Minimum Faces in Complexes Not Embeddable in Manifolds

Definitions and mathematical statement

For a simplicial complex $K$, let $|K|$ denote its geometric realization and let \[ f_k(K)=\#\{\sigma\in K:\dim\sigma=k\} \] be the number of its $k$-dimensional simplices, called its $k$-faces. A simplex has dimension $k$ when it has $k+1$ vertices. A complex has dimension $k$ when it has a $k$-simplex and no simplex of larger dimension.

The supplied problem asks, for a $2k$-dimensional manifold $M$, for the smallest number of $k$-faces of a $k$-dimensional simplicial complex that does not embed in $M$. Its precise formulation requires an admissible class of complexes and a meaning of embedding, which the record does not specify.

More explicitly, fix a nonnegative integer $k$, a manifold $M$ of dimension $2k$, an admissible class $\mathcal C_k$ of simplicial complexes of dimension exactly $k$, and an embedding category $\mathsf e$. Write $|K|\hookrightarrow_{\mathsf e}M$ to mean that an embedding in this category exists. If every member of $\mathcal C_k$ has finitely many $k$-faces and at least one member does not embed in $M$, the extremal quantity requested is \[ \mu_{\mathcal C_k,\mathsf e}(M) = \min\left\{f_k(K):K\in\mathcal C_k,\ |K|\not\hookrightarrow_{\mathsf e}M\right\}. \] Determine this quantity for the intended choices of $k$, $M$, $\mathcal C_k$, and $\mathsf e$. If the displayed set is empty, it has no minimum; the source supplies no convention for that case.

For the topological interpretation, an embedding is a continuous map $h:|K|\to M$ that is a homeomorphism onto its image with the subspace topology. This definition describes one possible interpretation; the supplied record does not select topological versus piecewise-linear embedding. It also does not specify whether complexes must be finite, the permitted range of $k$, or whether $M$ is connected, compact, closed, orientable, or allowed to have boundary. If $M$ has boundary, restrictions on how the embedded complex may meet that boundary are likewise unstated. The displayed formulation is therefore conditional on these choices, rather than a uniquely specified extremal problem. The initial wording about a given complex does not explain whether that complex imposes any further restriction on the complexes over which the minimum is taken.

Short English statement

For a specified $2k$-dimensional manifold $M$, ask how few $k$-faces a $k$-dimensional simplicial complex can have while failing to embed in $M$. The source leaves the admissible complexes, embedding category, manifold hypotheses, and range of $k$ unresolved. Its open status is supplied dated curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003468$, exact source problem code catalogue entry, title: Minimum Faces in Complexes Not Embeddable in Manifolds. The record belongs to the Oberwolfach Reports Open Problems collection, lists proposed year $2017$, and supplies no named proposer. Its labels open and corrected_verified are registry metadata; the supplied verification description concerns preservation of mathematical content and repairs to notation, typography, and sentence structure.
• Supplied report citation: Discrete Geometry (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/19, pp. 1135--1205. Supplied source URL and literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2017/19. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied citation year $2018$, proposed year $2017$, and DOI containing $2017$ are retained without resolving their relationship.
• Supplied literature-source label: "Paták and Tancer on embeddings": https://ems.press/content/serial-article-files/46683. No full bibliographic citation or specific supporting result is supplied for this link.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the extremal number is unknown in the stated generality, with existing work providing methods and special cases rather than a general minimum. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record explicitly requests independent verification of time-sensitive status and cited claims. No linked source has been independently inspected for this draft.