The homotopy groups of \(S_{E(2)}\) at \(p\geq 5\) revisited (Q424555)

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.