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A Characterization of Poset-Based Connected Manifolds and Discrete Surfaces via Cubically Normal Pseudomanifolds

This paper establishes a fundamental correspondence between the global combinatorial structure of finite regular cubical complexes and poset-based connected manifolds by proving that a complex's face poset is an n-PCM if and only if the complex itself is a cubically normal pseudomanifold, thereby providing a recognition algorithm for embedded voxel complexes.

Original authors: Jihun Bae, Yeonho Bae, Jinglu Hu

Published 2026-08-11

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CC BY 4.0

Original authors: Jihun Bae, Yeonho Bae, Jinglu Hu

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are looking at a digital photograph. To your eyes, it's a smooth picture of a cat or a mountain, but to the computer, it's just a grid of tiny squares (pixels) or cubes (voxels) glued together. In the world of digital geometry, these grids are like LEGO structures. Sometimes, you can build a shape that looks fine from the outside but has a weird, "spiky" secret inside where the pieces don't fit together nicely. These messy spots are called topological singularities, and they can confuse computer programs trying to measure the shape or count its holes.

To fix this, scientists have developed two different ways of thinking about these shapes. The first way is like a local rulebook: "Make sure every single pixel has a nice, tidy neighborhood." This stops the obvious messes but doesn't tell you if the whole structure is a single, connected object. The second way is more like a map of relationships: it ignores the grid entirely and just looks at how the pieces are ordered and connected to each other, like a family tree of the shape's parts. This approach is great for understanding the big picture but sometimes misses the specific rules of the grid. The big question has been: Can we find a perfect translation between these two languages? Can we look at the grid rules and know for sure that the relationship map will be a perfect, smooth shape?

This paper, written by Jihun Bae, Yeonho Bae, and Jinglu Hu, acts as that perfect translator. They discovered a specific set of rules for building with digital cubes that guarantees the resulting shape will be mathematically "perfect" in the relationship map sense. They call these special shapes "cubically normal pseudomanifolds." Think of it as a recipe for a digital cake: if you follow these four specific steps—making sure the cake is the right height everywhere, that the layers connect properly, that the whole thing is one piece, and that the frosting on the inside is smooth—then you are guaranteed to have a cake that is a true, smooth manifold.

The authors proved that if you have a digital structure built from cubes (of dimension 2 or higher), it will form a perfect "poset-based connected manifold" (a fancy term for a shape that behaves like a smooth surface or volume in its relationship map) if and only if it follows these four rules. It's a two-way street: if the shape is perfect, it must have followed the rules; if it followed the rules, it must be perfect. They also found that for simpler shapes (like lines or points), the rules change slightly or don't work the same way, which they explain separately.

Most importantly, this isn't just a theory; the authors turned these rules into a step-by-step checklist. If you have a 3D digital model, like a voxel-based character in a video game, you can run this checklist to get a "pass" or "fail" certificate. If it passes, you know the shape is topologically sound and ready for analysis. If it fails, you know exactly which part of the structure is causing the trouble. This bridges the gap between the messy, pixelated world of digital images and the clean, mathematical world of smooth shapes, giving computer scientists a reliable way to ensure their digital objects are well-behaved.

Technical Summary: A Characterization of Poset-Based Connected Manifolds and Discrete Surfaces via Cubically Normal Pseudomanifolds

Problem Statement
The paper addresses a gap in digital topology between two distinct approaches to characterizing manifold-like structures. The first approach, local well-composedness, imposes pixel- or voxel-local constraints on digital grids to prevent topological singularities but fails to describe the global combinatorial structure of the resulting boundary. The second approach, involving discrete surfaces and poset-based connected manifolds (PCMs), classifies manifold-like structures intrinsically on partially ordered sets (posets) without reference to an embedding.

The central problem is the lack of a precise correspondence between these viewpoints: it is unclear which specific combinatorial and incidence conditions on a finite regular cubical complex $K$ guarantee that its face poset $F(K)$ possesses the desired manifold properties (being an $n$ -PCM or a discrete $n$ -surface).

