Topological singularities in the codimension-two stratum of noncollapsed Ricci limits

We construct classical noncollapsed Ricci limits with a topologically singular point whose unique tangent cone has a topologically Euclidean vertex. Fix an integer n≥4 and a closed connected oriented smooth manifold Q^n such that, for some smoothly embedded closed n-disc D^n⊂Q, Q∖int D^n carries a positive-Ricci metric with unit-round strictly convex boundary. For all sufficiently small q>0, there is a complete pointed (n+1)-dimensional Gromov–Hausdorff limit (Y, d, p), realized by complete smooth boundaryless (n+1)-manifolds with nonnegative Ricci curvature and uniformly positive based unit-ball volumes, with unique tangent cone ℝ^(n−1)×Cone(S¹_(2πq)) at p, where Cone(S¹_(2πq)) is the cone over the circle of length 2πq. A neighbourhood of p is homeomorphic to the open topological cone over Q#(−Q), where −Q has opposite orientation, and H_j(Y, Y∖{p};A)≅H̃_(j−1)(Q#(−Q);A) for every abelian group A and j≥0. If Q#(−Q) is not an integral homology n-sphere, then p is topologically singular. At p, the density is q, and Y has empty Kapovitch–Mondino boundary. For each integer N≥5, there is an N-dimensional example where p is the unique nonmanifold point and every tangent cone at every point splits ℝ^(N−2). Thus the Cheeger–Colding codimension-four singular stratum is empty although Y is not a topological manifold, disproving the Cheeger–Colding conjecture that the interior of its complement is a topological manifold.