Kirby Problem 4.127

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 connected manifold $M$ is aspherical if its universal cover is contractible. For a smooth self-diffeomorphism $f:M\to M$, its mapping torus is \[ M_f=(M\times[0,1])/((x,1)\sim(f(x),0)). \] Here $M$ has dimension four. The fiber is either closed, or compact with boundary components that admit flat three-dimensional metrics. In the closed case a hyperbolic structure is a metric of constant sectional curvature $-1$ on $M_f$. In the boundary case the hyperbolic structure is interpreted on the interior of $M_f$, with complete finite-volume cusp ends and the boundary regarded as cusp cross sections.

Statement recorded in the supplied source.

Give necessary and sufficient conditions on $f$ for $M_f$ to admit the indicated hyperbolic structure, for every aspherical four-manifold $M$ in either of these two classes. The criterion must characterize the possible monodromies, not merely construct isolated fibering hyperbolic five-manifolds. [S2]

Short English statement

Characterize the diffeomorphisms of aspherical four-manifolds whose five-dimensional mapping tori are hyperbolic, including the finite-volume cusp case.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3003 (KP-4.127). 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 4.127, author preliminary version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.