Towards the $K(2)$–local homotopy groups of $Z$

Abstract

Previously (Adv. Math. 360 (2020) art. id. 106895), we introduced a class $\widetilde{\mathcal{𝒵}}$ of $2$–local finite spectra and showed that all spectra $Z\in \widetilde{\mathcal{𝒵}}$ admit a $v_2$–self-map of periodicity $1$. The aim here is to compute the $K\left(2\right)$–local homotopy groups ${\pi }_{\ast }{L}_{K\left(2\right)}Z$ of all spectra $Z\in \widetilde{\mathcal{𝒵}}$ using a homotopy fixed point spectral sequence, and we give an almost complete answer. The incompleteness lies in the fact that we are unable to eliminate one family of $d_3$–differentials and a few potential hidden $2$–extensions, though we conjecture that all these differentials and hidden extensions are trivial.