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

Abstract

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.

Cited by in corpus (12)

• Chromatic splitting for the $K(2)$-local sphere at $p=2$
• The telescope conjecture at height 2 and the tmf resolution
• Invertible $K(2)$-Local $E$-Modules in $C_4$-Spectra
• Chromatic Cyclotomic Extensions
• Picard groups and duality for Real Morava $E$-theories
• The topological modular forms of $\mathbb{R}{P}^2$ and $\mathbb{R}{P}^2\wedge \mathbb{C}{P}^2$
• Towards the $K(2)$-local homotopy groups of $Z$
• Gross-Hopkins Duals of Higher Real K-theory Spectra
• Morava K-theory and Filtrations by Powers
• Exotic Picard groups and chromatic vanishing via the Gross-Hopkins duality
• Computations of Orbits for the Lubin-Tate Ring
• On the tmf-Hurewicz image of $A_1$