Bounding Immersed Curves by Stable Singular Disk Maps

Definitions and mathematical statement

Let $S_g$ be the oriented surface appearing in the source record, and let $\gamma\colon S^1\to S_g$ parameterize the given generically immersed closed curve. In the usual smooth interpretation, a generic immersion has only transverse double points as self-intersections. Write $D^2=\{x\in{\mathbb{R}}^2:\|x\|\leq 1\}$, and identify $\partial D^2$ with $S^1$.

The supplied existence question is whether, for every such pair $(S_g,\gamma)$, there exist a compact $3$-manifold $M$, an identification $h\colon\partial M\to S_g$, and a proper map $f\colon D^2\to M$ having only stable singularities, with boundary tracing $\gamma$. Interpreting the source's boundary condition as requiring the disk interior to lie in the manifold interior gives \[ h\circ f|_{\partial D^2}=\gamma, \qquad f^{-1}(\partial M)=\partial D^2. \] Consequently, \[ h\left(f(D^2)\cap\partial M\right)=\gamma(S^1). \] Here topological properness means that inverse images of compact sets are compact; it is automatic for a continuous map from the compact disk into a Hausdorff manifold. The condition $f^{-1}(\partial M)=\partial D^2$ is an additional boundary condition, whose interpretation requires confirmation.

Unresolved qualifications. The displayed formulation makes explicit an intended interpretation of the source notation $\partial D=D\cap S_g=\gamma$, which conflates the disk domain, its mapped image, and the parameterized boundary curve. The source does not specify the differentiability category, the allowed stable singularities, the equivalence notion defining stability, or whether stability is relative to the prescribed boundary map. These choices remain part of the unresolved formulation. It also does not explicitly state that $S_g$ is compact and without boundary, although those properties are necessary for an identification with the entire boundary of a compact ordinary $3$-manifold. No orientation of $M$, nor compatibility with the given orientation of $S_g$, is explicitly required in the source.

Short English statement

Given a generically immersed closed curve on an oriented surface, the supplied problem asks whether it bounds a disk map with only stable singularities inside some compact $3$-manifold whose entire boundary is that surface. The meaning of stable singularities and the precise boundary requirements need clarification. The supplied open status is a dated curation label, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30002597$, source problem code catalogue entry, title Bounding Immersed Curves by Stable Singular Disk Maps. The supplied record belongs to the Oberwolfach Reports Open Problems collection and names no individual proposer.
• Supplied report citation: Algebraic Structures in Low-Dimensional Topology (2015), Oberwolfach Reports, pp. 1403--1458, DOI 10.4171/owr/2014/26. https://doi.org/10.4171/owr/2014/26. The record separately gives the proposed year as $2014$; the supplied citation year is retained without reconciliation.
• Supplied literature reference: Manturov, Parity and cobordism of free knots. https://doi.org/10.1070/SM2012v203n02ABEH004219.
• Supplied literature reference: Oberwolfach Report 26, stable-singularity disk problem. https://doi.org/10.4171/owr/2014/26.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open, reports parity obstructions and substantial partial results for free knots, and says that no general existence criterion for a bounding $3$-manifold and stable-singularity disk map for every immersed curve was found. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.