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Codimension-four manifold regularity conjecture for noncollapsed Ricci limit spaces

Establish that noncollapsed Ricci limit spaces are homeomorphic to manifolds away from a closed subset of Hausdorff codimension at least 4.

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Background

The codimension-four conjecture asserts manifold structure off a set of codimension ≥4 in the presence of lower Ricci bounds; it is proven under two-sided Ricci bounds (Cheeger–Naber) but remains open in the lower-bound-only setting.

The authors’ results provide partial progress and dimensional sharpness in several cases, motivating a resolution of the general conjecture with lower Ricci bounds.

References

Conjecturally, noncollapsed Ricci limit spaces might be homeomorphic to manifolds away from a closed subset of Hausdorff codimension at least 4: see [31, Conjecture 0.7], [28, Remark 10.23] and [33, Remark 1.19].

— Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below (2405.03839 - Bruè et al., 6 May 2024) in Section 1 (Introduction)