Topological regularity of spaces with an upper curvature bound

Abstract

We prove that a locally compact space with an upper curvature bound is a topological manifold if and only if all of its spaces of directions are homotopy equivalent and not contractible. We discuss applications to homology manifolds, limits of Riemannian manifolds and deduce a sphere theorem.

References in corpus (1)

• Perelman's Stability Theorem

Cited by in corpus (3)

• Structure of Submetries
• Volume pinching theorems for CAT(1) spaces
• A lower bound for the curvature integral under an upper curvature bound