Topological regularity of Busemann spaces of nonpositive curvature

TL;DR

This work extends topological results known for CAT(0) spaces to BNPC/GNPC spaces by developing strainers and extended strainers as the core tool. It proves local regularity: a locally BNPC space is a topological n-manifold iff punctured balls around each point have the homotopy type of S^{n-1}, and it shows that locally BNPC homology manifolds have discrete singular sets; it also establishes that globally BNPC topological 4-manifolds are homeomorphic to Euclidean space. The approach relies on constructing strainer maps with open, locally trivial-fibration behavior despite the GNPC setting’s angle asymmetry, and extends these results to G-spaces with a topological stability theorem. The paper also discusses generalizations to spaces with convex geodesic bicombings and to Busemann-curvature bounds, and outlines several open problems related to tangent cones, dimension theory, and Berwald-type regularity in GNPC settings.

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.

Figures (1)

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Theorems & Definitions (169)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 2.1
• Remark 2.2