Almost half-maximal diameter and threshold realizability

Definitions and mathematical statement

Fix an integer $n\geq2$. For a closed connected Riemannian $n$-manifold $(M,g)$, write $\sec_g\geq1$ when all tangent two-planes have sectional curvature at least one, and let $\operatorname{diam}_g(M)$ be the supremum of geodesic distances between pairs of points. A homeomorphism means a continuous bijection with continuous inverse; no differentiability requirement is imposed on it.

Does there exist a number $\varepsilon(n)>0$ such that every $(M,g)$ satisfying \[ \sec_g\geq1,\qquad \operatorname{diam}_g(M)\geq\frac\pi2-\varepsilon(n) \] is homeomorphic to some closed smooth $n$-manifold $M'$ admitting a Riemannian metric $g'$ with \[ \sec_{g'}\geq1,\qquad \operatorname{diam}_{g'}(M')\geq\frac\pi2? \] The replacement manifold and metric may depend on $M,g$, while the allowed diameter loss depends only on dimension. The source also asks for the weaker version with a prescribed positive volume lower bound: for every $V>0$, may one choose $\varepsilon(n,V)>0$ when additionally $\operatorname{Vol}_g(M)\geq V$? The volume condition is imposed on the original metric; the question does not demand that the replacement retain that volume. Neither formulation requires a small metric deformation or a bound on the distance between the two metrics. The target is a topological stability statement near the diameter threshold, permitting a new realization of the same homeomorphism type exactly at or above that threshold.

Short English statement

In each fixed dimension, are manifolds just below the diameter threshold $\pi/2$ topologically realizable at the threshold, either in general or with an additional volume bound?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20003128 (AIM-TOPOLOGY-0216), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Manifolds with nonnegative sectional curvature, item 3. Original workshop question underlying AIM-TOPOLOGY-0216. https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf.