Geometric Simplicial Complexes

• Geometric simplicial complexes are combinatorial structures built from vertices and higher-order simplices, realized geometrically in spaces like Euclidean space.
• Metric and probabilistic models enrich their study by using L1 metrics and measurable mappings to analyze spectral and operator properties.
• Algebraic frameworks employing Dirac and Hodge operators link these complexes to homological invariants and lattice structures, facilitating robust topological analysis.

A geometric simplicial complex is a combinatorial object constructed from vertices, higher-order simplices (edges, triangles, tetrahedra, etc.), and gluing rules that respect face relations, with a canonical embedding as a subset of a Euclidean space, a metric completion, or a probabilistic or functional representation. These complexes underpin modern topological combinatorics, spectral graph theory, topological data analysis, and discrete geometric analysis. Research on geometric realizations, metric models, topological-lattice structures, operator theory, and algebraic frameworks such as the strong ring emphasizes both foundational aspects and their analytic, probabilistic, and algebraic properties.

1. Construction and Geometric Realizations

A finite abstract simplicial complex $\mathcal{K}$ is a finite collection of nonempty sets (simplices) closed under taking nonempty subsets: if $x\in\mathcal{K}$ and $\varnothing\neq y\subset x$, then $y\in\mathcal{K}$. The combinatorial realization is often encoded as the Whitney complex of a simple graph (the set of all cliques), barycentric refinement, or via the order complex of a finite lattice.

The classical geometric realization $|\mathcal{K}|$ is given as a subcomplex of a Euclidean simplex in $\mathbb{R}^{|S|}$, where $S$ is the vertex set. Each $k$-simplex is realized as the convex hull of its vertices (represented by standard basis vectors). The union forms a polyhedral subset glued along shared faces. However, unless $\mathcal{K}$ is locally finite, $|\mathcal{K}|$ is not metrizable.

In the context of finite lattices, Bergman introduced a construction where the geometric realization of the order complex $\Delta(L)$ can be identified with a space of functions $f:L\to[0,1]$ whose threshold sets $f_t=\{x\mid f(x)\ge t\}$ are principal ideals. This space carries a natural topological-lattice structure with continuous meet and join operations, and can be embedded as a subdirect product of copies of $L$ (Bergman, 2016).

2. Metric and Probabilistic Models

The metric realization of a simplicial complex was further developed by Marin (Marin, 2017) via probabilistic constructions. For a complex $\mathcal{K}$ on vertex set $S$, with $(\Omega,\mu)$ a standard nonatomic probability space, the space $X(\mathcal{K})=L(\Omega,\mathcal{K})$ consists of measurable maps $f:\Omega\to S$ (identified almost everywhere) such that the essential support $\{s\in S \mid \mu(f^{-1}(s))>0\}$ is a simplex in $\mathcal{K}$. The $L^1$-metric is given by $d(f,g)=\mu\{t \mid f(t)\ne g(t)\}$, i.e., the probability of disagreement.

The associated metric completion $\overline{X(\mathcal{K})}$ is the closure of $L(\Omega,\mathcal{K})$ in $L^1(\Omega,S)$, consisting of all $S$-valued random variables whose essential supports have every finite subset contained in $\mathcal{K}$. The "law" (or mass-function) map $\Psi:L(\Omega,\mathcal{K})\to|\mathcal{K}|_1$ sends $f$ to its empirical distribution (probability mass function), with $|\mathcal{K}|_1$ the $\ell^1$-metric simplex of mass functions supported on simplices of $\mathcal{K}$.

$\Psi$ is a Serre fibration and a weak homotopy equivalence, with contractible infinite-dimensional fibers corresponding to the group of measure-preserving permutations of $\Omega$. The metric space $X(\mathcal{K})$ thus forms a contractible "fibered thickening" of the standard simplex, endowing geometric realization with a canonical metric and a probabilistic interpretation (Marin, 2017).

3. Algebraic Structures and Homological Invariants

A systematic algebraic framework for the manipulation and analysis of simplicial complexes is provided by the strong ring $R$ (Knill, 2017). $R$ is generated by finite abstract simplicial complexes with operations:

• Addition ($\oplus$): disjoint union (Grothendieck group).
• Multiplication ($\times$): set-theoretic product of simplices.
• The ring $R$ embeds as a subring of the Stanley–Reisner ring and as a subring of the strong Sabidussi ring.

To each $G\in R$ one associates:

• Dirac operator $D$: $D = d + d^*$, where $d$ is the (signed) boundary operator on the chain complex.
• Hodge Laplacian $H$: $H = D^2 = dd^* + d^*d$, with blocks $H_k$ acting on $k$-forms.
• Connection matrix $L$: $L = I + A_{G'}$, with $A_{G'}$ the adjacency matrix of the connection graph (vertices=simplices, edges=nontrivial intersection). $L$ is always unimodular.

