$\mathbb{R}$-motivic $v_1$-periodic homotopy

TL;DR

This paper computes the $v_1$-periodic $R$-motivic stable homotopy groups using the effective slice spectral sequence, also analyzing related $C$-motivic and $ta$-periodic cases, advancing understanding in motivic homotopy theory.

Contribution

It provides the first detailed computation of $v_1$-periodic $R$-motivic stable homotopy groups and extends the analysis to $C$-motivic and $ta$-periodic contexts.

Findings

Computed $v_1$-periodic $R$-motivic stable homotopy groups.

Analyzed $C$-motivic and $ta$-periodic $v_1$-periodic homotopy.

Demonstrated effectiveness of the slice spectral sequence in motivic computations.

Abstract

We compute the $v_1$-periodic $\mathbb{R}$-motivic stable homotopy groups. The main tool is the effective slice spectral sequence. Along the way, we also analyze $\mathbb{C}$-motivic and $\eta$-periodic $v_1$-periodic homotopy from the same perspective.