Vertex-Symmetry Criteria for Mod-Two Tightness

Definitions and mathematical statement

Let $X$ be a simplicial complex with vertex set $V(X)$ and dimension $2$; thus every simplex has at most three vertices and at least one simplex has three vertices. For $W\subseteq V(X)$, define the induced subcomplex by \[ X[W]=\{\sigma\in X:\sigma\subseteq W\}. \] No finiteness, purity, manifold, or connectedness hypothesis is explicitly supplied.

The source uses a quantity $c_Y(y_1,y_2)$ associated with an induced subcomplex $Y$ and an ordered pair of distinct vertices of $Y$. It supplies neither a definition nor a codomain for this quantity. Consequently, its symmetry condition can only be retained symbolically.

Question as supplied. For every two-dimensional simplicial complex $X$, does the implication \[ \left[ \begin{gathered} \text{for every }W\subseteq V(X)\text{ and all }y_1,y_2\in W,\\ y_1\ne y_2\ \Longrightarrow\ c_{X[W]}(y_1,y_2)=c_{X[W]}(y_2,y_1) \end{gathered} \right] \quad\Longrightarrow\quad X\text{ is }\mathbb Z_2\text{-tight} \] hold?

For clarity, a conventional homological interpretation of $\mathbb Z_2$-tightness requires $X$ to be connected and every inclusion $X[W]\hookrightarrow X$ to induce injective maps \[ H_i(X[W];\mathbb F_2)\longrightarrow H_i(X;\mathbb F_2) \qquad(W\subseteq V(X),\ i\geq 0), \] where $\mathbb F_2=\mathbb Z/2\mathbb Z$. This is an explicitly qualified possible interpretation: the supplied record does not define tightness or specify whether finiteness or other ambient conventions are assumed. The intended definition of $c_Y$, and the applicable tightness conventions, must be recovered before the displayed question constitutes a fully specified mathematical statement.

Short English statement

The supplied question asks whether symmetry of $c_Y$ on every ordered pair of distinct vertices in every induced subcomplex forces a two-dimensional simplicial complex to be $\mathbb Z_2$-tight. The quantity $c_Y$ is undefined in the record, so the question remains incomplete. The supplied open label and statement-verification label are curation claims, not independently verified conclusions.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002778$, title "Vertex-Symmetry Criteria for Mod-Two Tightness," source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2015; no proposer is named.
• Supplied report citation: Geometric and Algebraic Combinatorics (2015), Oberwolfach Reports, pp. 285--368, DOI 10.4171/owr/2015/5. Supplied source URL: https://doi.org/10.4171/owr/2015/5. This report reference is supplied provenance; its contents have not been independently inspected here, and no exact page or contribution for this problem is supplied.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/owr/2015/5.
• Literature assessment dated 2026-08-21: supplied dated curation, not independently verified. It labels the problem open and states that the extracted record contains only "We conjecture the following," omitting the conjectured statement, so no determinate mathematical claim can be verified. The background attributes this triage to the supplied OWR curation and requests independent verification of time-sensitive status and cited claims. This assessment conflicts with the explicit question present in the supplied statement fields.
• The supplied statement status is corrected_verified; its accompanying verification text says that the question was explicitly identified and its notation and publication prose repaired without changing its substance. This is a supplied editorial assertion, not independent mathematical or source verification.