A $\mathbb{C}$-motivic $w_1$-self-map of periodicity 1

Abstract

Let $\mathrm{DY} = \mathbb{1}/(\eta, \nu) \in \mathrm{SH}(\mathbb{C})$. We construct a $w_1$-self-map of periodicity 1 $w:\Sigma^{5,3}\mathrm{DY} \to \mathrm{DY}.$ The cofiber is a realization of $DA(1)$, the double of the subalgebra $A(1)$ of the $\mathbb{C}$-motivic Steenrod algebra $A$.