Simplicially Stable Spaces

• Simplicially stable spaces are moduli constructions defined by using simplicial complexes to encode stability conditions and compactify nodal curves.
• They integrate combinatorial data with Bridgeland stability, yielding explicit PL-homeomorphisms that capture wall-crossing phenomena in categorical settings.
• This framework underpins robust methods for computing cohomological invariants and offers new perspectives on the topology and stratification of moduli spaces.

Simplicially stable spaces are a class of moduli spaces and categorical constructions distinguished by their encoding of stability conditions and their intricate interactions with combinatorics, algebraic geometry, and homological algebra. Emerging from the interplay between Bridgeland stability conditions on triangulated categories and combinatorial data such as simplicial complexes, these spaces provide alternative compactifications and topological models generalizing classical moduli spaces and cluster complexes, with profound implications for the structure of moduli, wall-crossing phenomena, and cohomological invariants.

1. Formal Definitions and Foundational Constructions

A simplicially stable space is defined via the data of a finite set $S$ and an abstract simplicial complex $\mathcal{K}$ on $S$ (Newman, 23 Jan 2026). The moduli stack $\overline{M}_{g,\mathcal{K}}$ is the Deligne–Mumford compactification parametrizing families $(\pi: C \to T, \{p_s: T \to C\}_{s \in S})$ of nodal, pointed curves of genus $g$ such that, for each geometric fiber $C_t$:

• Any set of labels colliding at a point of $C_t$ forms a face of $\mathcal{K}$.
• Each rational tail carries a marking-set not contained in $\mathcal{K}$.

For $g=0$ and $\mathcal{K}$ triparted (every partition into faces has at least three parts), $\overline{M}_{0,\mathcal{K}}$ is a smooth, proper variety. The construction recovers Hassett’s moduli spaces of weighted stable curves by choosing rational weights $0<a_s \leq 1$ and setting $$\mathcal{K}_\mathbf{a} = \{I \subseteq S ~|~ \sum_{s \in I} a_s < 1\}$$, so that $$\overline{M}_{g,\mathcal{K}_\mathbf{a}} \simeq \overline{M}_{g,\mathbf{a}}$$ (Newman, 23 Jan 2026).

2. Simplicial Complexes and Categorical Stability

Simplicially stable spaces generalize the combinatorial structure of stability conditions on triangulated categories, most notably 2-Calabi–Yau categories. Under a Bridgeland stability condition $\sigma=(Z,\mathcal{P})$ on a triangulated category $\mathcal{C}$, the objects of interest are the $\sigma$–semistable spherical objects, leading to the construction of a simplicial complex $\Sigma_\sigma$ with the vertex set $$V_\sigma = \{\text{semistable spherical objects in }\mathcal{C}\}/[1]$$ (Bapat et al., 17 Sep 2025). A face $F \subset V_\sigma$ is present if and only if there exists a spherical object $X$ whose Harder–Narasimhan (HN) filtration has semistable factors exactly given by $F$.

This construction reflects the compatibility conditions for stability filtrations and establishes a complex encoding the relations among stable objects within the category. As the stability condition varies, the combinatorial structure of $\Sigma_\sigma$ records the wall-crossing behavior of stable objects, forming a system of piecewise-linear (PL) spheres or balls tracking semistable sphericals and their HN compatibility (Bapat et al., 17 Sep 2025).

3. Boundary Stratification and Moduli-theoretic Aspects

The boundary of $\overline{M}_{0,\mathcal{K}}$ is stratified via $\mathcal{K}$–stable graphs. Such a graph $\Gamma$ consists of a tree with vertices labeled by genera (all zero in genus $0$), half-edges, and a labeling $\ell: S \rightarrow$ legs of $\Gamma$. The data must satisfy:

• For each leg, the set of coinciding labels is a face of $\mathcal{K}$.
• Each vertex has valence $n(v)\geq 3$ and satisfies $2g(v)-2+n(v)\geq 0$.
• Rational tails cannot have label multiplets forming a face of $\mathcal{K}$.

The codimension of a stratum $\overline{M}^\Gamma$ is $|E(\Gamma)| + (|S| - \#\text{legs of }\Gamma)$. Divisorial boundary strata ($\operatorname{codim}=1$) are of two types: partition divisors (corresponding to a bi-partition of $S$ not in $\mathcal{K}$) and collision divisors (points with multiple coincident labels in a face of $\mathcal{K}$). These combinatorial structures govern the geometry and intersection theory of the space (Newman, 23 Jan 2026).

