Low-dimensional Milnor–Witt stems over $\mathbb R$

Abstract

We compute some motivic stable homotopy groups over $ℝ$. For $0\le p-q\le 3$, we describe the motivic stable homotopy groups ${\widehat{\pi }}_{p,q}$ 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.