Low dimensional Milnor-Witt stems over R

TL;DR

This paper computes the first four Milnor-Witt stems of motivic stable homotopy groups over R, using spectral sequences starting from known Ext groups over C, providing new insights into low-dimensional motivic homotopy theory.

Contribution

It introduces a method to compute low-dimensional motivic stable homotopy groups over R by leveraging Ext groups over C and spectral sequences, filling a gap in the understanding of these groups.

Findings

Computed the first four Milnor-Witt stems over R

Applied rho-Bockstein spectral sequence to Ext groups over C

Spectral sequence collapses in low-dimensional range

Abstract

This article computes some motivic stable homotopy groups over R. For 0 <= p - q <= 3, we describe the motivic stable homotopy groups of a completion of the motivic sphere spectrum. These are the first four Milnor-Witt stems. We start with the known Ext groups over C and apply the rho-Bockstein spectral sequence to obtain Ext groups over R. This is the input to an Adams spectral sequence, which collapses in our low dimensional range.