Cohomology of the Morava stabilizer group through the duality resolution at $n=p=2$

Agnes Beaudry, Irina Bobkova, Paul G. Goerss, Hans-Werner Henn, Viet-Cuong Pham, Vesna Stojanoska

Published 2022-10-28Version 1

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 $0\leq t < 12$, 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, in particular for the analysis of the $K(2)$-local Picard group.