R$\mathbb {R}$‐motivic stable stems
Eva Belmont, Daniel C. Isaksen
Abstract
We compute some R $\mathbb {R}$ -motivic stable homotopy groups. For s − w ⩽ 11 $s - w \leqslant 11$ , we describe the motivic stable homotopy groups π s , w $\pi _{s,w}$ of a completion of the R $\mathbb {R}$ -motivic sphere spectrum. We apply the ρ $\rho$ -Bockstein spectral sequence to obtain R $\mathbb {R}$ -motivic Ext $\operatorname{Ext}$ groups from the C $\mathbb {C}$ -motivic Ext $\operatorname{Ext}$ groups, which are well understood in a large range. These Ext $\operatorname{Ext}$ groups are the input to the R $\mathbb {R}$ -motivic Adams spectral sequence. We fully analyze the Adams differentials in a range, and we also analyze hidden extensions by ρ $\rho$ , 2, and η $\eta$ . As a consequence of our computations, we recover Mahowald invariants of many low-dimensional classical stable homotopy elements.