The second twisted Betti number and the convergence of collapsing Riemannian manifolds

Let M_i →^dGH X denote a sequence of n-manifolds converging to a compact metric space, X, in the Gromov-Hausdorff topology such that the sectional curvature is bounded in absolute value and dim(X) < n. We prove the following stability result: If the fundamental groups of M_i are torsion groups of uniformly bounded exponents and the second twisted Betti numbers of M_i vanish, then there is a manifold, M, and a sequence of diffeomorphisms from M to a subsequence of {M_i} such that the distance functions of the pullback metrics converge to a pseudo-metric in C⁰-norm. Furthermore, M admits a foliation with leaves diffeomorphic to flat manifolds (not necessarily compact) such that a vector is tangent to a leaf if and only if its norm converges to zero with respect to the pullback metrics. These results lead to a few interesting applications.