Kirby Problem 4.128

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A smooth four-manifold has positive scalar curvature (PSC) if it admits a Riemannian metric $g$ with $\operatorname{Scal}_g(x)>0$ at every point. Uniform positivity means $\inf_x\operatorname{Scal}_g(x)>0$; completeness means metric completeness. Write $\overline{\mathbb{CP}}^{2}$ for the complex projective plane with reversed orientation, and $\#$ for connected sum. An empty connected sum in the displayed closed simply connected classification is $S^4$.

Statement recorded in the supplied source.

Resolve the following four questions.

1. (a) Is every closed simply connected smooth PSC four-manifold diffeomorphic to \[\#^{a}\mathbb{CP}^2\#^{b}\overline{\mathbb{CP}}^{2}\#^{c}(S^2\times S^2)\] for some integers $a,b,c\geq0$?
2. (b) Characterize the closed smooth PSC four-manifolds with nontrivial fundamental group.
3. (c) Characterize the smooth four-manifolds with boundary admitting PSC metrics, and the boundary-condition variants described in the source: metrics that are products near the boundary, and metrics with positive outward mean curvature at the boundary.
4. (d) Characterize the noncompact smooth four-manifolds admitting complete metrics with uniformly positive scalar curvature.

[S2]

Short English statement

Classify positive-scalar-curvature four-manifolds in the simply connected, non-simply connected, boundary, and complete noncompact settings.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3004 (KP-4.128). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295, American Mathematical Society (2026), Problem 4.128, author preliminary version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.