Obstructions to nonnegative curvature and rational homotopy theory

We establish a link between rational homotopy theory and the problem which vector bundles admit complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively curved manifolds that, if C lies in the class and T is a torus of positive dimension, then "most" vector bundles over the product of C and T admit no complete nonnegatively curved metric.