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

TL;DR

This paper establishes a resolution of the $ a$-periodic motivic sphere spectrum over fields of characteristic not 2, leading to new computations of motivic stable stems and cobordism groups, and introduces Adams operations on Hermitian K-theory.

Contribution

It provides a novel 2-term resolution of the $ a$-periodic motivic sphere spectrum and constructs Adams operations on Hermitian K-theory spectra.

Findings

Resolved the $ a$-periodic motivic sphere spectrum using shifts of Witt K-theory.

Determined $ a$-periodized motivic stable stems and cobordism groups.

Established new completeness results for motivic spectra over fields of finite virtual 2-cohomological dimension.

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.