Pinching-gap reduction and dimension-six finiteness

Definitions and mathematical statement

For a fixed integer $n\geq2$, let $\mathcal C_n$ be the class of closed connected smooth $n$-manifolds $M$ for which \[ \pi_1(M)=0,\qquad \pi_2(M)=0, \] and for which some smooth Riemannian metric has strictly positive sectional curvature. The homotopy groups are based at any point; for a connected simply connected manifold the stated vanishing is independent of that choice. These hypotheses are often expressed by saying that $M$ is two-connected.

Is $\mathcal C_n$ finite up to diffeomorphism? If smooth finiteness is too strong, does finiteness at least hold up to homeomorphism? The source asks for a finiteness result without specifying the equivalence relation, so these distinguish the natural smooth and topological versions. The number of types may depend on $n$, but no curvature pinching, diameter, injectivity-radius, or volume bound is imposed uniformly across the class. Each manifold merely needs to admit one positively curved metric. Two different metrics on the same underlying manifold do not count as different types. Conversely, a counterexample to the smooth formulation requires infinitely many pairwise nondiffeomorphic manifolds of one fixed dimension satisfying all three conditions. An infinite family spread across increasing dimensions does not contradict the assertion. The question concerns the topology of the admitted manifolds, not compactness or the number of components of their spaces of positive-curvature metrics.

Short English statement

Are there only finitely many smooth, or at least topological, types of two-connected positively curved closed manifolds in each dimension?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20003179 (AIM-TOPOLOGY-0267), 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 54. Original workshop question underlying AIM-TOPOLOGY-0267. https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf.