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.
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Alexander Lytchak, Koichi Nagano, Topological regularity of spaces with an upper curvature bound. J. Eur. Math. Soc. 24 (2022), no. 1, pp. 137–165DOI 10.4171/JEMS/1091