$RO(G)$-graded homotopy fixed point spectral sequence for height $2$ Morava $E$-theory

TL;DR

This paper computes the $G$-homotopy fixed point spectral sequences for height 2 Morava $E$-theory at prime 2, using advanced equivariant techniques for specific finite subgroups of the Morava stabilizer group.

Contribution

It provides complete computations of the $G$-homotopy fixed point spectral sequences for certain subgroups, employing recent equivariant methods.

Findings

Explicit spectral sequence computations for $Q_8$, $SD_{16}$, $G_{24}$, and $G_{48}$.

New graded spectral sequences for $Q_8$ and $SD_{16}$ involving non-trivial representations.

Abstract

We consider $G=Q_8,SD_{16},G_{24},$ and $G_{48}$ as finite subgroups of the Morava stabilizer group which acts on the height $2$ Morava $E$-theory $\mathbf{E}_2$ at the prime $2$. We completely compute the $G$-homotopy fixed point spectral sequences of $\mathbf{E}_2$. Our computation uses recently developed equivariant techniques since Hill, Hopkins, and Ravenel. We also compute the $(*-\sigma_i)$-graded $Q_8$- and $SD_{16}$-homotopy fixed point spectral sequences, where $\sigma_i$ is a non-trivial one-dimensional representation of $Q_8$.