Kirby Problem 4.3

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.

For an oriented smooth closed $4$-manifold $X$, an exotic smooth structure is a smooth manifold orientation-preservingly homeomorphic to $X$ but not diffeomorphic to its standard smooth structure. Write $\#_kX$ for the connected sum of $k$ copies, with $\#_0X=S^4$ when a zero summand occurs. Let $\mathbb{CP}^2$ have its complex orientation and $\overline{\mathbb{CP}}^{\,2}$ the opposite orientation. A standard K3 surface is a simply connected compact complex surface with trivial canonical bundle, with its usual complex-oriented smooth structure. Use the usual product smooth structure on $S^2\times S^2$.

Statement recorded in the supplied source.

Determine which of the following closed simply connected manifolds admit exotic smooth structures: \[ \begin{array}{ll} \text{(a)}&\#_k\mathbb{CP}^2,\quad k\ge1;\\[2pt] \text{(b)}&\#_m(\mathbb{CP}^2\#\overline{\mathbb{CP}}^{\,2}),\quad1\le m\le8, \quad\text{and }\#_n(S^2\times S^2),\quad1\le n\le10;\\[2pt] \text{(c)}&(\#_p\mathrm{K3})\#(\#_q(S^2\times S^2)),\quad q\ge0,\quad p\ge q+5. \end{array} \] If such structures exist, determine the smallest possible $k,m,n$ in the indicated families, and the largest possible $p,q$ in (c), or whether those parameters are unbounded. The standard connected-sum smooth structures are the structures against which exoticness is measured. [S2]

Short English statement

Do exotic smooth structures exist on positive-definite sums of complex projective planes, small zero-signature connected sums, or K3 sums in the specified spin range, and what are the smallest or largest parameters admitting them?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2879 (KP-4.3). 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 4.3, p. 193 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.