Spanier–Whitehead duality in the $K(2)$-local category 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 $\mathbb {G}_2$ 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 $p=2$.