Nonnegative curvature, elliptic genus and unbounded Pontryagin numbers

We discuss the cobordism type of spin manifolds with nonnegative sectional curvature. We show that in each dimension $4k\ge 12$, there are infinitely many cobordism types of simply connected and nonnegatively curved spin manifolds. Moreover, we raise and analyze a question about possible cobordism obstructions to nonnegative curvature.