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 η-periodic motivic sphere spectrum over a field 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 α1-periodic slice spectral sequence over determines the α1-periodic slice spectral sequence over all extensions  k. This leads to a speculation on the role of a “connective Witt-theoretic J-spectrum” in η-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
Authors
Kyle Ormsby
Department of Mathematics
Reed College
Portland, OR
United States
Oliver Röndigs
Institut für Mathematik
Universität Osnabrück
Osnabrück
Germany