Poset-Based Connected Manifolds

• Poset-based connected manifolds are countable, locally-finite posets that satisfy recursive local-manifold conditions via combinatorial neighborhoods and links.
• They distinguish border elements by having non-discrete local neighborhoods, yielding a stratified structure analogous to manifolds with boundary.
• The framework establishes equivalences among n-PCM’s, discrete surfaces, and normal pseudomanifolds, guiding efficient algorithms for local link verification.

Searching arXiv for recent and foundational papers on poset-based connected manifolds and closely related terminology. Poset-based connected manifolds, sometimes called $n$-PCM’s, are countable, locally-finite posets equipped with a recursive local-manifold condition formulated in terms of combinatorial neighborhoods and links. In the formulation explained by Boutry, an $n$-PCM is explicitly a boundary-bearing object: interior elements have local neighborhoods that are discrete $(n-1)$-surfaces, while border elements have local neighborhoods that are $(n-1)$-PCM’s. This places the notion between discrete surfaces from mathematical morphology and normal pseudomanifolds from discrete geometry and topological data analysis, while also showing that even simplicial $n$-PCM’s need not be smooth without additional local hypotheses (Boutry, 3 Aug 2025).

1. Formal definition

Let $|X|=(X,\alpha_X)$ be a countable, locally-finite poset. Its rank is defined by

$\rk(|X|)=\max_{h\in X}\{\rk(h,|X|)\},$

the largest length of a chain in $X$. For each $h\in X$, the combinatorial neighborhood is

$${}_X(h)=\{\,h'\in X\mid h\in \alpha_X(h')\text{ or }h'\in \alpha_X(h)\,\},$$

with lower link $n$0 and upper link $n$1. The source further records the elementary rank relations

$n$2

It also states that, by induction, $n$3 has rank $n$4 when $n$5 has rank $n$6 (Boutry, 3 Aug 2025).

The border is defined for a poset of rank $n$7 by

$n$8

and the interior is $n$9, denoted $(n-1)$0. A discrete $(n-1)$1-surface is a CF-order of rank $(n-1)$2 in which every point’s lower link is a $(n-1)$3-surface, with the empty order declared to be the unique $(n-1)$4-surface and two points with no order between them the unique $(n-1)$5-surface.

With this preparation, the definition of an $(n-1)$6-PCM is recursive. If $(n-1)$7, then $(n-1)$8 and $(n-1)$9 is a $(n-1)$0-PCM. If $(n-1)$1, then $(n-1)$2 is a single element, and $(n-1)$3 is a $(n-1)$4-PCM. If $(n-1)$5, then $(n-1)$6 is an $(n-1)$7-PCM exactly when it is path-connected, its border satisfies $(n-1)$8, and for every $(n-1)$9,

$n$0

2. Border, interior, and recursive local structure

The border condition is not auxiliary; it is built into the definition. An $n$1-PCM is therefore not the borderless analogue of a manifold-like object. Rather, the borderless case is separated out and treated as a discrete $n$2-surface. This distinction is central to the later equivalence with normal pseudomanifolds with boundary versus without boundary (Boutry, 3 Aug 2025).

The recursive formulation has immediate structural consequences. The source states that any $n$3-PCM is pure, meaning that all maximal chains have length $n$4. It is also homogeneous, in the sense that all ranks $n$5 occur in every local neighborhood. In addition, the lower link of a point has the expected ranks on each side of that point. These properties are obtained by induction from the defining local condition on $n$6.

A common misconception is to treat the border merely as a set of exceptional points. In this framework, that interpretation is too weak. Border points are precisely those at which the neighborhood is not a discrete $n$7-surface but instead an $n$8-PCM, so the border is itself defined by a recursive manifold-with-boundary condition. This makes the theory intrinsically local and stratified rather than merely global.

3. Equivalence with discrete surfaces and normal pseudomanifolds

Two theorems organize the relation between $n$9-PCM’s, discrete surfaces, and normal pseudomanifolds. Let $|X|=(X,\alpha_X)$0 be a pure simplicial complex of dimension $|X|=(X,\alpha_X)$1. Then the source gives the equivalences

$|X|=(X,\alpha_X)$2

and

$|X|=(X,\alpha_X)$3

Moreover, when $|X|=(X,\alpha_X)$4, exactly those vertices in $|X|=(X,\alpha_X)$5 play the role of boundary faces of the pseudomanifold (Boutry, 3 Aug 2025).

