Universal Triangulations with Prescribed Boundary

Definitions and mathematical statement

Let $d\geq 2$ be an integer, covering the dimensional cases explicitly mentioned in the supplied statement. A finite simplicial $(d-1)$-sphere is a finite simplicial complex $T$ whose geometric realization $|T|$ is homeomorphic to $S^{d-1}$. A linear embedding $f:|T|\hookrightarrow{\mathbb{R}}^d$ is an embedding whose restriction to each simplex is affine. A simplicial $d$-ball is a finite simplicial complex $U$ triangulating a closed $d$-dimensional ball; prescribing boundary $T$ requires an identification $\partial U\cong T$ as simplicial complexes.

Requested problem, with unresolved specifications. Given such an embedded sphere $f:|T|\hookrightarrow{\mathbb{R}}^d$, characterize the triangulations called universal in the source, give an algorithm constructing a universal simplicial $d$-ball $U$ with prescribed boundary $T$, and establish bounds on its size. The supplied record does not define universality: neither the class of objects over which its quantifiers range nor the required extension, realization, or comparison property is specified. It also does not specify whether $U$ must admit a linear embedding extending $f$, whether boundary subdivision is permitted, which combinatorial quantity measures size, or the parameters controlling the requested bound. Consequently, this is an incomplete problem specification, not a fully quantified mathematical assertion. The boundary identification above expresses the literal prescribed-boundary requirement; any permission to subdivide it needs clarification.

The supplied record reports that the case $d=2$ is known and that higher-dimensional cases remain open. These are supplied status claims, not independently verified mathematical facts. The record does not explicitly address dimensions below $2$. Its analytic-torsion literature assessment concerns a different mathematical problem and cannot supply the missing definition of universality.

Short English statement

For a triangulated sphere embedded linearly in ${\mathbb{R}}^d$, the requested task is to characterize universal triangulations, algorithmically construct a universal triangulated ball with that boundary, and bound its size. The meaning of universal and the size measure are missing, so a precise conjecture cannot yet be stated. The reported planar result, higher-dimensional open status, and unrelated analytic-torsion assessment require source review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002069$, title Universal Triangulations with Prescribed Boundary, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The proposed year is recorded as $2012$; no proposer is identified.
• Supplied source citation: Triangulations (2013), Oberwolfach Reports, pp. 1405--1486, DOI 10.4171/owr/2012/24. Supplied source URL: https://doi.org/10.4171/owr/2012/24. The title/year and DOI are retained as supplied; their bibliographic correspondence has not been independently checked.
• Supplied literature reference labeled "Lesch analytic-torsion paper": https://doi.org/10.2140/apde.2013.6.221.
• Supplied literature reference labeled "original OWR report": https://doi.org/10.4171/owr/2012/25. This DOI differs from the report DOI in the source citation.
• Literature assessment dated 2026-08-21: supplied dated curation, not independently verified. It labels its subject open and states that Lesch's published singular-space gluing formula assumes product structure near the cutting hypersurface, while later non-product formulas for compact manifolds do not establish the conjectured singular result. The supplied background describes this as dated literature triage derived from OWR curation and calls for independent verification of time-sensitive status and cited claims. This assessment concerns analytic torsion and does not establish the status of the triangulation problem.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting that the cleanup is self-contained and adds no unsupported source-dependent content. These supplied review labels do not resolve the missing mathematical specifications. No source was visited for this draft.