Problem 3.5 — Is there a natural quotient complex of any one of the complexes discussed in §1 which might be useful for the constru…

Definitions and mathematical statement

Let $M_g$ denote the mapping class group in the source and let $X$ be one of the complexes discussed in its unprovided $\S1$.

Problem 3.5. Is there a natural quotient complex of such an $X$ useful for constructing nonfaithful representations of $M_g$? A possible precise setup is an $M_g$-equivariant quotient $q:X\to Y$ inducing \[ \rho_Y:M_g\longrightarrow\operatorname{Aut}(Y) \] with nontrivial kernel, followed by any further construction required for the intended representation target. The source does not say whether "representation" means this combinatorial action, a linear representation, or another kind. It also gives no naturalness or usefulness criterion. Collapsing $X$ to a point formally produces a trivial action, but does not by itself meet the stated research purpose; no such construction is asserted as an answer.

The accompanying discussion proposes using answers to Problems 3.1, 3.2, or 3.5 to obtain invariants distinguishing Heegaard splittings. It recalls a successful symplectic-representation invariant in [4] separating inequivalent minimal splittings, says later student attempts with other representations were unsuccessful, and proposes revisiting that direction. These are source history, not a theorem that other representations cannot work.

For a fixed genus and handlebody marking, let $\mathcal H\leq M_g$ be the handlebody subgroup. The source identifies the equivalence class of a splitting $H\cup_\varphi H'$ with the double coset \[ \mathcal H\varphi\mathcal H\in\mathcal H\backslash M_g/\mathcal H. \] A representation $\rho$ correspondingly gives image double-coset data $\rho(\mathcal H)\rho(\varphi)\rho(\mathcal H)$, but usefulness requires genuinely distinguishing intended splitting classes. The exact orientation, side-ordering, and handlebody-identification conventions are omitted. The source recalls that different splittings of one $3$-manifold can become equivalent after stabilization and asks broadly about the collection of all genera; it does not require the proposed splitting invariant to identify or distinguish classes after stabilization.

Short English statement

Seek a natural quotient of a surface complex that yields useful nonfaithful mapping-class representations. The source hopes to use such representations to distinguish Heegaard double cosets, following an earlier symplectic invariant, but omits the complexes and precise representation target.

Sources

• Local UnsolvedMath registry, permanent ID 11000176, source code catalogue entry, Problem 3.5, PDF page 161. No individual proposer is supplied; page distinguishes repeated local labels.
• 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 contributed chapters.
• The statement refers to its $\S1$ complexes, Problems 3.1 and 3.2, and reference [4] for the earlier symplectic splitting invariant, none supplied in full.
• Supplied status open and OPEN-TRIAGE. The research summary says only "Cannot verify." Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.