Stability of tori under lower sectional curvature

X; d X / be a Gromov-Hausdorff converging sequence of Riemannian manifolds with Sec g i 1, diam.M i / D, and such that the M n i are all homeomorphic to tori T n .Then X is homeomorphic to a k-dimensional torus T k for some 0 k n.This answers a question of Petrunin in the affirmative.We show this result is false if the M n i are homeomorphic to tori, but are only assumed to be Alexandrov spaces.When n D 3, we prove the same toric stability under the weaker condition Ric g i 2.