CAT(0) spaces are metric spaces where triangles are thinner than Euclidean triangles — a generalization of nonpositive curvature. A natural question: if a CAT(0) space is quasi-isometric to Euclidean space R^n (meaning it looks like R^n at large scales), must it be homeomorphic to R^n (meaning it actually is R^n topologically)? Cavallucci and Sambusetti (arXiv: 2603.23768) give three answers, and each draws a different boundary.
First boundary: yes, if the space is a homology manifold. A proper, geodesically complete CAT(0) homology manifold quasi-isometric to R^n must be homeomorphic to R^n. Large-scale geometry determines topology when the local structure is sufficiently manifold-like.
Second boundary: no, in general. They construct proper, geodesically complete CAT(0) spaces that are quasi-isometric to R^n but not homeomorphic to it. Without the homology manifold condition, the topological rigidity breaks. Spaces can look like R^n from far away while being something entirely different up close.
Third boundary: topological manifolds are not open in this class. They exhibit a sequence of non-manifold CAT(0) spaces converging (in the Gromov-Hausdorff sense) to a topological manifold. You can approach a manifold through non-manifolds without ever being one. The manifold property is not stable under geometric limits.
The through-claim: the question “does large-scale geometry determine topology?” has a boundary that can be precisely mapped. With enough local regularity (homology manifold): yes. Without it: no. And the transition is not smooth — you can't detect the manifold property from nearby approximations. Three results, each sharpening one edge of the same knife.
Cavallucci & Sambusetti, 2603.23768. Metric geometry / CAT(0) spaces / quasi-isometry / topological rigidity / Gromov-Hausdorff convergence.