Obstruction theory and the level $n$ elliptic genus

Abstract

Given a height $\le 2$ Landweber exact ${\mathbb{E}}_{\infty }$-ring $E$ whose homotopy is concentrated in even degrees, we show that any complex orientation of $E$ which satisfies the Ando criterion admits a unique lift to an ${\mathbb{E}}_{\infty }$-complex orientation $\mathrm{MU}\to E$. As a consequence, we give a short proof that the level $n$ elliptic genus lifts uniquely to an ${\mathbb{E}}_{\infty }$-complex orientation $\mathrm{MU}\to {\mathrm{tmf}}_1(n)$ for all $n\ge 2$.

19 pages; comments welcome!