The Brauer spectrum conjecture for complex K-theory

From papers

Let KUKU be complex topological KK-theory, let br(KU)\operatorname{br}(KU) be its Brauer spectrum, and let 4IZ{}_4I_{\mathbb{Z}} denote the relevant truncation of the Anderson dual of the sphere spectrum.

Brauer spectrum conjecture. There is an equivalence

br(KU)[0,4]4IZ.\operatorname{br}(KU)[0,4]\simeq{}_4I_{\mathbb{Z}}.

In particular,

π0(br(KU))Z/24.\pi_0(\operatorname{br}(KU))\cong\mathbb{Z}/24.

The conjecture is motivated by the identification of truncated Picard spectra with truncated Anderson duals and by the fact that π4(IZ)Z/24\pi_{-4}(I_{\mathbb{Z}})\cong\mathbb{Z}/24. It predicts the low-degree homotopy type of the Brauer spectrum, beyond the already understood relation between units, Picard spectra, and Brauer spectra.

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Sources & referencesView supporting material

Primary source

Jonathan Beardsley, Kiran Luecke and Jack Morava, “Brauer-Wall Groups and Truncated Picard Spectra of K-theory”, arXiv:2306.10112 (2023).

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