Topological regularity of Busemann spaces of nonpositive curvature

Abstract

We extend the topological results of Lytchak-Nagano and Lytchak-Nagano-Stadler for CAT(0) spaces to the setting of Busemann spaces of nonpositive curvature, i.e., BNPC spaces. We give a characterization of locally BNPC topological manifolds in terms of their links and show that the singular set of a locally BNPC homology manifold is discrete. We also prove that any (globally) BNPC topological 4-manifold is homeomorphic to Euclidean space. Applications include a topological stability theorem for locally BNPC G-spaces. Our arguments also apply to spaces admitting convex geodesic bicombings.

added figures, references, and Remarks 6.11, 6.14, 7.3, 8.8; slightly revised the statement of Proposition 6.12 and the proof of Proposition 6.23; made other minor changes and corrections