The homotopy groups of the η-periodic motivic sphere spectrum

Abstract

We compute the homotopy groups of the η-periodic motivic sphere spectrum over a finite-dimensional field k with characteristic not 2 and in which -1 a sum of four squares. We also study the general characteristic 0 case and show that the η-periodic slice spectral sequence over Q determines the η-periodic slice spectral sequence over all extensions of Q. This leads to a speculation on the role of a "connective Witt-theoretic J-spectrum" in η-periodic motivic homotopy theory.

14 pages, 2 figures, comments very welcome! v2: hypotheses weakened to finite-dimensional level 4 fields, discussion of connective Witt-theoretic J-spectrum added