Volume 20, issue 3 (2020)

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

Prasit Bhattacharya and Philip Egger

Algebraic & Geometric Topology 20 (2020) 1235–1277
Abstract

Previously (Adv. Math. 360 (2020) art. id. 106895), we introduced a class $\stackrel{˜}{\mathsc{𝒵}}$ of $2$–local finite spectra and showed that all spectra $Z\in \stackrel{˜}{\mathsc{𝒵}}$ 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 $Z\in \stackrel{˜}{\mathsc{𝒵}}$ 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.

Keywords
$K(2)$–local homotopy of $Z$, stable homotopy, $v_2$–periodicity
Mathematical Subject Classification 2010
Primary: 55N20, 55Q10, 55Q51