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

Abstract

The Joker is an important finite cyclic module over the mod-$2$ Steenrod algebra $\mathcal{A}$. 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 ($2$-compact groups, topological modular forms) and may be of independent interest.

arXiv comment11 pages, to appear in Algebraic and Geometric Topology

References in corpus (1)

• Computation of the homotopy of the spectrum tmf