Nilpotency, almost nonnegative curvature and the gradient flow
  We show that almost nonnegatively curved m-dimensional manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have C(m)-nilpotent fundamental groups. We also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanifolds. 
Vitali Kapovitch, Anton Petrunin and Wilderich Tuschmann