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

Kyle Ormsby and Oliver Röndigs

Vol. 306 (2020), No. 2, 679–697

DOI: 10.2140/pjm.2020.306.679

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 $\mathbb{Q}$ determines the ${\alpha }_1$-periodic slice spectral sequence over all extensions  $\mathsf{k}∕\mathbb{Q}$. 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

Mathematical Subject Classification 2010

Primary: 14F42

Secondary: 55Q45

Milestones

Received: 19 September 2019

Revised: 10 January 2020

Accepted: 19 January 2020

Published: 13 July 2020