Question 2.2 — (Donaldson).

Definitions and mathematical statement

Let $p:X\to B$ be a Lefschetz fibration with a given monodromy factorization. In the usual matching-path setting, an embedded path in the base joins two critical values, its interior avoids all critical values, and the vanishing cycles transported from its endpoints match in a common regular fiber under the required isotopy or Hamiltonian-isotopy convention. This is a conditional description: the source does not define matching, specify symplectic/complex data, or give the fiber and base dimensions.

Donaldson's Question 2.2 asks whether all matching paths can be enumerated from the given monodromy factorization. A precise formulation must fix whether paths are counted as actual maps, up to isotopy/homotopy relative endpoints and critical values, or with further equivalences induced by the fibration. It must also specify an encoding of the monodromy data and the desired enumeration: a possibly infinite effective list, a finite classification, or a decision procedure for candidate paths are different tasks. No finiteness assertion is supplied.

The factorization normally records vanishing-cycle monodromies after choices of reference fiber and distinguished paths. The record does not state which such choices are part of the input, or whether the matching criterion is determined by that input alone in the intended category. Thus the request remains to enumerate the actual source-defined matching paths, not an arbitrarily chosen set of arcs between critical values.

Short English statement

Can one enumerate every matching path of a Lefschetz fibration from its monodromy factorization? The matching condition, path equivalence, fibration category and computational input are missing, so completeness and the meaning of enumeration require source clarification.

Sources

• Local UnsolvedMath registry, permanent ID $11000153$, exact source code catalogue entry, Question 2.2, attributed in the heading to Donaldson, PDF page 142. No proposed year or first name is supplied.
• 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.
• 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; the research summary explicitly cannot verify status. No separate authoritative matching-path reference 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.