  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.
