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

Baker, Andrew; Bauer, Tilman

doi:10.2140/agt.2020.20.2129

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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

DOI: 10.2140/agt.2020.20.2129

Abstract

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.

Keywords

stable homotopy theory, Steenrod algebra

Mathematical Subject Classification 2010

Primary: 55P42

Secondary: 55S10, 55S20

References

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

Volume 20, issue 4 (2020)