Iterated doubles of the Joker and their realisability

TL;DR

This paper investigates the realizability of the Joker module and its iterated doubles within the Steenrod algebra framework, establishing new results on when these modules can be realized as cohomology of spectra.

Contribution

It introduces new realizability and non-realisability results for Joker modules and their iterated doubles in the context of the Steenrod algebra, extending previous understanding.

Findings

Realisability results for n=2,3

Non-realisability results for n≥4

Analysis of stable and unstable modules

Abstract

Let $\mathcal{A}(1)^*$ be the subHopf algebra of the mod~$2$ Steenrod algebra $\mathcal{A}^*$ generated by $\mathrm{Sq}^1$ and $\mathrm{Sq}^2$. The \emph{Joker} is the cyclic $\mathcal{A}(1)^*$-module $\mathcal{A}(1)^*/\mathcal{A}(1)^*\{\mathrm{Sq}^3\}$ which plays a special r\^ole in the study of $\mathcal{A}(1)^*$-modules. We discuss realisations of the Joker both as an $\mathcal{A}^*$-module and as the cohomology of a spectrum. We also consider analogous $\mathcal{A}(n)^*$-modules for $n\geq2$ and prove realisability results (both stable and unstable) for $n=2,3$ and non-realisability results for $n\geq4$.