Problem 2.14 — A 3-manifold is fibered if it admits a surface bundle structure.

Definitions and mathematical statement

In the source's usual $3$-manifold sense, $M$ is fibered if it is a surface bundle over $S^1$, and virtually fibered if some finite-sheeted cover of $M$ has such a bundle structure. The allowed surfaces, compactness, boundary and hyperbolic-cusp conventions are not specified in the extract.

For a Heegaard splitting $M=H_+\cup_\Sigma H_-$, the usual Hempel distance is the distance in the curve complex of $\Sigma$ between the two disk sets, whose vertices are essential boundary curves of compressing disks in the two sides. Surface complexity and compression-body conventions must be fixed when this definition needs modification.

Problem 2.14 deliberately asks a vague question: can the distance and special form of the Heegaard splitting associated with a surface bundle provide information about whether a nonfibered manifold might nevertheless be virtually fibered? It specifies no associated-splitting construction, numerical criterion, necessary/sufficient implication, algorithm or manifold class for such a test. Thus the task is a research program linking splitting geometry to virtual fibering, not a supplied inequality or exact recognition theorem.

The source places this in Thurston's question [49] whether every finite-volume hyperbolic $3$-manifold is virtually fibered. It calls that an outstanding open problem and reports Leininger's [28] first examples of hyperbolic knots with virtually fibered but nonfibered complements, twenty years after the question was posed. Crucially it dates its discussion explicitly to February 2005 and says much remains to learn then. These are historical source assessments and priority reports, not present-day status claims. The registry's generic open-triage label does not independently establish the current status of the historical Thurston question or of the broader Heegaard program.

Short English statement

Investigate whether the distance and special structure of bundle-associated Heegaard splittings help recognize virtual fibering of nonfibered manifolds. The source gives no exact criterion and explicitly dates its surrounding virtual-fibering discussion to February 2005.

Sources

• Local UnsolvedMath registry, permanent ID $11000171$, exact source code catalogue entry, Problem 2.14, PDF page 159. No individual proposer or proposed year is supplied; the title is truncated.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The volume has 329 explicitly typeset numbered items across chapters, repeated local numbering distinguished by PDF page.
• Internal references: Thurston [49] for the finite-volume hyperbolic virtual-fibering question; Leininger [28] for the reported knot-complement examples. Full entries are absent. The source explicitly says it is writing in February 2005 and reports a twenty-year interval from the question to those examples.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and requires a current primary-literature audit. Registry status open and classification OPEN-TRIAGE; research summary says only "Literature status: Truncated." No modern resolution assessment is supplied.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.