Topological regularity of spaces with an upper curvature bound

Asymptotic topological regularity of CAT(0) spaces

“…(2) If we have G n 0 (X) < 3/2, then a volume sphere theorem of Lytchak-Nagano [30,Theorem 8.3] for CAT(1) spaces implies that ∂ T X is homeomorphic to S n−1 .…”

“…From the properties (1)-( 4) listed above, we can derive Theorems 1.1-1.4. In the proofs of Theorems 1.2, 1.3, and 1.4, when we prove that X is a topological n-manifold, we use the local topological regularity theorem of Lytchak-Nagano [30,Theorem 1.1]. In the proof of Theorem 1.3, in order to determine the geometric structure of X, we describe a volume regularity condition for CAT (1) spaces to be almost isometric to a compact spherical building (Proposition 6.4).…”

“…Proposition 2.5. ( [30]) Let X be a proper, geodesically complete, geodesic CAT(κ) space. Let W be a connected open subset of X.…”

“…We say that a triple of points in a CAT(1) space is a tripod if the three points have pairwise distance at least π. Lytchak-Nagano [30] proved a capacity sphere theorem for CAT(1) spaces stating that if a compact, geodesically complete CAT(1) space admits no tripod, then it is homeomorphic to a sphere ([30, Theorem 1.5]). As its application, [30] showed the following volume sphere theorem for CAT(1) spaces ([30, Theorem 8.3], and [33] for the case of m = 2]):…”

“…Thurston [40,Theorem 3.3] showed that every homology 3-manifold with an upper curvature bound is a topological 3-manifold. Lytchak-Nagano [30,Theorem 1.2] proved that for every homology n-manifold M with an upper curvature bound there exists a locally finite subset E of M such that M − E is a topological n-manifold.…”