Strong Formal Subdivision Theory

• Strong formal subdivision is an order-preserving, rank-increasing surjective map between lower Eulerian posets, defined by a strict local Euler characteristic condition.
• It arises from a canonical mapping cylinder construction that establishes a bijection with join-admissible triples, linking combinatorial, topological, and algebraic properties.
• Applications include polytope subdivisions, fan morphisms, and computations of invariants such as the cd-index and local h-polynomials in both standard and equivariant contexts.

A strong formal subdivision is an order-preserving, rank-increasing surjective morphism between lower Eulerian posets that satisfies a strict local Euler characteristic condition. This concept simultaneously abstracts polyhedral subdivisions of polytopes and proper surjective morphisms of fans, encapsulating their combinatorial, topological, and algebraic properties. The framework naturally leads to a canonical bijection with certain join-admissible triples in lower Eulerian posets via the non-Hausdorff mapping cylinder construction, resulting in deep connections to flag invariants, the cd-index, local $h$-polynomials, and Kazhdan–Lusztig–Stanley theory.

1. Lower Eulerian Posets and Strong Formal Subdivisions

A finite poset $P$ is ranked if there exists a rank function $\rho:P\to\mathbb{Z}$ such that $\rho(y) = \rho(x) + 1$ whenever $y$ covers $x$. The poset is locally Eulerian if for every interval $[x,y]$, the number of elements of even rank equals that of odd rank; it is lower Eulerian if, in addition, a unique minimal element $\hat{0}$ exists. A poset is Eulerian if it is lower Eulerian with a unique maximal element $\hat{1}$ (Stapledon, 20 Nov 2025).

Let $X$ and $Y$ be lower Eulerian posets with rank functions $\rho_X$ and $\rho_Y$. A map $\sigma: X\to Y$ is called a strong formal subdivision if it is:

• order-preserving,
• rank-increasing: $\rho_X(x)\le\rho_Y(\sigma(x))$ for all $x\in X$,
• strongly surjective: for every $x\in X$, $y\in Y$ with $\sigma(x)\le y$, there is $x'\ge x$ in $X$ so that $\rho_X(x')=\rho_Y(y)$ and $\sigma(x')=y$,
• and for every $x\in X$ and $y\in Y$ with $\sigma(x)\le y$,

$$\sum_{\substack{x\le x'\in X\\sigma(x')=y}}(-1)^{\rho_Y(y)-\rho_X(x')} = 1.$$

These properties are designed to guarantee that the fibers over points in $Y$ have the right local Euler characteristic, ensuring compatibility with subdivisions arising in geometry or topology (Stapledon, 20 Nov 2025, Stapledon, 20 Nov 2025).

2. The Canonical Cylinder Bijection and Join-Admissible Triples

Each strong formal subdivision $\sigma:X\to Y$ corresponds canonically to a triple $(\Gamma, \rho_\Gamma, q)$, where $\Gamma$ is a lower Eulerian poset with rank function $\rho_\Gamma$ and $q\ne \hat{0}_\Gamma$ is join-admissible (i.e., $z\vee q$ exists for all $z\in\Gamma$). This relationship is established via the non-Hausdorff mapping cylinder construction:

• The mapping cylinder $\mathrm{Cyl}(\sigma)$ is the disjoint union $X\sqcup Y$ with the original orders and additional relations $x\le y$ whenever $x\in X$, $y\in Y$, and $\sigma(x)\le y$.
• The rank function is defined by $\rho_{\mathrm{Cyl}}(z) = \rho_X(z)$ for $z\in X$, $\rho_{\mathrm{Cyl}}(z)=\rho_Y(z)+1$ for $z\in Y$.
• The triple $(\Gamma, \rho_\Gamma, q)$ is $$(\mathrm{Cyl}(\sigma), \rho_{\mathrm{Cyl}}, \hat{0}_Y)$$.

Conversely, starting from such a triple, the strong formal subdivision is recovered as

$$\sigma: \Gamma \setminus \Gamma_{\ge q} \longrightarrow \Gamma_{\ge q},\quad x \mapsto x\vee q$$

with the appropriate (possibly shifted) rank functions. This bijection underpins the combinatorial abstraction of geometric subdivision phenomena and provides a natural vehicle for structural translation across discrete and geometric settings (Stapledon, 20 Nov 2025, Stapledon, 20 Nov 2025).

3. Examples and Structural Significance

Strong formal subdivisions naturally model a range of examples:

• Polytope Subdivisions: A refinement $S'$ of a polyhedral subdivision $S$ of a polytope $P$ induces

$\sigma:\mathrm{face}(S')\to\mathrm{face}(S)$

mapping each cell to the minimal $S$-cell containing it, giving a strong formal subdivision of rank $0$ (Stapledon, 20 Nov 2025, Stapledon, 20 Nov 2025).

• Fan Morphisms: A proper surjective map of fans $\Sigma'\to\Sigma$ induces $\mathrm{face}(\Sigma')\to\mathrm{face}(\Sigma)$, a strong formal subdivision of rank equal to the kernel dimension of the underlying vector space map.
• Identity Map: For any lower Eulerian poset $B$, the identity map $\mathrm{id}_B: B\to B$ is a strong formal subdivision, whose cylinder is the pyramid poset $\mathrm{Pyr}(B)$.
• One-Point Adjunctions: The join-admissible element $q$ in the cylinder construction corresponds combinatorially to joining with an "adjoined" minimum.

The mapping cylinder provides a categorical 'join' operation interpolating between domain and codomain posets, supporting recursion and uniform proofs for invariants (Stapledon, 20 Nov 2025).

