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We provide a topological duality resolution for the spectrum    E  2   h   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   h  Fwhere    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 ( Ann. of Math . (2) 162 (2005) 777–822) 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.

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