# 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.

## Authors

Department of Mathematics, Northwestern University, Evanston, IL 60208, United States

Hans-Werner Henn

Institut de Recherche Mathématique Avancée, Université Louis Pasteur, 67084 Strasbourg, France

Mark Mahowald

Department of Mathematics, Northwestern University, Evanston IL 60208, United States

Charles Rezk

Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, United States