Iterated doubles of the Joker and their realisability

Abstract

Let $\mathcal{A}(1{)}^{\ast }$ be the subHopf algebra of the mod~$2$ Steenrod algebra ${\mathcal{A}}^{\ast }$ generated by ${\mathrm{Sq}}^1$ and ${\mathrm{Sq}}^2$. The \emph{Joker} is the cyclic $\mathcal{A}(1{)}^{\ast }$-module $\mathcal{A}(1{)}^{\ast }/\mathcal{A}(1{)}^{\ast }\{{\mathrm{Sq}}^3\}$ which plays a special rôle in the study of $\mathcal{A}(1{)}^{\ast }$-modules. We discuss realisations of the Joker both as an ${\mathcal{A}}^{\ast }$-module and as the cohomology of a spectrum. We also consider analogous $\mathcal{A}(n{)}^{\ast }$-modules for $n\ge 2$ and prove realisability results (both stable and unstable) for $n=2,3$ and non-realisability results for $n\ge 4$.

Minor changes and corrections. A version will appear in Homology, Homotopy and Applications