4. Invariants: cd-Index and Flag Polynomials

For Eulerian posets, the flag $f$-polynomial

$$\Psi(B;a,b) = \sum_{S\subseteq[n]} f_S\,w_S,\quad w_S = \prod_{i=1}^n [i\in S\mapsto b;\;i\notin S\mapsto (a-b)]$$

encodes face incidence data. The unique polynomial $\Phi(B;c,d)$ in noncommuting $c$ and $d$ satisfying $\Phi(a+b, ab+ba) = \Psi(B;a,b)$ is the cd-index.

Under the strong formal subdivision setting and cylinder construction, the cd-index can be computed recursively:

$$\Phi(\Gamma) = \ell^\Phi(X) + \tfrac{1}{2}\bigl[\Phi(\overline{\partial X})\,c + \Phi\bigl(\overline{X_{\le \hat0_Y}}\bigr)c\,\Phi(Y) + \sum_{\hat0_Y<y<\hat1_Y} \bigl(\ell^\Phi(X_{\le y})c + \Phi(\overline{X_{<y})}d\bigr)\,\Phi([y,\hat1_Y]) \bigr]$$

where $\ell^\Phi(X)$ is the local cd-index, and overbars indicate certain quotient posets (Stapledon, 20 Nov 2025).

This recurrence applies to standard constructions, such as pyramids, prisms, and bipyramids, and demonstrates the algorithmic tractability of computing the cd-index via strong formal subdivision theory.

5. Local $h$-Polynomials and Kazhdan–Lusztig–Stanley Invariants

Given a strong formal subdivision $\phi: P\to Q$ with cylinder $\Gamma$, the local $h$-polynomial $\ell_\phi$ is constructed via the incidence algebras associated to the Eulerian kernel $\kappa(x,y) = (t-1)^{\rho(y)-\rho(x)}$. For $P, Q$ Eulerian,

$\ell_\phi = \ell_\phi(\hat0_P, \hat1_Q)$

is symmetric and unimodal, as proved by Karu.

The main connection to Kazhdan–Lusztig–Stanley (KLS) invariants is as follows. Let $g_\Gamma$, $g_Q$ be the left-KLS functions for $\Gamma$, $Q$ respectively, and define the involution and $\Delta$ operator on symmetric polynomials

$$p^{\mathrm{rev}}(x,y;t) = t^{r_\Gamma(x,y)} p(x,y; t^{-1}),\qquad \Delta p = (p^{\mathrm{rev}} - p)/(t-1)$$

Then,

$$g_\Gamma(x,y) = \sum_{y'} \Delta \ell_\phi(x, y')\, g_Q(y', y)$$

with analogous statements for right-KLS and the $Z$-function (Stapledon, 20 Nov 2025).

This formalism further unifies local $h$-polynomials, KLS functions, and relative $g$-polynomials. For instance, when $Q$ is a polytope face lattice and $P$ arises from a projective subdivision,

$g(Q,F) = \Delta\, \ell_\phi(\hat0_P, \hat1_Q)$

demonstrating that Braden–MacPherson's relative $g$-polynomials coincide with these local $h$-polynomials (Stapledon, 20 Nov 2025).

6. Equivariant Generalizations and Ehrhart Theory Applications

If a finite group $W$ acts compatibly on the posets, all incidence algebra constructions lift to the equivariant setting, resulting in equivariant KLS functions and local $h$-polynomials valued in representation rings. For a $W$-invariant lattice polytope $P$ subdivided by a $W$-invariant subdivision $S$,

$$h^*(P,W;t) = \sum_{F\in S} \frac{|W_F|}{|W|} \,\mathrm{Ind}_{W_F}^W \left( \ell^*(F,W_F;t)\,h_\phi(F, \hat1)\right)$$

where $h^*(P,W;t)$ is the equivariant $h^*$-series, and $\ell^*$ denotes the local equivariant $h^*$-series. For unimodular triangulations, $h^*(P,W;t)$ equals the equivariant $h$-polynomial $h_\phi(\hat0, \hat1)$ (Stapledon, 20 Nov 2025).

This framework unifies and generalizes combinatorial and cohomological results: Stanley's theory of subdivisions, intersection cohomology-based proofs of local $h$ symmetry and unimodality, and the equivariant Braden–MacPherson $g$-polynomials—all subsumed via the strong formal subdivision theory and the canonical cylinder bijection.

7. Summary Table: Key Structures in Strong Formal Subdivision Theory

Structure	Description	Origin/Example
Lower Eulerian poset	Ranked poset with unique $\hat0$, locally Eulerian intervals	Face lattice of a polytope, fan poset
Strong formal subdivision	Order-preserving, rank-increasing surjection satisfying $(*)$	Polytope/fan subdivision morphism
Mapping cylinder $\mathrm{Cyl}(\sigma)$	Poset on $X \sqcup Y$ with induced and bridging order	Used in canonical bijection construction
Join-admissible element $q$	Non-minimum $q$ s.t.\ $z\vee q$ exists for all $z$ in $\Gamma$	Pointed face in a face lattice
Local $h$-polynomial	Derived from fibers of $\sigma$ in incidence algebra; symmetric	Tracks refined face structure
cd-index	Encodes flag $f$-vector information via noncommuting variables $c,d$	Provides recursive invariants
KLS-invariants	Kazhdan–Lusztig–Stanley functions; recover local $h$ via subdivision	Enables intersection cohomology linkage
Equivariant versions	Invariance under group actions; values in representation rings	Ehrhart theory, group-labeled polytopes

Strong formal subdivisions provide a unified combinatorial structure for analyzing subdivisions in discrete geometry and topology. This theory supports explicit calculation of invariants relevant in enumerative, geometric, and representation-theoretic contexts, and facilitates their extension to equivariant and cohomological settings (Stapledon, 20 Nov 2025, Stapledon, 20 Nov 2025).