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

We investigate the topological regularity and stability of noncollapsed Ricci limit spaces (Mᵢⁿ, gᵢ, pᵢ) (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/ₓ, and ₓ 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: (i) A new manifold recognition theorem for noncollapsed RCD (-2, 3) spaces. (ii) A cone rigidity result ruling out noncollapsed Ricci limit spaces of the form R^n-3 C (RP²).