Stability of Tori under Lower Sectional Curvature

Let $(M_i^n,g_i)\to (X,d_X)$ be a Gromov-Hausdorff converging sequence of Riemannian manifolds with ${\mathrm{S}\mathrm{e}\mathrm{c}}_{g_i}\ge -1$, $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(M_i)\le D$, and such that the $M_i^n$ are all homeomorphic to tori $T^n$. Then $X$ is homeomorphic to a $k$-dimensional torus $T^k$ for some $0\le k\le n$. This answers a question of Petrunin in the affirmative. We show this result is false is the $M_i^n$ are homeomorphic tori which are only assumed to be Alexandrov spaces. When $n=3$, we prove the same tori stability under the weaker condition ${\mathrm{R}\mathrm{i}\mathrm{c}}_{g_i}\ge -2$.