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/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: (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 Rⁿ⁻³× C(RP²).