Kirby Problem 4.20

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 a closed topological four-manifold $M$, write $\operatorname{ks}(M)\in\mathbb Z/2\mathbb Z$ for the Kirby--Siebenmann obstruction evaluated on its mod-two fundamental class; it obstructs a compatible piecewise-linear structure and therefore a smooth structure. Smoothable means homeomorphic to a smooth manifold.

Let $\star\mathbb{RP}^4$ denote the unique homeomorphism class of topological manifolds homotopy equivalent to $\mathbb{RP}^4$ with $\operatorname{ks}=1$. This is Ruberman's nonsmoothable model. Its identification is given in K3, Problem 4.87, Remark (1), and Problem 4.88, Remark (2), PDF page 261. The star symbol here names that model; it is not a claim that every star construction satisfies a connected-sum cancellation identity.

Let $\mathrm{En}$ be the complex Enriques surface, obtained as the quotient of a K3 surface by a free holomorphic involution. A K3 surface is a simply connected compact complex surface with trivial canonical bundle. Let $\star\mathrm{En}$ be its oriented topological star partner: it has the oriented homotopy type of $\mathrm{En}$ and opposite Kirby--Siebenmann invariant, hence $\operatorname{ks}=1$. Existence of this named partner is part of the original K3 statement; uniqueness within that oriented homotopy type follows from Teichner's classification theorem for fundamental group $\mathbb Z/2\mathbb Z$. [S3]

Statement recorded in the supplied source.

Are the topological connected sums \[ (\star\mathbb{RP}^4)\#(\star\mathbb{RP}^4),\qquad (\star\mathrm{En})\#(\star\mathrm{En}) \] smoothable? The source also records the same question for the mixed sum \[ (\star\mathbb{RP}^4)\#(\star\mathrm{En}). \] Each sum has zero Kirby--Siebenmann invariant by additivity, but the requested conclusion is smoothability without adding $S^2\times S^2$ factors. [S2]

Short English statement

Are the doubles of the nonsmooth projective-space model and the Enriques star partner smoothable, and is their mixed connected sum smoothable?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2896 (KP-4.20). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3—A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), author version, Problem 4.20 and Remarks, PDF page 205; also Problems 4.87--4.88, PDF page 261. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 Peter Teichner, On the star-construction for topological 4-manifolds, Introduction and Theorem 1, pages 1--2. https://math.berkeley.edu/~teichner/Papers/star.pdf.