Problem 2.4 — Knowing that d(φ)≤ 1, can we decide whether d(φ) = 0?

Definitions and mathematical statement

Let $M=V\cup_S W$ be the Heegaard splitting represented by the source's gluing map $\varphi$. In the usual genus-at-least-two curve-graph convention, let $\mathcal D(V),\mathcal D(W)\subset\mathcal C(S)$ be the sets of boundary classes of compressing disks on the two sides, and set \[d(\varphi)=d_{\mathcal C(S)}(\mathcal D(V),\mathcal D(W)).\] A compressing disk has boundary an essential curve on $S$. The source does not reproduce this definition or its low-complexity conventions, but the stated geometric restatement identifies the intended splitting-distance question.

A splitting is weakly reducible if there exist opposite-side compressing disks with disjoint boundaries. It is reducible if there are opposite-side compressing disks whose boundaries are the same essential curve (or can be isotoped to coincide). In the indicated curve-graph setting, these correspond to $d(\varphi)\leq1$ and $d(\varphi)=0$, respectively.

The problem asks for an algorithm with the promise $d(\varphi)\leq1$, deciding whether $d(\varphi)=0$; equivalently, decide whether a given weakly reducible Heegaard splitting is reducible. In this promise class the alternatives are distance zero and distance one. The decision concerns the given splitting, not merely whether the ambient three-manifold contains an essential reducing sphere. No general exact-distance algorithm or bound beyond the promise is demanded.

A full algorithmic statement must fix finite input data, such as an effective gluing-map description with the two handlebody/compression-body markings, or a triangulated manifold with specified splitting. The excerpt does not fix that encoding, the genus as a parameter, closed/boundary category or exceptional low-genus distance conventions. Those must not be omitted when translating the decision request.

Short English statement

Given a weakly reducible Heegaard splitting, decide whether it is reducible. In the intended distance convention this distinguishes distance zero from distance one under the promise that distance is at most one.

Sources

• Local UnsolvedMath registry, permanent ID $11000161$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 2.4, PDF page 156: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Registry access 2026-07-29; pdf-font-structure extraction. No separate algorithm or distance reference is supplied.