A resolution of the K(2)-local sphere at the prime 3

Abstract

We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava $K$-theory $K(2)$. At the prime $3$, we write the spectrum $L_{K(2)}S^0$ as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form $E_2^{hF}$ where $F$ is a finite subgroup of the Morava stabilizer group and $E_2$ is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case $n=2$ at $p=3$ represents the edge of our current knowledge: $n=1$ is classical and at $n=2$, the prime $3$ is the largest prime where the Morava stabilizer group has a $p$-torsion subgroup, so that the homotopy theory is not entirely algebraic.