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

The Joker is an important finite cyclic module over the mod-2 Steenrod algebra 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.

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stable homotopy theory, Steenrod algebra
Mathematical Subject Classification 2010
Primary: 55P42
Secondary: 55S10, 55S20
Received: 29 May 2019
Revised: 12 September 2019
Accepted: 17 October 2019
Published: 20 July 2020
Andrew Baker
School of Mathematics and Statistics
University of Glasgow
United Kingdom
Tilman Bauer
Institutionen för Matematik
Kungliga Tekniska Högskolan