Homology, Homotopy and Applications

Volume 20 (2018)

Number 2

Iterated doubles of the Joker and their realisability

Pages: 341 – 360

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


Andrew Baker (School of Mathematics & Statistics, University of Glasgow, Scotland)


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 $n \geqslant 4$.


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

Published 11 July 2018