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)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.

💡 Research Summary

The paper develops a concrete resolution of the K(2)-local sphere at the prime 3, presenting the spectrum L_{K(2)}S^0 as the inverse limit of a four‑stage tower of fibrations. Each stage’s fiber is a homotopy fixed‑point spectrum E_2^{hF}, where E_2 denotes the second Lubin–Tate (Morava) spectrum and F runs over four distinguished finite subgroups of the Morava stabilizer group G_2(3). The chosen subgroups—G_{24}, the semidihedral group SD_{16}, the product C_3·C_4, and the cyclic group C_3—capture all the 3‑torsion phenomena present in G_2(3).

To compute the homotopy groups of the fibers, the authors construct homotopy fixed‑point spectral sequences (HFSS) with E_2‑terms H^s(F;π_tE_2). Because π_*E_2 is a 2‑periodic module over the Witt vectors W(F_9)