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 $n\ge4$ and a closed connected oriented smooth manifold $Q^n$ such that, for some smoothly embedded closed $n$-disc $D^n\subset Q$, $Q\setminus\operatorname{int}D^n$ 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 $\R^{n-1}\times\Cone(S^1_{2\pi q})$ at $p$, where $\Cone(S^1_{2\pi q})$ is the cone over the circle of length $2\pi q$. A neighbourhood of $p$ is homeomorphic to the open topological cone over $Q\#(-Q)$, where $-Q$ has opposite orientation, and $H_j(Y, Y\setminus\{p\};A)\cong\widetilde H_{j-1}(Q\#(-Q);A)$ for every abelian group $A$ and $j\ge0$. 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 $N\ge5$, 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.