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

TL;DR

This paper proves topological regularity and stability properties of noncollapsed Ricci limit spaces, confirming conjectures in specific dimensions and introducing new recognition and rigidity results that advance understanding of Ricci curvature limits.

Contribution

It confirms a conjecture about tangent cone cross-sections in four dimensions and extends similar results to higher dimensions under symmetry assumptions, also establishing manifold structure in certain cases.

Findings

Cross-sections at points in 4D are homeomorphic to spherical space forms.

$(n-3)$-symmetric Ricci limits are topological manifolds.

New recognition theorem for noncollapsed RCD spaces and cone rigidity results.

Abstract

We investigate the topological regularity and stability of noncollapsed Ricci limit spaces $(M_i^n,g_i,p_i)\to (X^n,d)$. 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 $\mathbb{R}^{n-3}\times C(\mathbb{RP}^2)$.