Problem 2.11 — Hempel’s distance function was chosen so that it would capture the geometry, and indeed it does that very well, yet i…

Definitions and mathematical statement

For a genus-$g$ Heegaard surface $S$, retain Hempel's distance and the source's Hatcher--Thurston complex $HT(S)$, pants complex $P(S)$, and mapping class group $M$. In a usual Hatcher--Thurston model vertices are cut systems, and in a pants complex vertices are pants decompositions, but the exact vertex, edge, and metric conventions are not supplied.

Problem 2.11. Can Heegaard distance be redefined using $HT(S)$, or possibly $P(S)$ or another mapping-class-group complex, and can such a replacement yield new information? The source motivates $HT(S)$ by saying pairs of vertices determine Heegaard diagrams and every genus-$g$ diagram arises this way. A proposed distance must specify its data and graph metric, account for the choices of diagrams or disk systems, and state the equivalence of Heegaard splittings under which it is to be an invariant. The source gives no single replacement formula or required comparison theorem, and allows the alternative complexes.

The remaining extraction starts separate bundle-related context. Given $\alpha\in M_g$, it forms \[ M_\alpha=(S\times[0,1])/\left((p,0)\sim(\alpha(p),1)\right). \] It says bundle structures need not be unique and describes bundle equivalence by conjugacy of monodromies. That criterion requires the intended fiber and base-orientation convention; it is not a classification of underlying $3$-manifolds, and allowing reversal of the base can change the monodromy comparison.

Citing [1], it chooses $p,q\in S$ with $p\neq q$ and $p\neq\alpha(q)$, disjoint closed regular neighborhoods $P,Q$ of $p\times[0,1/2]$ and $q\times[1/2,1]$, and gives schematic pieces \[ H=\left(S\times[0,1/2]\setminus Q\right)\cup P, \qquad H'=\left(S\times[1/2,1]\setminus P\right)\cup Q. \] The displayed source parentheses are malformed, and these neighborhood placements and attachments must be checked before using the formulas as a precise construction; no interchange of $P,Q$ is silently made. The source reports that the pieces are genus-$2g+1$ handlebodies giving a splitting $H\cup_\beta H'$ for some $\beta\in M_{2g+1}$, called the bundle-related Heegaard splitting. The further problems about that construction are not included and are not added to the distance question.

Short English statement

Seek a useful Heegaard-distance theory based on the Hatcher--Thurston complex, pants complex, or another surface complex. The following mapping-torus construction is retained as separate context, including its genus-$2g+1$ claim and unresolved neighborhood formulas.

Sources

• Local UnsolvedMath registry, permanent ID 11000168, source code catalogue entry, Problem 2.11, PDF page 158. The chapter header identifies J. Birman; no individual proposer field is supplied.
• Farb, Problems on Mapping Class Groups and Related Topics. Supplied URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf; recorded access 2026-07-29; extraction pdf-font-structure. The volume contains 329 explicitly typeset headings across chapters, with repeated labels disambiguated by page.
• Reference [1] is cited for the mapping-torus-related Heegaard construction, but its full entry is absent. The following promised bundle-related problems are not supplied.
• Supplied status open and OPEN-TRIAGE. The summary says only "Truncated." Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.