Poincaré‑free proof of topological rigidity for RCD(0,3) manifolds

Develop a proof of Theorem 1.9—that any noncollapsed RCD(0,3) topological manifold with Euclidean volume growth is homeomorphic to R^3—that does not rely on Perelman’s solution of the Poincaré conjecture.

Background

The current proof of Theorem 1.9 uses the Poincaré conjecture to conclude that contractible 3‑manifolds with the stated geometric properties are homeomorphic to R3. A Poincaré‑free argument would provide an intrinsic geometric approach within the RCD framework.

Finding such a proof could illuminate new techniques for topological rigidity in synthetic curvature settings and potentially extend to higher‑dimensional or more general contexts.

References

Question 1.18. Is there a proof of Theorem 1.9 that does not rely on the solution to the Poincaré conjecture?

— Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below (2405.03839 - Bruè et al., 6 May 2024) in Section 1.6 (Question 1.18)