Minimal Simplicial Complexes

• Minimal simplicial complexes are defined as triangulations that use the fewest vertices to represent a space's homotopy type.
• They exhibit sharp extremal vertex-count bounds linked to homological properties and boast complete classifications for small vertex counts in group presentations.
• Advanced algorithmic pipelines enable practical construction and enumeration of near-minimal complexes, expanding applications in topology and combinatorics.

Minimal simplicial complexes are a central object of study in algebraic topology, combinatorics, and geometric group theory, where they represent triangulations of spaces or homotopy types with the smallest possible number of vertices. The concept encompasses vertex-minimal triangulations for specific homological properties, minimal Cohen-Macaulay complexes, minimal connected covers (generalizing trees), minimal Taylor-resolved complexes, and minimal triangulations for prescribed fundamental groups. Recent research has established extremal vertex-count bounds, complete group classifications for complexes on few vertices, precise combinatorial invariants, and practical computational pipelines for constructing near-minimal complexes. The following sections survey key definitions, classification theorems, enumeration results, homological and group-theoretic constraints, and algorithmic advances.

1. Foundational Definitions and Extremal Criteria

A minimal simplicial complex is typically defined by the property that no triangulation of the same space or homotopy type uses fewer vertices. Explicitly, for a path-connected CW complex $X$, $v(X)$ is the minimal number of vertices among all simplicial complexes $K$ with $|K|\simeq X$ (Govc et al., 4 Nov 2025). The minimal triangulation problem is closely tied to invariants such as the covering type $ct(X)$ and the Karoubi–Weibel invariant $KW(G)$, which for a group $G$ is the minimal number of vertices in a $K(G,1)$ complex realizing $G$ as $\pi_1$ with trivial higher homotopy (Govc et al., 4 Nov 2025).

Extremal vertex-count bounds are precise in several homological regimes. For pure $d$-dimensional complexes with nontrivial $k$-th homology, the vertex count satisfies

$$f_0(X) \ge \left\lceil \frac{(d+1)(k+2)}{k+1} \right\rceil$$

and this bound is tight, with explicit constructions realizing equality (Kogan, 21 Jan 2025). If the complex is strongly connected in top dimension, the minimal vertex count refines to

$f_0(X) \ge d+1 + \frac{d}{k}$

again sharp up to rounding (Kogan, 21 Jan 2025). These results extend classical lower bounds such as the minimal $d$-ball being the simplex $\Delta^d$, and pure $d$-sphere being the boundary $\partial\Delta^{d+1}$.

2. Minimal Triangulations for Group Presentations

Minimal complexes representing a given group arise in the classification of $K(G,1)$ homotopy types. For a finitely presented group $G$, one seeks complexes $K$ with $\pi_1(K) \cong G$ and minimal $|V(K)|$. Recent work provides complete classifications for complexes with up to eight vertices:

• For $n \leq 7$, non-free groups realized are limited to cyclic $\mathbb{Z}_m$ ($m \le 4$) and $\mathbb{Z}\times\mathbb{Z}$ (Govc et al., 4 Nov 2025).
• For $n=8$, the only possible non-free indecomposable factors are $$H \in \{\mathbb{Z}_2, \mathbb{Z}_3, \mathbb{Z}_4, \mathbb{Z} \times \mathbb{Z}, \mathbb{Z} \rtimes \mathbb{Z}, B_3\}$$ (where $B_3$ is the braid group on three strands) (Govc et al., 4 Nov 2025).
• With $n=9$, the palette expands to include groups such as $\mathbb{Z}_m$ ($5 \le m \le 9$), $D_\infty$, $D_6$, $Q_8$, Baumslag–Solitar and various surface groups (Govc et al., 4 Nov 2025).
• For free abelian groups, sharp bounds for the minimal vertex number are $V(n) \le 8n + O(1)$ and $V(n) \ge c n^{3/4}$ (Frick et al., 2021).

The table below summarizes realizable non-free groups with $n\leq 8$:

Vertices	Realizable Non-Free Groups	Notes
$\leq 7$	$\mathbb{Z}_m$ ($m\le4$),	All up to free factor;
	$\mathbb{Z}\times\mathbb{Z}$	free group ranks $\le15$
8	$\mathbb{Z}_2$, $\mathbb{Z}_3$, $\mathbb{Z}_4$	Free product with $F_r$ allowed, $r\le21$
	$\mathbb{Z}\times\mathbb{Z}$, $\mathbb{Z} \rtimes \mathbb{Z}$, $B_3$

This sharpens previous lower bounds and provides recognition criteria for fundamental groups arising from small complexes, leading to applications in the minimal triangulation of manifolds such as the Poincaré homology sphere (Govc et al., 4 Nov 2025).

3. Minimal Cohen–Macaulay Complexes and Enumeration

A simplicial complex $\Delta$ is minimal Cohen–Macaulay (CM) if removal of any facet destroys the CM property. Equivalently, no CM subcomplex exists upon deletion of a top-dimensional face. Every CM complex is shelled over a minimal CM subcomplex, these minimal cores serving as "atoms" for the CM category (Dao et al., 2019). Rigidity results include:

• For pure $(d-1)$-dimensional minimal CM complexes that are $f$-simplicial, the vertex count is exactly $n=2d$ (Wang et al., 2021).
• Minimal CM complexes are $(d-1)$-acyclic and must exhibit specific acyclicity and boundary ridge structures (Dao et al., 2019).
• The join or suitably glued union of minimal CM complexes yields new minimal CM complexes, facilitating explicit constructions in higher dimensions (Dao et al., 2019).

Enumerative combinatorics for abstract complexes is governed by the Dedekind numbers $d_n$, which grow super-exponentially, with $d_n \sim 2^{2^n}$ as $n \to \infty$ (Govc et al., 4 Nov 2025). For 2-pure complexes, $h_3(n) \approx 2^{2^n}$ as well, implying the landscape of minimal complexes is vastly complex even at modest $n$.

4. Minimal Path-Connected Covers and Higher-Dimensional Trees

Minimal connected covers, or minimally path-connected complexes, generalize the notion of trees to higher dimensions. A minimal connected cover $Y$ is pure $r$-dimensional, connected, and removing any facet disconnects $Y$ (Mead, 2019). Key properties include:

• $Y$ has vanishing $H_r(Y;\mathbb{Z})$ (no top-dimensional homology).
• Each facet carries a free codimension-1 face, so $Y$ collapses to a lower-dimensional complex.
• Lower and upper bounds on the number of minimal connected covers on $n$ labeled vertices are $A_r^n n^n \leq M_r(n) \leq B_r^n n^n$ for constants $A_r,B_r$ (Mead, 2019).
• Complexes may realize arbitrary homology in degrees $i\le r-2$, contrasting with Kalai's $Q$-acyclic trees, which are topologically much simpler.

Random models and threshold results for connectivity in pure random $r$-complexes show sharp phase transitions at $p^*(n)= \frac{r! \log n}{n^r}$ (Mead, 2019).

5. Alexander Duality, Non-Pure Minimal Complexes, and Taylor Resolutions

Non-homogeneous minimal balls and spheres are defined via the minimality of the number of maximal simplices, with the combinatorial Alexander dual operation playing a central role (Capitelli, 2014). Classification theorems characterize minimal NH-spheres and NH-balls precisely as those whose iterated Alexander duals stabilize to boundaries of simplices or full simplices, respectively. This framework characterizes all minimal non-pure balls and spheres without explicit reference to manifold decompositions.

In commutative algebra, minimal Taylor-resolved simplicial complexes are those whose Stanley-Reisner ring admits a minimal Taylor resolution. Four conditions are equivalent for such complexes: the strong gcd condition on minimal non-faces, Golodness of the face ring, the moment-angle complex being a wedge of spheres, and the desuspension of the BBCG polyhedral product decomposition (Iriye et al., 2015). Pairwise-intersecting minimal non-faces enforce a complete intersection graph, and minimality dictates specific combinatorial configurations.

6. Algorithmic Construction and Computational Approaches

Recent advances enable practical construction of minimal or near-minimal triangulations from CW data. The pipeline involves weak simplicial approximation with generalized barycentric or edgewise subdivisions, Delaunay-based geometric simplifications, edge contractions under link condition, and a compact two-layer mapping-cone gluing process (Tinarrage, 2021). Empirical results show vertex counts for projective spaces and lens spaces within a small factor of known minimal values:

Space	Vertices (before contraction)	Vertices (after contraction)
$P^1$	4	3
$P^2$	11	6
$P^3$	73	15
$P^4$	2664	708

Although provable minimality is not guaranteed, the approach consistently yields compact triangulations amenable to direct calculations of homology and fundamental groups (Tinarrage, 2021).

7. Applications, Recognition Criteria, and Future Directions

Minimal simplicial complexes underpin recognition of PL manifolds: if all vertex-links in a $d$-manifold triangulation are homology spheres with at most 12 (or 13) vertices in dimension 3 (or 4), the manifold must be PL (Govc et al., 4 Nov 2025). For homology spheres, no nontrivial topology is possible below $d+10$ vertices (Govc et al., 4 Nov 2025). Progress on the Björner–Lutz conjecture raises minimality bounds for the Poincaré homology sphere to 13 or 14 vertices, depending on bistellar-flippability.

Understanding minimality phenomena informs algebraic invariants, optimizes computational topology pipelines, and advances enumeration and recognition algorithms for both theoretical and applied purposes. Open problems include extending group classification to $n\geq10$, sharpening bounds for $KW(G)$ on infinite families, and characterizing minimal complexes under broader algebraic and combinatorial conditions.

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The theory and construction of minimal simplicial complexes unify extremal combinatorics, algebraic topology, homological algebra, and geometric group theory, enabling precise enumeration, classification, and computation of topological invariants for spaces and groups. The field remains replete with open problems concerning complexity growth, algebraic structure, and algorithmic minimality.