Tate-valued Characteristic Classes II: Applications
Shachar Carmeli, Kiran Luecke
TL;DR
This work builds a general sharp construction to realize $\mathrm{MU}$-based $\mathbb{E}_\infty$ maps into Tate fixed-point objects, yielding explicit Frobenius formulas and cyclotomic lifts. It develops an obstruction theory for $\mathbb{E}_n$ complex orientations, obtaining concrete nonexistence results at small primes and clarifying the rigidity of $\mathrm{MU}$ as a cyclotomic base. The results unify representations, Todd-type coordinates, and Tate theories to constrain possible orientations, and extend to applications such as height restrictions, Steenrod-type operations on $\mathrm{MU}$, and computational tools. The framework provides a computational pathway to analyze orientation problems across chromatic heights and shows that Frobenius lifts and cyclotomic structures impose strong structural constraints on $\mathrm{MU}$-based oriented rings.
Abstract
We present a construction that manufactures $\E_\infty$ 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 $\MU$ as well as certain lifts of Frobenius. We prove a rigidity property of $\MU$ as a \emph{cyclotomic} object. We construct a general obstruction theory for $\E_n$ complex orientations and establish various non-existence results for $p$-typical $\E_n$ orientations for low values of $p$ and $n$. We end with some miscellaneous further applications.