Equality Cases in Lower Bounds for Triangulated 3-Manifolds

Definitions and mathematical statement

A closed $3$-manifold means a compact topological manifold of dimension $3$ without boundary. A finite simplicial triangulation of $M$ is a finite simplicial complex $\Delta$ whose geometric realization $|\Delta|$ is homeomorphic to $M$. Write $f_0(\Delta)$ and $f_1(\Delta)$ for its numbers of vertices and edges, respectively, and set \[ \Gamma(M)=\min_{\substack{\Delta\text{ a finite simplicial complex}\\ |\Delta|\cong M}} \left(f_1(\Delta)-4f_0(\Delta)+10\right). \] Here $\cong$ denotes homeomorphism. For a specified coefficient field $k$, write \[ \beta_1(M;k)=\dim_k H_1(M;k). \] The source writes only $\beta_1(M)$; it does not specify its coefficient convention.

Let $B$ denote the nonorientable total space of the nontrivial $S^2$-bundle over $S^1$. The source identifies connected sums of copies of $S^2\times S^1$ and $B$ as known equality examples for the bound it attributes to Novik, \[ \Gamma(M)\geq 10\beta_1(M). \] Question as supplied. Determine which closed $3$-manifolds, beyond those connected sums, satisfy \[ \Gamma(M)=10\beta_1(M). \]

Unresolved qualifications. A precise classification requires the intended coefficient convention for $\beta_1$ and the hypotheses of the attributed lower bound. The record does not explicitly state connectedness, any orientability or homological orientability requirement, or whether an empty connected sum is included among the listed examples. Consequently, the displayed bound and its claimed example family are recorded as supplied assertions, without asserting their validity for every closed $3$-manifold or every coefficient field. If field coefficients are intended, the equality question becomes $\Gamma(M)=10\beta_1(M;k)$ for the intended fixed field $k$ and admissible class of manifolds; neither is silently selected here.

Short English statement

The question asks which closed $3$-manifolds attain equality in a supplied lower bound comparing the minimum of $f_1-4f_0+10$ over triangulations with ten times the first Betti number, beyond the stated connected sums of sphere bundles over the circle. The coefficient convention and exact hypotheses need clarification. The supplied open status is dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30001703$, source problem code catalogue entry, title Equality Cases in Lower Bounds for Triangulated 3-Manifolds; supplied complete source record associated with data/unsolvedmath/problems.json and catalog/problems.json.
• Topological and Geometric Combinatorics (2011), Oberwolfach Reports, pp. 349--423, DOI 10.4171/owr/2011/08. Supplied source URL and literature-source link, labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2011/08. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• The supplied background attributes the lower bound to Novik and says that the report leaves open other equality cases beyond connected sums of sphere bundles. No proposer is named, and no separate citation for the lower bound or problem-specific page is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied status is "open", and the assessment states that Novik's lower bound is known while other equality cases remain open in the report. This is supplied dated curation derived from the OWR record, not independently verified literature or source verification; time-sensitive status and cited claims require independent checking.
• The registry labels the statement corrected_verified, with a supplied note reporting manual semantic, typographic, and LaTeX review. This records the registry's review claim and does not establish mathematical correctness or resolve the qualifications above.