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

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 $(*-σ_i)$-graded $Q_8$- and $SD_{16}$-homotopy fixed point spectral sequences, where $σ_i$ is a non-trivial one-dimensional representation of $Q_8$.

Figures (17)

• Figure 1: $d_5$, $d_{13}$, $d_{23}$-differentials
• Figure 2: The $E_1$-page of the integer/$(*-\sigma_i)$-graded 2BSS.
• Figure 3: The $E_\infty$-page of the integer-graded 2BSS.The dotted lines are hidden $h_2$ extensions.
• Figure 4: The $E_\infty$-page of the $(*-\sigma_i)$-graded 2BSS. The dotted lines are hidden $h_1$ and $h_2$ extensions.
• Figure 5: The $E_3$-page of the integer-graded $Q_8$-HFPSS($\mathbf{E}_2$). The red lines are $d_3$-differentials.

Theorems & Definitions (171)

• Theorem A
• Theorem B
• Theorem C: \ref{['thm:sharpvanishingline']}
• Lemma 2.1: Ull13,DLS2022
• Definition 2.2
• Proposition 2.3
• Definition 2.4
• Proposition 2.5
• Lemma 2.6
• Theorem 2.8