The realizability of some finite-length modules over the Steenrod algebra by spaces


The Joker is a finite cyclic module over the mod-$2$ Steenrod algebra $\mathcal A$ which appears frequently in Adams resolutions as well as in Picard groups. In a previous paper, the first author commenced a study of when the Joker and its iterated Steenrod doubles are realizable as cohomologies of spectra or spaces. In this paper, we complete this study by showing that every version of the Joker is realizable by a space of as low a dimension as the unstability condition of modules over the Steenrod algebra permits. We construct these spaces using, on the one hand, spaces that have Dickson algebras as their cohomology rings (classifying spaces of the Lie groups $\operatorname{SO}(3)$, $G_2$, as well as the exotic $2$-compact group $\operatorname{DW}_3$), and on the other hand, ring spectra having with the cohomology of Hopf quotients if the Steenrod algebra (mod-$2$ cohomology, real $K$-theory, and topological modular forms).

Algebraic & Geometric Topology 20 (2020), pp. 2129-2143, arXiv:1903.10288