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Mondino’s orbifold conjecture for noncollapsed RCD(−2,3) spaces

Establish that any noncollapsed RCD(−2,3) metric measure space is homeomorphic to an orbifold, possibly with boundary; equivalently, show that the local topology is determined by the topology of tangent cones.

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Background

The authors note Mondino's conjecture predicting that local topology of noncollapsed RCD(−2,3) spaces should be encoded by their tangent cones, yielding an orbifold structure.

Their manifold recognition theorem proves the conjecture in the special case where all tangent cones are Euclidean; the general case remains open and would significantly advance the understanding of RCD spaces.

References

A conjecture due to Mondino predicts that a noncollapsed RCD(−2,3) space should be homeomorphic to an orbifold, possibly with a boundary. The conjecture might be rephrased by saying that the local topology should be determined by the topology of tangent cones.

— Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below (2405.03839 - Bruè et al., 6 May 2024) in Section 1.3 (The topology of three-dimensional RCD spaces)