On the periodic $v_2$–self-map of $A_1$

Abstract

The spectrum $Y:=M_2\left(1\right)\wedge C\eta$ admits eight $v_1$-self-maps of periodicity $1$. These eight self-maps admit four different cofibers, which we denote by $A_1\left[ij\right]$ for $i,j\in \left\{0,1\right\}$. We show that each of these four spectra admits a $v_2$-self-map of periodicity $32$.