Brauer–Wall groups and truncated Picard spectra of K-theory

Beardsley, Jonathan; Luecke, Kiran; Morava, Jack

doi:10.2140/agt.2026.26.1749

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

Author index

To appear

Other MSP journals

Brauer–Wall groups and truncated Picard spectra of $K$-theory

Jonathan Beardsley, Kiran Luecke and Jack Morava

Algebraic & Geometric Topology 26 (2026) 1749–1780

DOI: 10.2140/agt.2026.26.1749

Abstract

We compute the first two $k$ -invariants of the Picard spectra of $K U$ and $K O$ by analyzing their Picard groupoids and constructing their unit spectra as global sections of sheaves on the category of manifolds. This allows us to determine the $𝔼_{\infty}$ -structures of their truncations $Pic (K U) [0, 3]$ and $Pic (K O) [0, 2]$ . It follows that these truncated Picard spaces represent the Brauer groups of $ℤ ∕ 2$ -graded algebra bundles of Donovan, Karoubi, Moutuou and Maycock; the Brauer groups of super 2-lines; and the $K$ -theory twists of Freed, Hopkins and Teleman. Our results also imply that these spaces represent twists of $S t r i n g$ - and $S p i n$ -structures on manifolds and can be used to twist $tmf$ -cohomology. Finally, we are able to identify $pic (K U) [0, 3]$ with a cotruncation of the Anderson dual of the sphere spectrum.

Keywords

$K$-theory, Brauer groups, Picard spectra, twisted $K$-theory

Mathematical Subject Classification

Primary: 19L50, 55N15, 55P42

References

Publication

Received: 31 January 2024

Revised: 23 April 2025

Accepted: 8 June 2025

Published: 23 May 2026

Authors

Jonathan Beardsley
Department of Mathematics and Statistics University of Nevada, Reno Reno, NV United States

Kiran Luecke
Department of Mathematics University of Illinois Urbana-Champaign Urbana, IL United States

Jack Morava
Department of Mathematics The Johns Hopkins University Baltimore, MD United States

Open Access made possible by participating institutions via Subscribe to Open.

Volume 26, issue 5 (2026)