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

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)S⁰ as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E₂^hF where F is a finite subgroup of the Morava stabilizer group and E₂ 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.