The second twisted Betti number and the convergence of collapsing Riemannian manifolds |
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Authors: | Fuquan Fang Xiaochun Rong |
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Institution: | (1) Nankai Institute of Mathematics, Nankai University, Tianjing 300071, PRC (e-mail: ffang@nankai.edu.cn), CN;(2) Mathematics Department, Beijing Normal University, Beijing, PRC, CN;(3) Mathematics Department, Rutgers University, New Brunswick, NJ 08903, USA (e-mail: rong@math.rutgers.edu), US |
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Abstract: | Let M
i
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.
Oblatum 17-I-2002 & 27-II-2002?Published online: 29 April 2002 |
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Keywords: | |
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