$E_n$ genera

Abstract

Let $R$ be an $E_2$ ring spectrum with zero odd-dimensional homotopy groups. Every map of ring spectra $MU\to R$ is represented by a map of $E_2$ ring spectra. If $2$ is invertible in ${\pi }_{0}R$, then every map of ring spectra $MSO\to R$ is represented by a map of $E_2$ ring spectra.