TOPOLOGICAL SINGULARITIES IN THE CODIMENSION-TWO STRATUM OF NONCOLLAPSED RICCI LIMITS
Abstract
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 Qn such that, for some smoothly embedded closed n-disc Dn⊂ Q, Q∖intDn 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 Rn−1×Cone(S12π q) at p, where Cone(S12π q) is the cone over the circle of length 2π q. A neighbourhood of p is homeomorphic to the open topological cone over Q#(−Q), where −Q has opposite orientation, and Hj(Y,Y∖{p};A)≅H̃j−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 RN−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.
Related
- Codimension-three regularity of noncollapsed Ricci limit spaces under an integral volume-deficit bound
- Codimension-four regularity of noncollapsed Ricci limit spaces under an integral volume-deficit bound
- Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below
- Homological obstructions to local contractibility in noncollapsed Ricci limits
- Fundamental-group epimorphisms in noncollapsed Ricci limits
BibTeX
@misc{anass2026topological,
title = {{TOPOLOGICAL SINGULARITIES IN THE CODIMENSION-TWO STRATUM OF NONCOLLAPSED RICCI LIMITS}},
author = {Nifa Anass},
howpublished = {HAL},
year = {2026},
month = jul,
doi = {10.5281/zenodo.22670411},
url = {https://hal.science/hal-05746087v2},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - TOPOLOGICAL SINGULARITIES IN THE CODIMENSION-TWO STRATUM OF NONCOLLAPSED RICCI LIMITS
AU - Nifa Anass
PY - 2026
DA - 2026/07/15
PB - HAL
DO - 10.5281/zenodo.22670411
UR - https://hal.science/hal-05746087v2
AB - 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∖intD^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(S^1_2π q) at p, where Cone(S^1_2π q) is the cone over the circle of length 2π q. 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.
ER -