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

TL;DR

This paper calculates the continuous cohomology of the Morava stabilizer group at p=2 for certain degrees, and determines differentials in the spectral sequence for the K(2)-local sphere, advancing understanding in stable homotopy theory.

Contribution

It provides explicit computations of cohomology groups and differentials using the Algebraic Duality Spectral Sequence at p=2, which were previously unknown.

Findings

Computed $H^*( ext{Morava stabilizer group}, E_t)$ for 0 ≤ t < 12 at p=2.

Determined the $d_3$-differentials in the homotopy fixed point spectral sequence.

Enhanced understanding of the $K(2)$-local sphere spectrum in stable homotopy theory.

Abstract

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.