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Peer-reviewedCC BYAlgebraic & Geometric Topology

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

  • Jonathan Beardsley1
  • Kiran Luecke2
  • Jack Morava3
  1. 1Department of Mathematics and Statistics, University of Nevada, Reno, Reno, NV, United States
  2. 2Department of Mathematics, University of Illinois Urbana-Champaign, Urbana, IL, United States
  3. 3Department of Mathematics, The Johns Hopkins University, Baltimore, MD, United States
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Abstract

We compute the first two k-invariants of the Picard spectra of KU and KO 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 1 -structures of their truncations Pic.KU/Œ0; 3 and Pic.KO/Œ0; 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 Stringand Spin-structures on manifolds and can be used to twist tmf-cohomology.Finally, we are able to identify pic.KU/Œ0; 3 with a cotruncation of the Anderson dual of the sphere spectrum.

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Cite

BibTeXRIS
BibTeX
@article{beardsley2026brauerwall,
  title = {{Brauer–Wall groups and truncated Picard spectra of K-theory}},
  author = {Jonathan Beardsley and Kiran Luecke and Jack Morava},
  journal = {Algebraic \& Geometric Topology},
  year = {2026},
  month = may,
  doi = {10.2140/agt.2026.26.1749},
  url = {https://doi.org/10.2140/agt.2026.26.1749},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Brauer–Wall groups and truncated Picard spectra of K-theory
AU  - Jonathan Beardsley
AU  - Kiran Luecke
AU  - Jack Morava
PY  - 2026
DA  - 2026/05/23
JO  - Algebraic & Geometric Topology
DO  - 10.2140/agt.2026.26.1749
UR  - https://doi.org/10.2140/agt.2026.26.1749
AB  - We compute the first two k-invariants of the Picard spectra of KU and KO 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 1 -structures of their truncations Pic.KU/Œ0; 3 and Pic.KO/Œ0; 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 Stringand Spin-structures on manifolds and can be used to twist tmf-cohomology.Finally, we are able to identify pic.KU/Œ0; 3 with a cotruncation of the Anderson dual of the sphere spectrum.
SN  - 1472-2739
SN  - 1472-2747
ER  -