Iterated doubles of the Joker and their realisability

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 $n\geq4$.