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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 𝒵˜ of 2–local finite spectra and showed that all spectra Z 𝒵˜ admit a v2–self-map of periodicity  1. The aim here is to compute the K(2)–local homotopy groups πLK(2)Z of all spectra Z 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 d3–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
References
Publication
Received: 17 April 2018
Revised: 22 August 2019
Accepted: 6 September 2019
Published: 27 May 2020
Authors
Prasit Bhattacharya
Department of Mathematics
University of Virginia
Charlottesville, VA
United States
Philip Egger
Center for Neuroprosthetics
Swiss Federal Institute of Technology (EPFL)
Geneva
Switzerland