Kirby Problem 4.7

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.

Let $M_1,M_2$ be smooth closed oriented $4$-manifolds that are orientation-preservingly homeomorphic. For $k\ge0$, let $M_i\#_k(S^2\times S^2)$ mean stabilization by the connected sum with $k$ standard copies of the product; for $k=0$ it is $M_i$. Equivalence of stabilized manifolds means oriented diffeomorphism. The manifolds need not be simply connected. The related twisted stabilization uses the nontrivial oriented $S^2$ bundle over $S^2$, diffeomorphic to $\mathbb{CP}^2\#\overline{\mathbb{CP}}^{\,2}$.

Statement recorded in the supplied source.

(a) Is $M_1\#(S^2\times S^2)$ always diffeomorphic to $M_2\#(S^2\times S^2)$? (b) Is there a single integer $n>0$, independent of the pair, for which \[ M_1\#_n(S^2\times S^2)\cong M_2\#_n(S^2\times S^2) \] for every homeomorphic closed oriented pair $M_1,M_2$? The source also asks the same one-stabilization and universal-bound questions with the twisted $S^2$ bundle in place of $S^2\times S^2$. The bound in (b) must be uniform across all pairs; the known existence of a pair-dependent stabilization number is weaker. [S2]

Short English statement

Does one product stabilization make every homeomorphic closed oriented four-manifold pair diffeomorphic, or is there at least a universal finite number of stabilizations? Ask the corresponding questions for twisted-sphere-bundle stabilization as well.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2883 (KP-4.7). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (editors), K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), Problem 4.7, p. 196 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.