Johnson-Filtration Images of Stable Heegaard Splittings

Definitions and mathematical statement

An integral homology $3$-sphere is a closed connected $3$-manifold $M$ with $H_i(M;{\mathbb{Z}})\cong H_i(S^3;{\mathbb{Z}})$ for every $i\geq0$. A genus-$g$ Heegaard splitting expresses $M$ as the union of two genus-$g$ handlebodies along their common boundary. Stabilization of such a splitting adds the standard genus-one splitting of $S^3$.

The supplied record refers to sets $V_k(3)$ as Johnson-filtration images associated with stable Heegaard splittings of integral homology $3$-spheres. It does not define their domain, target, image map, or stabilization identifications. Retaining this notation, the first requested problem is to describe $V_k(3)$ and determine whether \[ \forall k\in{\mathbb{Z}}_{\geq1},\qquad V_k(3)=V_1(3). \]

The second requested problem uses a group $\pi$, subgroups $\Gamma_{k+1}$, and groups denoted $A_{g,1}(k)$, $B_{g,1}$, and $AB_{g,1}$, whose elements act on $\pi$. For a normal subgroup $\Gamma_{k+1}\trianglelefteq\pi$, write $a\equiv b\pmod{\Gamma_{k+1}}$ when $a\Gamma_{k+1}=b\Gamma_{k+1}$ in $\pi/\Gamma_{k+1}$. With the intended groups and actions fixed, the requested assertion is \[ \begin{gathered} \forall\varphi\in A_{g,1}(k)\ \forall\psi\in B_{g,1},\\ \left[\forall x\in\pi,\quad \varphi(x)\equiv\psi(x)\pmod{\Gamma_{k+1}}\right]\\ \Longrightarrow \left[\exists\mu\in AB_{g,1}\ \forall x\in\pi,\quad \mu(x)\equiv\varphi(x)\equiv\psi(x)\pmod{\Gamma_{k+1}}\right]. \end{gathered} \] This is to be considered for the intended admissible genera $g$ and filtration indices $k$; the record explicitly specifies $k\geq1$ for the first question but does not separately specify the parameter ranges for the second.

Unresolved qualifications. The record does not identify $\pi$, define $\Gamma_\bullet$ or its indexing, specify the relevant Johnson filtration and handlebody subgroups, or explain the notation $AB_{g,1}$. In particular, its meaning as an intersection, product, or separately defined subgroup must not be presumed. The surface, boundary and basepoint conventions needed to interpret the actions on $\pi$ are also absent. Consequently, these formulas preserve the supplied targets but do not constitute a fully specified mathematical formulation without the missing definitions.

Short English statement

Describe the Johnson-filtration images $V_k(3)$ associated with stable Heegaard splittings of integral homology $3$-spheres, and ask whether every such image equals $V_1(3)$. Also ask whether the common action modulo $\Gamma_{k+1}$ of the specified elements $\varphi$ and $\psi$ can be realized by one element $\mu\in AB_{g,1}$. Several essential group and filtration definitions are missing from the supplied record, so the formulation requires human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30003289$, exact source problem code catalogue entry, title: Johnson-Filtration Images of Stable Heegaard Splittings; collection: Oberwolfach Reports Open Problems. The supplied proposed year is 2016; no individual proposer is identified.
• Supplied report citation: Surface Bundles (2017), Oberwolfach Reports, pp. 3149--3195, DOI 10.4171/owr/2016/56. https://doi.org/10.4171/owr/2016/56. The supplied background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports; no problem-specific page or passage is supplied.
• Supplied literature source: Oberwolfach Report 56 (2017). https://doi.org/10.4171/owr/2016/56. Literature checked at: 2026-08-21. This is supplied dated curation, not an independently verified assessment. It labels the problem open and states that Johnson-filtration and mapping-class-group methods give partial information, while the asserted equality of the "third-level images" and the requested simultaneous realization remain unresolved. The supplied background explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review, including a claim of self-containment. These are supplied editorial assertions; the missing definitions identified here remain unresolved, and no source visit or independent mathematical verification is asserted.