An $\mathbb{R}$-motivic $v_1-$self-map of periodicity $1$

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 ${\mathcal{Y}}^{{\mathrm{C}}_2}$.