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

## TL;DR

This paper characterizes which finite cyclic modules over the mod-2 Steenrod algebra, including the Joker and its iterates, can be realized by topological spaces, using advanced homotopy theory techniques.

## Contribution

It provides a complete classification of the realizability of Joker modules over the Steenrod algebra by spaces or spectra, extending previous work.

## Key findings

- Joker and its iterates are realizable by low-dimensional spaces
- Complete characterization of realizability of Joker modules
- Uses sporadic homotopy phenomena like 2-compact groups and topological modular forms

## 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.