Iterated doubles of the Joker and their realisability

Let $\mathcal{A}(1)^*$ be the subHopf algebra of the $\mathrm{mod} \: 2$ Steenrod algebra $\mathcal{A}^*$ generated by $\mathrm{Sq}^1$ and $\mathrm{Sq}^2$. The Joker is the cyclic $\mathcal{A}(1)^*$-module $\mathcal{A}(1)^* / \mathcal{A}(1)^* \lbrace \mathrm{Sq}^3 \rbrace$ which plays a special rôle 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 $n \geqslant 2$ and prove realisability results (both stable and unstable) for $n = 2, 3$ and non-realisability results for $n \geqslant 4$.

Keywords

stable homotopy theory, Steenrod algebra

2010 Mathematics Subject Classification

Primary 55P42. Secondary 55S10, 55S20.

Full Text (PDF format)

Received 25 March 2018

Received revised 26 April 2018

Accepted 3 May 2018

Published 11 July 2018