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arXiv · 2005.06778

$\eta$-periodic motivic stable homotopy theory over fields

Tom Bachmann·Michael J. Hopkins
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Abstract

Over any field of characteristic not 2, we establish a 2-term resolution of the $\eta$-periodic, 2-local motivic sphere spectrum by shifts of the connective 2-local Witt K-theory spectrum. This is curiously similar to the resolution of the K(1)-local sphere in classical stable homotopy theory. As applications we determine the $\eta$-periodized motivic stable stems and the $\eta$-periodized algebraic symplectic and SL-cobordism groups. Along the way we construct Adams operations on the motivic spectrum representing Hermitian K-theory and establish new completeness results for certain motivic spectra over fields of finite virtual 2-cohomological dimension. In an appendix, we supply a new proof of the homotopy fixed point theorem for the Hermitian K-theory of fields.

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Tom Bachmann, Michael J. Hopkins. 2020-05-14. $\eta$-periodic motivic stable homotopy theory over fields. https://arxiv.org/abs/2005.06778

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