Kirby Problem 3.13

Definitions and mathematical statement

Let $M$ be a closed connected hyperbolic three-manifold. An immersion $f:\Sigma\to M$ of a smooth surface is a smooth map whose differential is injective at every point. Thus it is locally an embedded surface, though different points of $\Sigma$ may have the same image. It is $\pi_1$-injective when the induced homomorphism of fundamental groups is injective.

Does there exist, for every such $M$, a closed connected surface $\Sigma\ne S^2$ and a $\pi_1$-injective immersion satisfying \[ |f^{-1}(x)|\leq2\qquad\text{for every }x\in M? \] This is the precise multiplicity condition meant by "only double points." The source's statement does not separately demand that all double intersections be transverse, so the cardinality bound is the defining requirement here. Embedded surfaces qualify, since their fibers have cardinality at most one.

The exclusion of the sphere prevents a trivial local construction. No genus bound is prescribed, and orientability of $\Sigma$ is not added to the source's assumptions. Producing an immersed essential surface with triple points does not meet the conclusion; those higher multiplicities must be absent everywhere in the proposed immersion.

Short English statement

Must every closed hyperbolic three-manifold contain an immersed essential closed surface that never passes through any point more than twice?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2811 (KP-3.13), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 3.13. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.