Cohomology of the Morava stabilizer group through the duality resolution at $n=p=2$
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We compute the continuous cohomology of the Morava stabilizer group with coefficients in Morava $E$-theory, $H^*(mathbb{G}_2, E_t)$, at $p=2$, for $0leq t < 12$, using the Algebraic Duality Spectral Sequence. Furthermore, in that same range, we compute the $d_3$-differentials in the homotopy fixed point spectral sequence for the $K(2)$-local sphere spectrum. These cohomology groups and differentials play a central role in $K(2)$-local stable homotopy theory.