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Wilson spaces, snaith constructions, and elliptic orientations

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Abstract

We construct a canonical family of even periodic \(\mathbb{E}_{\infty}\)-ring spectra, with exactly one member of the family for every prime \(p\) and chromatic height \(n\). At height 1 our construction is due to Snaith, who built complex \(K\)-theory from \(\mathbb{CP}^{\infty}\). At height 2 we replace \(\mathbb{CP}^{\infty}\) with a \(p\)-local retract of \(\mathrm{BU} \langle 6 \rangle \), producing a new theory that orients elliptic, but not generic, height 2 Morava \(E\)-theories.

In general our construction exhibits a kind of redshift, whereby \(\mathrm{BP}\langle n-1 \rangle \) is used to produce a height \(n\) theory. A familiar sequence of Bocksteins, studied by Tamanoi, Ravenel, Wilson, and Yagita, relates the \(K(n)\)-localization of our height \(n\) ring to work of Peterson and Westerland building \(E_{n}^{hS\mathbb{G}^{\pm}}\) from \(\mathrm{K}(\mathbb{Z},n+1)\).

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Acknowledgements

We thank Craig Westerland for some very enlightening discussions, particularly regarding Sects. 5 & 6. We also owe thanks to Eric Peterson, Piotr Pstrągowski, Doug Ravenel, Andrew Senger, and Steve Wilson, and to the anonymous referee for numerous clarifying and enlightening remarks. Most of all we thank Mike Hopkins, Jacob Lurie, and Haynes Miller, who served as the three authors’ three PhD advisors and endured countless conversations about this work over the last several years. Through the course of the work, the second author was supported by NSF Grant DMS-1803273, and the third author by NSF Grant DGE-1122374.

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  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Hood Chatham, Jeremy Hahn & Allen Yuan

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Chatham, H., Hahn, J. & Yuan, A. Wilson spaces, snaith constructions, and elliptic orientations. Invent. math. 236, 165–217 (2024). https://doi.org/10.1007/s00222-024-01239-3

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