The Joker is an important finite cyclic module over the mod- Steenrod algebra . We show that the Joker, its first two iterated Steenrod doubles, and their linear duals are realizable by spaces of as low a dimension as the instability condition of modules over the Steenrod algebra permits. This continues and concludes prior work by the first author and yields a complete characterization of which versions of Jokers are realizable by spaces or spectra and which are not. The constructions involve sporadic phenomena in homotopy theory (–compact groups, topological modular forms) and may be of independent interest.
"The realizability of some finite-length modules over the Steenrod algebra by spaces." Algebr. Geom. Topol. 20 (4) 2129 - 2143, 2020. https://doi.org/10.2140/agt.2020.20.2129