An $\mathbb{R}$-motivic $v_{1}-$self-map of periodicity $1$

TL;DR

This paper constructs and lifts a $v_1$-self-map in the context of $b R$-motivic and $b C_2$-equivariant spectra, revealing new relationships between motivic and equivariant stable homotopy theory.

Contribution

It demonstrates the lifting of a $v_1$-self-map from a type 1 spectrum to both $b R$-motivic and $b C_2$-equivariant spectra, and analyzes its properties.

Findings

A $v_1$-self-map of $b R$-motivic spectrum is constructed.

The self-map's cofiber realizes the subalgebra $b R$-motivic Steenrod algebra.

The $b C_2$-equivariant self-map is shown to be nilpotent on geometric fixed points.

Abstract

We consider a nontrivial action of $\mathrm{C}_2$ on the type $1$ spectrum $\mathcal{Y} := \mathrm{M}_2(1) \wedge \mathrm{C}(\eta)$, which is well-known for admitting a $1$-periodic $v_1-$self-map. The resultant finite $\mathrm{C}_2$-equivariant spectrum $\mathcal{Y}^{\mathrm{C}_2}$ can also be viewed as the complex points of a finite $\mathbb{R}$-motivic spectrum $\mathcal{Y}^\mathbb{R}$. In this paper, we show that one of the $1$-periodic $v_1-$self-maps of $\mathcal{Y}$ can be lifted to a self-map of $\mathcal{Y}^{\mathrm{C}_2}$ as well as $\mathcal{Y}^{\mathbb{R}}$. Further, the cofiber of the self-map of $\mathcal{Y}^{\mathbb{R}}$ is a realization of the subalgebra $\mathcal{A}^\mathbb{R}(1)$ of the $\mathbb{R}$-motivic Steenrod algebra. We also show that the $\mathrm{C}_2$-equivariant self-map is nilpotent on the geometric fixed-points of $\mathcal{Y}^{\mathrm{C}_2}$.