4. Wall-Crossing and Piecewise-Linear Manifold Structure

For a fixed stability condition, the simplicial complex $\Sigma_\sigma$ is determined by the stable objects and their HN decompositions. As $\sigma$ traverses the stability manifold $\operatorname{Stab}(\mathcal{C})$, the set of semistable sphericals $V_\sigma$ remains fixed except at real codimension-one walls, where the set jumps. Each wall-crossing induces a canonical PL homeomorphism $\kappa: \Sigma_\sigma \rightarrow \Sigma_{\sigma'}$ constructed by local PL subdivisions via the “splitting” and “fusing” of chains of sphericals, and gluing the local rules to a global homeomorphism (Bapat et al., 17 Sep 2025).

In the type $A_n$ 2-Calabi–Yau category, there exists an explicit combinatorial identification: $\Sigma_\sigma$ coincides with the complex $K^*(\mathbf{a})$ of pointed pseudo-triangulations of an $(n+1)$-point planar configuration $\mathbf{a}$. Consequently, $\Sigma_\sigma$ is PL-homeomorphic to $S^{2n-4}$, the boundary of an iterated cone over the cluster complex, confirming the spherical PL-manifold structure of these complexes (Bapat et al., 17 Sep 2025).

5. Algebraic Properties: Chow Rings and Generators

The Chow ring $\mathrm{CH}^*(\overline{M}_{0,\mathcal{K}})$ is canonically presented by divisorial classes indexed by $\mathcal{K}$–stable graphs of codimension $1$. Generators consist of partition divisors $D_I$ ($I \notin \mathcal{K}$, $|I| \geq 2$), and collision divisors $E_{st}$ ($\{s,t\} \in \mathcal{K}$, $s\neq t$). The relations are of two kinds:

• Empty-intersection relations: products of divisors vanish if the corresponding strata are disjoint.
• WDVV relations: for every four distinct labels $i,j,k,\ell$, the sum over divisors separating $\{i,j\}$ from $\{k,\ell\}$ equals that over divisors separating $\{i,k\}$ from $\{j,\ell\}$.

This structure specializes to Keel’s presentation for $\overline{M}_{0,n}$ in the case $\mathcal{K}$ is discrete (no collisions). For more general $\mathcal{K}$, the relations and ring structure interpolate continuously between the classical and full collision (pairwise) cases, allowing for explicit computation in a wide range of moduli settings (Newman, 23 Jan 2026).

6. Simplicial Stability in Homological and Topological Context

The notion of “simplicial stability” also arises in topological data analysis and simplicial homology, where it refers to the resistance of Betti numbers to small perturbations of weights or connections in the complex (Guglielmi et al., 2023). Structural stability is quantified as the smallest perturbation of weights needed to force a change (typically an increase) in the dimension of a homology group. Algorithmic analysis is cast as a spectral matrix-nearness problem for the higher-order Laplacians, solved via a bilevel (outer/inner level) gradient-flow algorithm.

This approach identifies critical simplices whose perturbation leads to topological change, providing insight into the robustness or fragility of combinatorial and geometric structures, and reflecting a closely related notion of “stability” in the sense of persistence and resilience to degenerations (Guglielmi et al., 2023).

7. Perspectives and Paradigmatic Role

The paradigm embodied by simplicially stable spaces is that categorical or moduli-theoretic stability data can be encoded and transported via simplicial complexes, whose topology evolves canonically under wall-crossing induced by parameter variation. In 2-Calabi–Yau and related settings, the parameter space of stability conditions gives rise to a family of PL spheres, whose combinatorics mirrors the evolution of semistable objects and their categorical (or geometric) compatibilities (Bapat et al., 17 Sep 2025, Newman, 23 Jan 2026).

This synthesis ties together combinatorial topology (pseudo-triangulation complexes), algebraic geometry (moduli of curves with collisions), and categorical representation theory (Bridgeland stability, spherical objects), providing both unifying language and powerful computational tools for understanding stability phenomena and their invariants in geometry and topology. The field remains open to further generalization to higher dimensions, non-Calabi–Yau categories, and broader classes of moduli spaces.

---

Key references:

• “A sphere of spherical objects” (Bapat et al., 17 Sep 2025)
• “Chow rings of moduli spaces of genus 0 curves with collisions” (Newman, 23 Jan 2026)
• “Quantifying the structural stability of simplicial homology” (Guglielmi et al., 2023)