Kirby Problem 4.121

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.

An open book on a closed smooth four-manifold $X$ consists of an embedded closed surface $L\subset X$ with trivial normal bundle and a smooth locally trivial fibration $f:X\setminus L\to S^1$. In a tubular neighborhood $L\times D^2$, it has the polar-coordinate form $f(x,re^{i\theta})=e^{i\theta}$ for $r>0$. The closures of fibers are the pages and $L$ is the binding. For an oriented $X$, $\sigma(X)$ is the signature of the intersection form on $H_2(X;\mathbb R)$.

Statement recorded in the supplied source.

Characterize the closed smooth four-manifolds admitting an open book. In particular, if $X$ is closed, simply connected and smooth with $\sigma(X)=0$, must $X$ admit an open book? [S2]

Short English statement

Which closed smooth four-manifolds have open books, and is zero signature sufficient when the manifold is simply connected?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2997 (KP-4.121). 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 4.121, author preliminary version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.