Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below

Source abstract

We investigate the topological regularity and stability of noncollapsed Ricci limit spaces (Min,gi,pi) →GH(Xn,d). We confirm a conjecture proposed by Colding and Naber in dimension n=4, showing that the cross-sections of tangent cones at a given point x∈X4 are all homeomorphic to a fixed spherical space form S3∕Γx, and Γx is trivial away from a 0-dimensional set. In dimensions n>4, we show an analogous statement at points where all tangent cones are (n−4)-symmetric. Furthermore, we prove that (n−3)-symmetric noncollapsed Ricci limits are topological manifolds, thus confirming a particular case of a conjecture due to Cheeger, Colding, and Tian. Our analysis relies on two key results, whose importance goes beyond their applications in the study of cross-sections of noncollapsed Ricci limit spaces: A new manifold recognition theorem for noncollapsed RCD(−2,3) spaces. A cone rigidity result ruling out noncollapsed Ricci limit spaces of the form Rn−3×C(RP2).