Methodology
The authors establish a bidirectional characterization linking the order-theoretic properties of the face poset $F(K)$ to the combinatorial properties of the underlying cubical complex $K$ . The methodology relies on:

Definitions: Introducing the concept of a cubically normal pseudomanifold for a finite regular cubical complex $K$ $K$ . This definition decomposes into four specific conditions:
- Purity: Every cell extends to a top-dimensional cell.
- Facet-degree condition: Every codimension-1 cell (facet) is incident to exactly 1 or 2 top-dimensional cells.
- Facet-adjacency connectivity: The graph of top-dimensional cells, connected via shared facets, is connected.
- Link normality: For every cell of codimension $\ge 2$ , the poset of its non-empty proper cofaces (which forms a simplicial complex) is itself a pseudomanifold.
Order-Theoretic Analysis: Utilizing recursive definitions of discrete surfaces and PCMs, which rely on the structure of strict neighborhoods ( $\theta'$ ) in the poset. The authors employ lemmas regarding the decomposition of neighborhoods into ordinal sums of lower and upper sets and the isomorphism between the poset of proper cofaces and the face poset of a simplicial link.
Dimensional Separation: The analysis is stratified by rank ( $n$ $n$ ):
- Rank $n \ge 2$ : The main characterization theorem.
- Rank $n = 1$ : A graph-theoretic dichotomy (non-branching intervals or circles).
- Rank $n = 0$ : A one-directional implication due to degeneracies in the base cases.

Key Contributions

Main Characterization Theorem (Rank $n \ge 2$ ): The paper proves that for a finite regular cubical complex $K$ $K$ of rank $n \ge 2$ $n \geq 2$ , the face poset $F(K)$ $F (K)$ is an $n$ $n$ -PCM or a discrete $n$ $n$ -surface if and only if $K$ $K$ is a cubically normal pseudomanifold.
- This equivalence unifies the intrinsic poset definitions with the incidence structure of cubical complexes.
- It demonstrates that the four combinatorial conditions (purity, facet degree, facet-adjacency connectivity, and link normality) are necessary and sufficient to recover the manifold properties in the poset.
Low-Dimensional Classifications:
- Rank 1: The authors provide a specific classification stating $F(K)$ is a 1-PCM or 1-surface iff $K$ is a non-branching cellular interval or a cellular circle.
- Rank 0: The paper clarifies that the two-way equivalence fails for rank 0; only the implication from a rank-0 cubical pseudomanifold to a 0-PCM holds, while the converse fails for configurations like two disconnected points.
Algorithmic Recognition: The characterization yields a direct algorithmic consequence. Since the four conditions of a cubically normal pseudomanifold are purely combinatorial and local or semi-local, they can be checked algorithmically. The authors illustrate this with a recognition procedure for embedded 3D voxel complexes that issues a structured pass/fail certificate for each condition.

Results

Forward Implication ( $\Rightarrow$ ): If $F(K)$ is an $n$ -PCM or discrete $n$ -surface, then $K$ satisfies purity, the facet-degree condition (1 or 2), facet-adjacency connectivity, and link normality. The proof establishes these properties as conclusions derived from the poset structure, rather than assuming them for $K$ .
Backward Implication ( $\Leftarrow$ ): If $K$ is a cubically normal pseudomanifold, then $F(K)$ satisfies the recursive definitions of an $n$ -PCM or discrete $n$ -surface. This is proven by analyzing the strict neighborhoods of elements in $F(K)$ and showing they decompose correctly into lower-dimensional surfaces or PCMs.
Border Correspondence: The paper establishes a correspondence between the border of the poset ( $\Delta_{F(K)}$ ) and the degree of facets in $K$ . Specifically, a facet $\tau$ belongs to the border of the poset if and only if its degree in $K$ is 1.

Significance and Scope
The paper claims to provide a "direct bridge" between the intrinsic order-theoretic language of poset-based manifolds and the incidence structure of cubical complexes, including those derived from digital images.

Theoretical Unification: It separates local, codimension-one, and global connectivity requirements, making explicit which features of a cubical complex are preserved when viewed through its face poset.
Limitations: The authors explicitly note that the main theorem applies to $n \ge 2$ . The rank 0 and rank 1 cases require separate statements and proofs due to structural degeneracies (e.g., the failure of the two-way converse at rank 0).
Practicality: The work offers a recognition procedure for embedded voxel complexes, allowing for the verification of manifold properties without relying on geometric embedding assumptions, provided the complex is regular and finite.

The paper does not propose new experimental applications or future research directions beyond the immediate algorithmic recognition of voxel complexes; its contribution is strictly the theoretical characterization and the resulting verification method.

Technical Summary: A Characterization of Poset-Based Connected Manifolds and Discrete Surfaces via Cubically Normal Pseudomanifolds

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