Tate-valued Characteristic Classes II: Applications

We present a construction that manufactures ${\text{E}}_{\mathrm{\infty }}$ orientations of Tate fixed-point objects together with useful formulas for these maps, and then give a number of applications. For example, we produce a formula for the Frobenius homomorphisms of Thom spectra such as $\text{MU}$ as well as certain lifts of Frobenius. We prove a rigidity property of $\text{MU}$ as a \emph{cyclotomic} object. We construct a general obstruction theory for ${\text{E}}_n$ complex orientations and establish various non-existence results for $p$-typical ${\text{E}}_n$ orientations for low values of $p$ and $n$. We end with some miscellaneous further applications.