The proof sketch proceeds in both directions. Starting from a discrete $|X|=(X,\alpha_X)$6-surface or an $|X|=(X,\alpha_X)$7-PCM, one first derives purity and the local incidence condition that each $|X|=(X,\alpha_X)$8-face belongs to one or two $|X|=(X,\alpha_X)$9-faces, distinguishing the interior and border cases. To obtain $\rk(|X|)=\max_{h\in X}\{\rk(h,|X|)\},$0-connectivity, the argument takes a path of $\rk(|X|)=\max_{h\in X}\{\rk(h,|X|)\},$1-simplices and homogenizes it into an $\rk(|X|)=\max_{h\in X}\{\rk(h,|X|)\},$2-path by systematically replacing low-dimensional vertices by short paths through their connected link; this works because every lower link is connected.

Conversely, starting from a normal pseudomanifold, one already has purity, the incidence condition on $\rk(|X|)=\max_{h\in X}\{\rk(h,|X|)\},$3-faces, and $\rk(|X|)=\max_{h\in X}\{\rk(h,|X|)\},$4-connectivity. The source then checks that every link of a face of codimension at least $\rk(|X|)=\max_{h\in X}\{\rk(h,|X|)\},$5 is again a pseudomanifold, hence connected, and uses induction to recover the local alternatives required in the definitions of a discrete $\rk(|X|)=\max_{h\in X}\{\rk(h,|X|)\},$6-surface and an $\rk(|X|)=\max_{h\in X}\{\rk(h,|X|)\},$7-PCM. In this sense, the simplicial theory is not merely analogous to normal pseudomanifolds; it coincides with it.

4. Connectivity and local-decision methodology

One algorithmic consequence is stated explicitly: to decide whether a pure simplicial complex $\rk(|X|)=\max_{h\in X}\{\rk(h,|X|)\},$8 of dimension $\rk(|X|)=\max_{h\in X}\{\rk(h,|X|)\},$9 is a normal pseudomanifold, with or without boundary, it suffices to verify the local link-conditions around each face, rather than to attempt the recursive blow-up tests of discrete surfaces or PCM’s (Boutry, 3 Aug 2025).

The source also describes a schematic procedure proving $X$0-connectivity of any $X$1-PCM or discrete $X$2-surface. Given a path of $X$3-simplices joining two facets, one first replaces every even-indexed simplex by a neighboring top-dimensional one, using purity. One then successively removes each low-dimensional “dip” $X$4 by passing through the connected link $X$5. This raises the minimal dimension appearing in the path until all simplices lie in ranks $X$6, thereby producing a true $X$7-path.

This local-to-global behavior is significant because it shows that the manifold-like connectivity is not imposed as an external axiom at every level. Instead, it is recovered from recursive neighborhood conditions. A plausible implication is that the theory is especially well-suited to settings where local incidence data are easier to compute than global topological invariants.

5. Failure of automatic smoothness

Even under the additional assumption that an $X$8-PCM is simplicial, smoothness does not follow automatically. The source identifies a concrete counterexample in dimension $X$9. One starts with a simplicial $h\in X$0-PCM $h\in X$1 that is topologically an annulus, described as a “simplicial band,” whose boundary consists of two disjoint $h\in X$2-cycles. One then forms the simplicial join with a singleton $h\in X$3, obtaining

$h\in X$4

The result is a simplicial $h\in X$5-PCM: every vertex of $h\in X$6 has either link a $h\in X$7-surface, for interior vertices, or link the $h\in X$8-PCM $h\in X$9, at $${}_X(h)=\{\,h'\in X\mid h\in \alpha_X(h')\text{ or }h'\in \alpha_X(h)\,\},$$0 itself (Boutry, 3 Aug 2025).

