Spanier--Whitehead duality in the K(2)-local category at p=2

TL;DR

This paper proves a duality property for certain spectra in the K(2)-local category at p=2, showing the dual is a shifted version of the original spectrum, extending known results at p=3.

Contribution

It establishes the K(2)-local Spanier--Whitehead duality for spectra associated with finite subgroups of the Morava stabilizer group at p=2, using the topological duality resolution spectral sequence.

Findings

The dual of $E_2^{hF}$ is $ ext{Σ}^{44}E_2^{hF}$ for any finite subgroup F.

Results are analogous to those at height 2 and p=3.

Utilizes the topological duality resolution spectral sequence at p=2.

Abstract

The fixed point spectra of Morava E-theory $E_n$ under the action of finite subgroups of the Morava stabilizer group $\mathbb{G}_n$ and their K(n)-local Spanier--Whitehead duals can be used to approximate the K(n)-local sphere in certain cases. For any finite subgroup F of the height 2 Morava stabilizer group at p=2 we prove that the K(2)-local Spanier--Whitehead dual of the spectrum $E_2^{hF}$ is $\Sigma^{44}E_2^{hF}$. These results are analogous to the known results at height 2 and p=3. The main computational tool we use is the topological duality resolution spectral sequence for the spectrum $E_2^{h\mathbb{S}_2^1}$ at p=2.