*Preprint*

**Inserted:** 8 may 2024

**Last Updated:** 8 may 2024

**Year:** 2024

**Abstract:**

We investigate the topological regularity and stability of noncollapsed Ricci limit spaces $(M^n_i,g_i,p_i)→(X^n,d)$. 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 $x\in X^4$ are all homeomorphic to a fixed spherical space form $S^3/\Gamma_x$, and $\Gamma_x$ 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 $\rm{RCD}(−2,3)$ spaces. (ii) A cone rigidity result ruling out noncollapsed Ricci limit spaces of the form $\mathbb{R}^{n-3}\times C(\mathbb{RP}^2)$.

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