Nonetheless, the boundary of $${}_X(h)=\{\,h'\in X\mid h\in \alpha_X(h')\text{ or }h'\in \alpha_X(h)\,\},$$1 is exactly $${}_X(h)=\{\,h'\in X\mid h\in \alpha_X(h')\text{ or }h'\in \alpha_X(h)\,\},$$2, and $${}_X(h)=\{\,h'\in X\mid h\in \alpha_X(h')\text{ or }h'\in \alpha_X(h)\,\},$$3 is not connected: it is the union of two disjoint $${}_X(h)=\{\,h'\in X\mid h\in \alpha_X(h')\text{ or }h'\in \alpha_X(h)\,\},$$4-surfaces. Hence $${}_X(h)=\{\,h'\in X\mid h\in \alpha_X(h')\text{ or }h'\in \alpha_X(h)\,\},$$5 is not a smooth $${}_X(h)=\{\,h'\in X\mid h\in \alpha_X(h')\text{ or }h'\in \alpha_X(h)\,\},$$6-PCM. The source describes the geometry as a “pinched simplicial box,” with the central vertex $${}_X(h)=\{\,h'\in X\mid h\in \alpha_X(h')\text{ or }h'\in \alpha_X(h)\,\},$$7 having a link consisting of two separate loops. In that description, $${}_X(h)=\{\,h'\in X\mid h\in \alpha_X(h')\text{ or }h'\in \alpha_X(h)\,\},$$8 is a genuine pinch point, and the pinch prevents the boundary from being a single discrete $${}_X(h)=\{\,h'\in X\mid h\in \alpha_X(h')\text{ or }h'\in \alpha_X(h)\,\},$$9-surface.

This example resolves a possible misunderstanding. Adding simplicial-complex structure and many standard topological properties does not force the boundary behavior expected from smooth manifold intuition. The obstruction is local and is expressed by the disconnectedness pattern visible in the relevant link.

6. Sufficient local criterion for smoothness

Boutry’s sufficient condition for smoothness is a border-local surface condition. For a simplicial $n$00-PCM $n$01 of rank $n$02, the hypothesis is

$n$03

The source glosses this as the requirement that the border of any neighborhood is already locally a discrete $n$04-surface (Boutry, 3 Aug 2025).

Under condition $n$05, the theorem states that $n$06 is a smooth $n$07-PCM. In particular, the boundary $n$08 is itself a discrete $n$09-surface, or a separated union of such, and locally at each $n$10 the link is a smooth $n$11-PCM or $n$12-surface. The paper also adds the low-dimensional remarks that any $n$13- or $n$14-PCM is automatically smooth, and that every simplicial $n$15-PCM is a smooth $n$16-PCM, being just an open path whose boundary is two distinct vertices.

The criterion is notable because it isolates exactly what the counterexample violates. The issue is not simpliciality, purity, or the basic PCM recursion by themselves, but the absence of a sufficiently strong local surface condition along the border. This suggests that smoothness, in this framework, is a second-order local property: it depends not only on neighborhoods in $n$17 but on neighborhoods internal to $n$18.

7. Related poset-topological contexts and terminological cautions

The terminology “connected” in poset-based connected manifolds should not be conflated with the role of connected opens in manifold calculus. Pryor, refining Weiss’s embedding-calculus framework, shows that when $n$19, one may take $n$20, the full subposet of connected opens, and a good cofunctor is linear, i.e. polynomial of degree $n$21, exactly when it is determined up to equivalence by its restriction to connected opens. There, however, the underlying objects are open subsets of a smooth manifold ordered by inclusion, not manifold-like objects realized as posets in the Boutry sense (Pryor, 2013).

A different neighboring use of poset topology appears in the work of Gaetz and Hersh on Bruhat interval polytope lattices. For intervals $n$22, the order complex $n$23 is homotopy equivalent to a sphere when $n$24 is a face of $n$25 and contractible otherwise; the source further states that this produces, for each interval, a piecewise-linear sphere or ball, and that these can be glued along common subintervals to build larger combinatorial manifolds encoded by the poset (Gaetz et al., 2024).

These adjacent developments clarify the scope of the $n$26-PCM notion. Poset-based connected manifolds are one specific framework in which manifold-like behavior is extracted from recursive local poset data. Other poset-based theories also produce sphere-, ball-, or manifold-like structures, but they do so from different inputs: open-set posets in manifold calculus, and interval lattices or order complexes in Bruhat polytope topology. This suggests a broader landscape of “poset-local” topology, with $n$27-PCM’s occupying the branch most directly tied to discrete surfaces, mathematical morphology, and normal pseudomanifolds.