Endotrivial modules for the quaternion group and iterated Jokers in chromatic homotopy theory

TL;DR

This paper explores the realization of an exceptional endotrivial module over the quaternion group within the height 2 chromatic homotopy theory, connecting algebraic Joker modules, iterated doubles, and Morava K-theory.

Contribution

It demonstrates that Morava K-theory of double Jokers realizes a unique endotrivial module over the quaternion group in the chromatic setting, revealing new algebraic and topological connections.

Findings

Morava K-theory of double Jokers realizes an exceptional endotrivial module.

Existence of this module depends on the field containing a primitive cube root of unity.

Connections established with Massey products in quaternion group cohomology.

Abstract

The algebraic Joker module was originally described in the 1970s by Adams and Priddy and is a $5$-dimensional module over the subHopf algebra $\mathcal{A}(1)$ of the mod $2$ Steenrod algebra. It is a self-dual endotrivial module, i.e., an invertible object in the stable module category of $\mathcal{A}(1)$. Recently it has been shown that no analogues exist for $\mathcal{A}(n)$ with $n>1$. Using iterated doubling this also gives an iterated double which is an $\mathcal{A}(n)$-module but not stably invertible. In previous work the author showed that for $n=1,2,3$ these iterated doubles were realisable as cohomology of CW spectra, but no such realisation existed for $n>3$. The main point of the paper is to show that in the height $2$ chromatic context, the Morava $K$-theory of double Jokers realise an exceptional endotrivial module over the quaternion group of order $8$ that only exists over a field of characteristic $2$ containing a primitive cube root of unity. This has connections with certain Massey products in the cohomology of the quaternion group.