#### Volume 20, issue 4 (2020)

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

### Andrew Baker and Tilman Bauer

Algebraic & Geometric Topology 20 (2020) 2129–2143
##### Abstract

The Joker is an important finite cyclic module over the mod-$2$ Steenrod algebra $\mathsc{𝒜}$. 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.

##### Keywords
stable homotopy theory, Steenrod algebra
##### Mathematical Subject Classification 2010
Primary: 55P42
Secondary: 55S10, 55S20
##### Publication
Received: 29 May 2019
Revised: 12 September 2019
Accepted: 17 October 2019
Published: 20 July 2020
##### Authors
 Andrew Baker School of Mathematics and Statistics University of Glasgow Glasgow United Kingdom http://www.maths.gla.ac.uk/~ajb Tilman Bauer Institutionen för Matematik Kungliga Tekniska Högskolan Stockholm Sweden https://people.kth.se/~tilmanb