The Ando-Hopkins-Rezk orientation is surjective

Abstract

We show that the map $\pi_\ast \mathrm{MString}\to \pi_\ast \mathrm{tmf}$ induced by the Ando-Hopkins-Rezk orientation is surjective. This proves an unpublished claim of Hopkins and Mahowald. We do so by constructing an $\mathbf{E}_1$-ring $B$ and a map $B \to \mathrm{MString}$ such that the composite $B \to \mathrm{MString} \to \mathrm{tmf}$ is surjective on homotopy. Applications to differential topology, and in particular to Hirzebruch's prize question, are discussed.