# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1706.06170