzed the inclusion of symmetric actions into all actions and their duality properties.

Abstract

We describe the variety of `symmetric' left actions of the mod 2 Steenrod algebra $\mathcal{A}$ on its subalgebra $\mathcal{A}(2)$. These arise as the cohomology of $\text{v}_2$ self maps $\Sigma^7 Z \longrightarrow Z$, as in arXiv:1608.06250 [math.AT]. There are $256$ $\mathbb{F}_2$ points in this variety, arising from $16$ such actions of $Sq^8$ and, for each such, $16$ actions of $Sq^{16}$. We describe in similar fashion the 1600 $\mathcal{A}$ actions on $\mathcal{A}(2)$ found by Roth(1977) and the inclusion of the variety of symmetric actions into the variety of all actions. We also describe two related varieties of $\mathcal{A}$ actions, the maps between these and the behavior of Spanier-Whitehead duality on these varieties. Finally, we note that the actions which have been used in the literature correspond to the simplest choices, in which all the coordinates equal zero.