Kirby Problem 4.129

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 PSC four-orbifold with isolated singularities is locally modeled at its singular points on $D^4/\Gamma$, where $\Gamma\leq O(4)$ is finite and acts freely on $S^3$, and has a Riemannian orbifold metric of positive scalar curvature. Its regular part is the complement of the singular points. In this isolated-singularity setting its orbifold fundamental group can be computed from the regular part, as in the source.

An ordinary $j$-surgery for $j=0,1$ replaces a framed $S^j\times\operatorname{int}D^{4-j}$ by $D^{j+1}\times S^{3-j}$. The allowed generalized zero-surgery also removes neighborhoods $D^4/\Gamma$ of two singular points of the same type and joins the resulting boundaries by $(S^3/\Gamma)\times[0,1]$.

Statement recorded in the supplied source.

For every closed smooth PSC four-manifold $M$, does there exist a possibly disconnected closed four-orbifold $M'$ with only isolated singularities such that:

1. $M'$ admits a PSC orbifold metric;
2. every connected component of $M'$ has finite orbifold fundamental group;
3. a finite sequence of ordinary zero- and one-surgeries and the generalized zero-surgeries just described transforms $M'$ into $M$?

[S2]

Short English statement

Can every closed positive-scalar-curvature four-manifold be assembled by zero- and one-surgeries from positive-scalar-curvature orbifold pieces with finite fundamental groups?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3005 (KP-4.129). 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 4.129, author preliminary version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.