Let X be a compact Gromove-Hausdorff limit space of a sequence of n-manifolds, Mi, of sectional curvature secMi ≥ −1, or Ricci curvature RicMi ≥ −(n− 1) and all points in Mi satisfy a local rewinding Reifenberg condition, respectively. We will report a recent work: if Mi is homeomorphic, or diffeomorphic to a nil-manifold, respectively, then X is homeomorphic to, or diffeomorphic to a nil-manifold, respec-tively. As an application, we generalize a recent result by Bur´e-Naber-Semola that a compact GH-limit of a sequence of Mi homeomorphic to a torus of secMi ≥ −1 is a homeomorphic to torus.