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Tuesday, October 22, 2013

**Abstract:** The homotopy of the $p$-local sphere spectrum $S$ is determined by a family of localizations $L_{K(n)}S$ with respect to Morava K-theories $K(n)$. We will discuss some computations when $p, n=2$. Considerable information here can be derived from the action of the Morava stabilizer group on the Lubin-Tate theory. Goerss, Henn, Mahowald and Rezk have constructed a resolution of the $K(2)$-local sphere at the prime 3 which allows to simplify computations of $\pi_*L_{K(2)}S$. We will discuss a generalization of their work to the prime 2 and construct a resolution of the spectrum $E^{h\mathbb{S}_2^1}$, which is closely related to the K(2)-local sphere at the prime 2.