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

TL;DR

This paper computes the $K(2)$-local homotopy groups of a specific class of 2-local finite spectra, using a homotopy fixed point spectral sequence, providing an almost complete description of their structure.

Contribution

It introduces a method to compute the $K(2)$-local homotopy groups for spectra in a new class, advancing understanding of their homotopical properties.

Findings

Computed $K(2)$-local homotopy groups for spectra in $ ilde{oldsymbol{ ext{Z}}}$

Identified key differentials in the spectral sequence, with some conjectural extensions

Provided evidence supporting the triviality of certain differentials and extensions

Abstract

Recently 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 of this article is to compute the $K(2)$-local homotopy groups $\pi_*L_{K(2)}Z$ of all spectra $Z \in \widetilde{\mathcal{Z}}$ using a homotopy fixed point spectral sequence, and we give an almost complete computation. The incompleteness lies in the fact that we are unable to eliminate one family of $d_3$-differentials and a few potential hidden extensions, though we conjecture that all these differentials and hidden extensions are trivial.