Ring and spectral properties include:

• Betti numbers $b_k(G) = \dim\ker H_k$, extended linearly to all $G\in R$.
• Poincaré polynomial $p_G(t) = \sum_{k\ge 0} b_k(G)t^k$ is a ring homomorphism $R\to\mathbb{Z}[t]$.
• Euler characteristic $\chi(G)=e_G(-1)$, with $e_G(t)=\sum_k v_k(G)t^k$ the $f$-vector.
• Wu characteristics as multilinear ring homomorphisms.
• Spectra of $H$ add: $\sigma(H(G\times H))=\sigma(H(G))+\sigma(H(H))$.
• Spectra of $L$ multiply: $$\sigma(L(G\times H))=\sigma(L(G))\cdot \sigma(L(H))$$.

The Künneth theorem is explicit: $$H^n(G\times H)\simeq\oplus_{i+j=n}H^i(G)\otimes H^j(H)$$. Gauss–Bonnet, Poincaré–Hopf, and Brouwer–Lefschetz theorems extend functorially to $R$ (Knill, 2017).

4. Discrete Hodge Theory and Essential Self-Adjointness

On weighted $n$-simplicial complexes $(V, m_i)$ (with vertex, edge, and higher simplex weights), the discrete exterior derivative and its adjoint define the Gauss–Bonnet operator $D = d + \delta$, and the discrete Hodge Laplacian $L = D^2 = \bigoplus_{i=0}^n L_i$ with $L_i = \delta_i d_i + d_{i-1}\delta_{i-1}$. Hilbert spaces of cochains $\ell^2(m_i)$ allow $D$ and $L$ to act as unbounded operators.

A geometric criterion for essential self-adjointness (ESA) of these operators is $\chi$-completeness: the existence of an exhaustion by finite sets and cut-off functions of uniformly bounded discrete energy. Global $\chi$-completeness guarantees ESA for all $L_i$; local $\chi$-completeness at level $\ell$ suffices for block $L_\ell$; strong ESA can also follow by a divergence estimate without $\chi$-completeness (Ennaceur et al., 21 Oct 2025).

Concrete examples include:

• Lattices and their perturbations (e.g., $\mathbb{Z}^d$ and its subcomplexes).
• Alternating triangulations and truncated simplicial trees.
• Weighted trees with superlinear offspring growths meeting the divergence (but not $\chi$-completeness) criterion.

The machinery connects combinatorial completeness to spectral properties analogous to those in Riemannian geometry.

5. Random and Topological Models

Stochastic geometry provides a probabilistic approach to geometric simplicial complexes by considering random configurations, typically Poisson point processes on a torus or Euclidean domain. The associated Čech complex $\mathcal{C}_\epsilon(\omega)$ is constructed by placing balls of radius $\epsilon$ at each random point; a simplex corresponds to a nonempty intersection of such balls.

Explicit formulas for high moments of simplex counts, as well as the mean and variance of the Euler characteristic, are available via Malliavin calculus. In high-intensity regimes ($\lambda\to\infty$ for intensity $\lambda$), Betti numbers and homological invariants concentrate around the deterministic values of the underlying space (e.g., the torus), with Gaussian fluctuations of order $\lambda^{-1/2}$ (Decreusefond et al., 2011). Central limit theorems and sharp concentration inequalities for Betti numbers and the Euler characteristic quantify the extent to which topological properties are faithfully recovered in the thermodynamic limit.

6. Lattice Structures and Topological Lattices

For a finite lattice $L$, its order complex $\Delta(L)$ can be geometrically realized as a simplicial complex in $\mathbb{R}^L$, whose space of functions $f:L\to[0,1]$ with suitable threshold properties forms a topological lattice via pointwise $\min/\max$ (meet/join) and principal ideals.

Key structural properties include:

• $\Delta(L)$ is modular/distributive if $L$ is, but there exist geometric complexes (e.g., $\Delta(M_3)$ for $M_3$ the modular non-distributive lattice) which are modular but not distributive as topological lattices (Bergman, 2016).
• The union of three 2-simplices ("triangles") along a common edge yields a 2-dimensional complex not embeddable in the plane, showing nonexistence of a distributive lattice structure.
• Variants include general closure operators, tensor-style extensions, up-down symmetric versions, thickened ("fattened") versions, and "stitching" constructions assembling lattices along a common chain.
• Counterexamples demonstrate the failure of distributivity, compactness, and planarity under various generalizations.

These constructions highlight deep connections between combinatorial, geometric, and lattice-theoretic properties of geometric simplicial complexes.

7. Infinite Lattice Limits and Analytic Features

The behavior of geometric simplicial complexes under infinite-volume limits (e.g., approximating $\mathbb{Z}^d$ via products of cycles) reveals stability and spectral gaps in operator theory. The connection Laplacian $L$ remains invertible with a uniform mass gap, ensuring the existence of almost-periodic Green's functions, unique solutions to the Poisson equation $Lu=\rho$, and stable PDE behavior for nonlinear or perturbed discrete equations (Knill, 2017). This analysis informs both foundational spectral theory and applications in discrete geometric analysis.

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References:

• (Marin, 2017) "Simplicial Random Variables"
• (Ennaceur et al., 21 Oct 2025) "Geometric Criteria for Essential Self-Adjointness of Discrete Hodge Laplacians on Weighted Simplicial Complexes"
• (Knill, 2017) "The strong ring of simplicial complexes"
• (Decreusefond et al., 2011) "Simplicial Homology of Random Configurations"
• (Bergman, 2016) "Simplicial complexes with lattice structures"