Stability of tori under lower sectional curvature

Let (Mi ^{n; g}i) ^GH→ (X; d _X) be a Gromov–Hausdorff converging sequence of Riemannian manifolds with Sec _gi > 1, diam(M _i) < D, and such that the Mi ⁿ 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 Mi ⁿ are homeomorphic to tori, but are only assumed to be Alexandrov spaces. When n = 3, we prove the same toric stability under the weaker condition Ric _gi > 2.