The homotopy groups of \(S_{E(2)}\) at \(p\geq 5\) revisited
type: Item
instance of: scholarly article
title: The homotopy groups of \(S_{E(2)}\) at \(p\geq 5\) revisited
zbMATH Open document ID: 1256.55004
published in: Advances in Mathematics
publication date: 2012-06-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/1002.0807
review text: This paper gives a conceptualization of the results on the homotopy groups \(\pi_*(S_{E(2)})\) at a prime \(p\geq 5\) of the \(E(2)\)-localized sphere spectrum given by Shimomura-Yabe. At the prime three, the homotopy groups \(\pi_*(S_{E(2)})\) are explained by the resolution and the projective Morava stabilizer groups developed by Goerss, Henn, Karamanov, Mahowald, and Rezk. The author adapts the projective Morava stabilizer groups technique to the case \(p\geq 5\) to analyze \(\pi_*(S_{E(2)})\). In the process, some errors in the original calculation of \(\pi_*(S_{E(2)})\) are identified and corrected. By this approach, the homotopy groups \(\pi_*(S_{K(2)})\) are determined and the chromatic splitting conjecture is verified. The patterns in which the new generators appear also show Gross-Hopkins duality.
reviewed by: Katsumi Shimomura
Mathematics Subject Classification ID: 55Q99
Mathematics Subject Classification ID: 55Q45
Mathematics Subject Classification ID: 55Q51
zbMATH DE Number: 6040337
zbMATH Keywords: stable homotopy groups of spheres
zbMATH Keywords: chromatic filtration
zbMATH Keywords: \(E(2)\)-localization
author: Mark Behrens
MaRDI profile type: Publication
OpenAlex ID: W2114041507
arXiv ID: 1002.0807
cites work: Congruences between modular forms given by the divided \(\beta \) family in homotopy theory
cites work: A resolution of the \(K(2)\)-local sphere at the prime 3
cites work: The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory
cites work: The homotopy of the \(K(2)\)-local Moore spectrum at the prime 3 revisited
cites work: Q4327004
cites work: Q4848243
cites work: Morava 𝐾-theories and localisation
cites work: On Hopkins' Picard group Pic\(_2\) at the prime 3
cites work: Periodic phenomena in the Adams-Novikov spectral sequence
cites work: The cohomology of the Morava stabilizer algebras
cites work: Q3747338
cites work: Q4278124
cites work: On the Adams-Novikov spectral sequence and products of \(\beta\)-elements
cites work: Non-triviality of some compositions of \(\beta\)-elements in the stable homotopy of the Moore spaces
cites work: The Adams-Novikov \(E_2\)-term for \(\pi_*(L_2S^0)\) at the prime 2
cites work: The homotopy groups \(\pi_*(L_2 S^0)\) at the prime 3
cites work: Q4288649
cites work: The homotopy groups \(\pi_ * (L_ 2 S^ 0)\)
DOI: 10.1016/J.AIM.2012.02.023
description: scientific article; zbMATH DE number 6040337