Real Orientations of Lubin--Tate Spectra and the Slice Spectral Sequence of a C4-Equivariant Height-4 Theory

Abstract

In this thesis, we show that Lubin--Tate spectra at the prime $2$ are Real oriented and Real Landweber exact. The proof is by application of the Goerss--Hopkins--Miller theorem to algebras with involution. For each height $n$, we compute the entire homotopy fixed point spectral sequence for $E_n$ with its $C_2$-action given by the formal inverse. We study, as the height varies, the Hurewicz images of the stable homotopy groups of spheres in the homotopy of these $C_2$-fixed points. We completely compute the slice spectral sequence of the $C_4$-spectrum $BP^{(\!(C_4)\!)}\langle 2 \rangle$. After periodization and $K(4)$-localization, this spectrum is equivalent to a height-4 Lubin--Tate theory $E_4$ with $C_4$-action induced from the Goerss--Hopkins--Miller theorem. In particular, our computation shows that $E_4^{hC_{12}}$ is 384-periodic.