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

Bhattacharya, Prasit; Egger, Philip

doi:10.2140/agt.2020.20.1235

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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

DOI: 10.2140/agt.2020.20.1235

Abstract

Previously (Adv. Math. 360 (2020) art. id. 106895), we introduced a class $\tilde{𝒵}$ of $2$ –local finite spectra and showed that all spectra $Z \in \tilde{𝒵}$ admit a $v_{2}$ –self-map of periodicity $1$ . The aim here is to compute the $K (2)$ –local homotopy groups $π_{*} L_{K (2)} Z$ of all spectra $Z \in \tilde{𝒵}$ 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

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

Volume 20, issue 3 (2020)