Kirby Problem 3.15

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 $M$ be a closed oriented hyperbolic $3$-manifold, with first Betti number $b_1(M)=\dim_{\mathbb R}H^1(M;\mathbb R)>0$. Its hyperbolic metric has sectional curvature $-1$. For an oriented real rank-two bundle $E\to M$, write $e(E)\in H^2(M;\mathbb Z)$ for its Euler class and $e(E)_{\mathbb R}$ for its image in real cohomology. A cooriented codimension-one foliation $\mathcal F$ is taut if every leaf meets a closed transverse curve; its Euler class is $e(T\mathcal F)$. A positive cooriented contact structure is a plane bundle $\xi=\ker\lambda$ with $\lambda\wedge d\lambda>0$, oriented by $d\lambda|_\xi$. It is tight if it has no overtwisted disk, and universally tight if its lift to the universal cover is tight. For an oriented nonsingular flow generated by $X$, use the Euler class of its oriented normal bundle $TM/\mathbb RX$. A pseudo-Anosov flow has transverse invariant stable and unstable two-dimensional foliations, with finitely many possible pronged singular periodic orbits, and contraction/expansion away from them. A quasigeodesic flow has constants $K\ge1,C\ge0$ such that every lifted orbit $\gamma$ in hyperbolic space, parametrized by arclength, satisfies \[ K^{-1}|s-t|-C\le d(\gamma(s),\gamma(t))\le K|s-t|+C. \] For $\rho:\pi_1(M)\to\operatorname{Homeo}^+(S^1)$, form the oriented flat circle bundle $E_\rho=(\widetilde M\times S^1)/\pi_1(M)$ using the diagonal action; its Euler class is $e(E_\rho)$.

Statement recorded in the supplied source.

For each such $M$, determine the following six subsets of $H^2(M;\mathbb R)$: \[ \begin{aligned} \mathcal E_{\rm taut}&=\{e(T\mathcal F)_{\mathbb R}:\mathcal F\text{ taut}\},\\ \mathcal E_{\rm tight}&=\{e(\xi)_{\mathbb R}:\xi\text{ tight}\},\\ \mathcal E_{\rm universal}&=\{e(\xi)_{\mathbb R}:\xi\text{ universally tight}\},\\ \mathcal E_{\rm pA}&=\{e(TM/\mathbb RX)_{\mathbb R}:X\text{ generates a pseudo-Anosov flow}\},\\ \mathcal E_{\rm qg}&=\{e(TM/\mathbb RX)_{\mathbb R}:X\text{ generates a quasigeodesic flow}\},\\ \mathcal E_{S^1}&=\{e(E_\rho)_{\mathbb R}:\rho:\pi_1(M)\to\operatorname{Homeo}^+(S^1)\}. \end{aligned} \] The question is to characterize all realizable classes, not only classes on the boundary of the dual Thurston norm ball. [S2]

Short English statement

Which real cohomology classes occur as Euler classes of taut foliations, tight or universally tight contact structures, pseudo-Anosov or quasigeodesic flows, and oriented circle actions on a closed hyperbolic three-manifold with positive first Betti number?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2813 (KP-3.15). 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 3.15, p. 141 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.