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

Baker, A. (2022) Endotrivial modules for the quaternion group and iterated Jokers in chromatic homotopy theory. arXiv, (doi: 10.48550/arXiv.2212.00437) (Unpublished)

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 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 A(1). Recently it has been shown that no analogues exist for A(n) with n>1. In previous work the author used doubling to produce an `iterated double Joker' which is an A(n)-module but not stably invertible. We also 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 this 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.

• Item Type: Articles
• Keywords: Stable homotopy theory, steenrod algebra, Lubin-Tate spectrum, Morava K-theory, endotrivial module.
• Status: Unpublished
• Refereed: No
• Glasgow Author(s) Enlighten ID: Baker, Dr Andrew
• Authors: Baker, A.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: arXiv
• Published Online: 01 December 2022
• Copyright Holders: Copyright © 2022 The Author
• Publisher Policy: Reproduced under a Creative Commons licence