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 This proves an unpublished claim of Hopkins and Mahowald.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.

18 pages; 4 figures; some fixes; comments very welcome