R-motivic stable stems

TL;DR

This paper computes R-motivic stable homotopy groups up to a certain range, using spectral sequences and Ext groups, and recovers classical stable homotopy invariants.

Contribution

It introduces a method to compute R-motivic stable homotopy groups using the $ ho$-Bockstein spectral sequence and analyzes Adams differentials and extensions.

Findings

Computed $ ho$-motivic stable homotopy groups for $s-w \, extleq 11$

Analyzed Adams differentials and hidden extensions in the spectral sequence

Recovered Mahowald invariants of low-dimensional classical elements

Abstract

We compute some R-motivic stable homotopy groups. For $s - w \leq 11$, we describe the motivic stable homotopy groups $\pi_{s,w}$ of a completion of the R-motivic sphere spectrum. We apply the $\rho$-Bockstein spectral sequence to obtain R-motivic Ext groups from the C-motivic Ext groups, which are well-understood in a large range. These Ext groups are the input to the R-motivic Adams spectral sequence. We fully analyze the Adams differentials in a range, and we also analyze hidden extensions by $\rho$, 2, and $\eta$. As a consequence of our computations, we recover Mahowald invariants of many low-dimensional classical stable homotopy elements.