Topological resolutions in K(2)-local homotopy theory at the prime 2

We provide a topological duality resolution for the spectrum $E_2^{h{\mathbb{S}}_2^1}$, which itself can be used to build the $K(2)$-local sphere. The resolution is built from spectra of the form $E_2^{hF}$ where $E_2$ is the Morava spectrum for the formal group of a supersingular curve at the prime $2$ and $F$ is a finite subgroup of the automorphisms of that formal group. The results are in complete analogy with the resolutions of Goerss, Henn, Mahowald, and Rezk at the prime $3$, but the methods are of necessity very different. As in the prime $3$ case, the main difficulty is in identifying the top fiber; to do this, we make calculations using Henn’s centralizer resolution.