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

Abstract

Let Mi\(\overset{d_{GH}}{\longrightarrow}\)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 Mi are torsion groups of uniformly bounded exponents and the second twisted Betti numbers of Mi vanish, then there is a manifold, M, and a sequence of diffeomorphisms from M to a subsequence of {Mi} 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.