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

TL;DR

This paper constructs a detailed framework for understanding the stable homotopy groups of the sphere at the prime 3, specifically after localization at the second Morava K-theory, by analyzing a tower of fibrations involving Morava stabilizer groups.

Contribution

It introduces a new tower of fibrations for the K(2)-local sphere at prime 3 and explicitly computes the homotopy groups of the fibers, advancing understanding at this complex prime.

Findings

Explicit calculation of homotopy groups of fibers

Development of a tower of fibrations for the K(2)-local sphere

Enhanced understanding of the homotopy theory at 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.