#### Vol. 306, No. 2, 2020

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The homotopy groups of the $\eta$-periodic motivic sphere spectrum

### Kyle Ormsby and Oliver Röndigs

Vol. 306 (2020), No. 2, 679–697
##### Abstract

We compute the homotopy groups of the $\eta$-periodic motivic sphere spectrum over a field $\mathsf{k}$ of finite cohomological dimension with characteristic not $2$ and in which $-1$ is a sum of four squares. We also study the general characteristic $0$ case and show that the ${\alpha }_{1}$-periodic slice spectral sequence over $ℚ$ determines the ${\alpha }_{1}$-periodic slice spectral sequence over all extensions  $\mathsf{k}∕ℚ$. This leads to a speculation on the role of a “connective Witt-theoretic $J$-spectrum” in $\eta$-periodic motivic homotopy theory.

##### Keywords
motivic homotopy theory, stable motivic homotopy sheaves, slice spectral sequence
Primary: 14F42
Secondary: 55Q45