Finiteness from Face-Number Bounds for Manifolds with Boundary

Definitions and mathematical statement

The source asks a finiteness question about manifolds with boundary, but does not specify the triangulation category or all required hypotheses. The following makes the face-number interpretation explicit while retaining these unresolved qualifications.

If the intended objects are finite simplicial triangulations, let $K$ be such a triangulation whose underlying space $|K|$ is a topological $d$-manifold with boundary. Let $f_0(K)$ and $f_1(K)$ denote its numbers of vertices and edges, respectively, and set \[ v_{\mathrm{int}}(K)=\#\{v\in K^{(0)}:v\notin\partial|K|\}, \qquad I_d(K)=f_1(K)-d f_0(K)+\binom{d+1}{2}-v_{\mathrm{int}}(K). \] Here $K^{(0)}$ is the vertex set, and $\partial|K|$ is the manifold boundary.

For each fixed intended dimension $d$, let $\mathcal T_d$ denote the intended class of admissible triangulated manifolds. The extracted question is: for every fixed bound $n$, is \[ \#\left\{[|K|]_{\mathrm{homeo}}: K\in\mathcal T_d,\ I_d(K)\le n\right\}<\infty? \] The notation $[|K|]_{\mathrm{homeo}}$ denotes the homeomorphism type of the underlying space. This formulation counts a type when it has an admissible triangulation satisfying the bound. The displayed quantity is defined from a triangulation; the source does not establish that it is independent of triangulation.

The class $\mathcal T_d$, the allowed dimensions, and the domain of $n$ remain unspecified in the supplied record. In particular, the record does not settle connectedness, whether the boundary must be nonempty, or whether triangulations must satisfy additional combinatorial or piecewise-linear manifold conditions. Finite simplicial triangulations impose compactness, which is an explicit interpretive assumption here rather than a supplied hypothesis. These omissions prevent this draft from identifying a uniquely specified conjecture.

Short English statement

For a fixed dimension, does an upper bound on the stated combination of edge, vertex, and interior-vertex counts allow only finitely many homeomorphism types of manifolds with boundary? The admissible triangulations and several manifold hypotheses require clarification. The supplied record labels the question open; that status has not been independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001704$, source problem code catalogue entry, titled Finiteness from Face-Number Bounds for Manifolds with Boundary, in the Oberwolfach Reports Open Problems collection.
• Supplied report citation: Topological and Geometric Combinatorics (2011), Oberwolfach Reports, pp. 349--423, DOI 10.4171/owr/2011/08. Source URL and supplied literature-source link, labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2011/08. The record attributes the extraction to workshop open-problem discussions and gives proposed year 2011; it supplies no individual proposer or precise location within the report.
• Literature assessment checked at 2026-08-21: the supplied dated curation labels the problem open and describes it as an extracted higher-dimensional extension question without a complete claim or general resolution in the cited literature. This is supplied dated curation, not an independently verified assessment; the record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review of its cleanup. These are supplied editorial assertions, not mathematical verification of the formulation in this draft.