Discreteness of points with non-spherical tangent cross-sections in noncollapsed Ricci limit spaces

Determine whether, for any noncollapsed Ricci limit space (X,d), the subset of points x ∈ X at which some tangent cone has a cross-section not homeomorphic to the 3-sphere S^3 is a discrete subset of X.

Background

The authors prove in Theorem 1.1 that, in dimension 4, the set of points where the cross-sections of tangent cones are not homeomorphic to S^3 has Hausdorff dimension 0. They conjecture a stronger property—discreteness—which would sharpen the zero-dimensional conclusion and align with the behavior seen in settings with two-sided Ricci curvature bounds or Kähler structures.

Establishing discreteness would refine the topological stratification of singular sets in noncollapsed Ricci limit spaces and complement known rectifiability and stability results.