Complex Orientations are Partial Strictifications of the Unit
Doron Grossman-Naples
TL;DR
The paper develops a higher-algebraic interpretation of complex orientations of ring spectra as $\mathbb{E}_2$-strictifications of the unit, leveraging the fact that $\mathbb{CP}^{\infty}$ is the free strict abelian group on $S^2$ to model the orientation data algebraically. It proves that complex orientations correspond to $\mathbb{E}_2$-lifts of $1\in\pi_0 R$ (Theorem A) and shows that higher $\mathbb{E}_n$-orientations ($n>2$) are essentially impossible in chromatic contexts, forcing $L_{K(n)}R\simeq0$ for all finite heights $n$; this is established via Ravenel–Wilson calculations of $K(n)_*K(\mathbb{Z},3)$. The work clarifies why even-periodicity plays a central role in chromatic homotopy theory and discusses potential generalizations to other groups and obstruction theories for lifting orientations. Overall, the results delineate the limits of higher strictifications in encoding orientation data and connect orientation theory to the structure of formal groups and moduli in higher algebra. The approach provides a conceptual bridge between complex orientation theory and $\infty$-categorical algebraic structures, with implications for how orientation data interacts with chromatic localization.
Abstract
We give a higher-algebraic interpretation of complex orientations of ring spectra as "$\mathbb{E}_2$ strictifications" of the identity element. We show that higher strictifications do not exist for most ring spectra of interest in chromatic homotopy theory.