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

• 成果类型: Article
• 署名作者: Fang, FQ; Rong, XC
• 署名单位: Nankai University; Beijing Normal University; Rutgers University System; Rutgers University New Brunswick
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-002-0230-2
• 发表日期: 2002
• 页码: 61-109
• 关键词: curvature; diameter; FINITENESS; geometry; FAMILY

摘要

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 C-0-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.