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# Homology, Homotopy and Applications

## Volume 20 (2018)

### Number 2

### Iterated doubles of the Joker and their realisability

Pages: 341 – 360

DOI: https://dx.doi.org/10.4310/HHA.2018.v20.n2.a17

#### Author

#### Abstract

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.

Received 25 March 2018

Received revised 26 April 2018

Published 11 July 2018