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Topological regularity theorems for Alexandrov spaces. (English) Zbl 0915.53021

Usually, Gromov-Hausdorff limits of Riemannian manifolds with curvature bounded uniformly from below are almost never Riemannian manifolds, but they always are Alexandrov spaces, i.e., finite dimensional inner metric spaces with a lower curvature bound in the sense of distance comparison [see, for example, K. Grove and P. Petersen [J. Differ. Geom. 33, 379-394 (1991; Zbl 0729.53045)] for more details).
The main purpose of the paper under review is to give some topological characterization of topological manifolds without boundary for Alexandrov spaces.
The main result is as follows:
Theorem 1.1. An \(n\)-dimensional Alexandrov space \(X\) is a topological manifold without boundary if and only if it satisfies the following two properties: (1) \(X\) is a homology manifold, that is, \(H_*(X,X-p)\cong H_*(\mathbb R^n, \mathbb R^n -\{0\})\) for any \(p\in X\), and (2) in the case that \(n\geq 4\), the space of directions \(\Sigma_p\) at \(p\) is simply connected for any \(p\in X\).
To prove Theorem 1.1, the author first proves that for an Alexandrov space \(X\) the following three statements are equivalent: (i) \(X\) is a homology manifold, (ii) \(X\times\mathbb R^k\) is a topological manifold without boundary for some \(k\geq 1\), and (iii) \(X\times\mathbb R\) is a topological manifold without boundary.
Some consequences of Theorem 1.1 are given. Special emphasis is given to the structure of compact nonnegatively curved Alexandrov spaces.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)

Citations:

Zbl 0729.53045
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