$f$-Polynomials, $h$-Polynomials, and $l^2$-Euler Characteristics

We introduce a many-variable version of the f-polynomial and h-polynomial associated to a finite simplicial complex. In this context the h-polynomial is actually a rational function. We establish connections with the l2-Euler characteristic of right-angled buildings. When L is a triangulation of a sphere we obtain a new formula for the l2-Euler characteristic.


f -polynomials and h-polynomials
The f -polynomial f (t) = f L (t) and the h-polynomial h(t) = h L (t) are defined by: Replacing t by t − 1 in the equation defining the h-vector and multiplying each side by t m we see that the relation between the f -polynomial

D. Boros
and h-polynomial can be written as: or, by replacing t by −t as: It is h(−t) that is of interest in l 2 -topology.
For example, the f -polynomial and h-polynomial for a triangulation of a 1-sphere as a 4-gon are: We now proceed to define the f -polynomial and h-polynomial in several variables. Given a finite simplicial complex L as above, denote by S(L) the set of simplices in L together with the empty set ∅. Let v 1 , v 2 , . . . , v n be the vertices of L and t = (t 1 , t 2 , . . . , t n ). If σ ∈ S(L) let We define the monomials: Similarly, we define: In this several variables context the correct version of the f -polynomial f (t) = f L (t) is defined by: For example, if L is a 4-gon then t = (t 1 , t 2 , t 3 , t 4 ) and When t 1 = t 2 = · · · = t m = t we obtain the one variable version of the f -polynomial but This version of the "h-polynomial" is useful for applications in l 2topology. In combinatorics, H(−t) could be of interest.

h-polynomials and l 2 -Euler characteristics
Associated to a finite simplicial complex L we have a right-angled Coxeter system (W L , S) and two other simplicial complexes, K L and the Davis complex Σ L . The right-angled Coxeter group associated to L, denoted W L , is defined as follows. The set of generators S is the vertex set of L and the edges of L give relations: s 2 = 1 and (st) 2 = 1, whenever {s, t} spans an edge in L. Note that W L depends only on the 1-skeleton of L. K L is the cone on the barycentric subdivision of L. Equivalently, K L can be viewed as the geometric realization of S(L), where S(L) denotes the poset of vertex sets of simplices of L (including the empty simplex). For each s ∈ S, K s is the closed star of the vertex corresponding to s in the barycentric subdivision of L. Equivalently, K s can be viewed as the geometric realization of the poset S(L) ≥s . Σ L is constructed by pasting together copies of K L along the mirrors K s , one copy of K L for each element of W L . For more details on this construction we refer the reader to Chapter 5 of [3]. Given t = (t i ) i∈S , where t i are positive integers, one can define a group G L and the Davis realization Σ(t, L) of a right-angled building of thickness t (whose apartments are copies of Σ L ). The group G L is defined as follows: Suppose we are given a family of groups (P i ) i∈S such that each P i is a cyclic group of order t i + 1. G L is defined as the graph product of the (P i ) i∈S with respect to L. Note that G L depends only on the 1-skeleton of L and the thickness t. Σ(t, L) is obtained by pasting together copies of K L , one for each element of G L , using the same construction used for defining Σ L . Explicitly, given x ∈ K L , let S(x) be the set of s ∈ S such that x belongs to K s . Then Σ(t, L) is the quotient of G L × K L by the equivalence relation ∼, where (g, x) ∼ (g ′ , x ′ ) if and only if x = x ′ and g, g ′ belong to the same coset of G S(x) . In this definition, Σ(t, L) is a simplicial complex. There is one orbit of simplices for each simplex σ in K L .
Let L be a finite simplicial complex and let G the group associated to Σ(t, L). One can define the l 2 -Euler characteristic by where the sum runs over all orbits of simplices in Σ(t, L) and G σ denotes the stabilizer of the simplex σ. Equivalently, the sum runs over all simplices σ of K L .
Lemma 2.1. Let X be an index set with n elements. Then: Proof: Dividing each side by in the formula from Lemma 3.1 we get Theorem 2.2. Let L be a finite simplicial complex and Σ(t, L) the Davis realization of a right-angled building of thickness t. Then: Proof: To prove this formula we divide K L into subcomplexes (not disjoint), namely the geometric realization of S(L) ≥T , where T ∈ S(L) and denoted K T . Each such subcomplex groups together simplices in K L that have the same isotropy group. Let B T be the subset consisting of the union of open simplices in one such subcomplex. The B T are disjoint. Let C T be the orbit of B T under G T . Then the orbits of C T partition Σ(t, L). If we take the signed sum of the simplices in C T , divided by the order of their isotropy subgroup, we get the following alternative formula for the l 2 -Euler characteristic (compare [2]).
where ∂K T denotes the boundary complex of K T , χ(∂K T ) denotes the ordinary Euler characteristic of ∂K T and |G T | denotes the order of the finite isotropy group G T . Note that there is a one-to-one correspondence where each T corresponds to a simplex σ in L.
On the other hand, in the formula that defines H(t), the sum runs over all simplices of L. Each term −t can be written as according to the previous lemma. The sum runs over all subsets of I(σ) including the empty set.
We sum now over all simplices σ in L and grouping terms with the same denominator it is easily seen that H(t) coincides with the alternative formula for χ t (L) written above.

Convex polytopes and l 2 -Euler characteristics
We now restrict our attention to triangulations of spheres which are dual to simple polytopes. We prove another formula for the l 2 -Euler characteristic but here we take a dual approach. Let P n be an n-dimensional simple convex polytope (an n-dimensional convex polytope is simple if the number of codimension-one faces meeting at each vertex is n; equivalently, P n is simple if the dual of its boundary complex is an (n − 1)-dimensional simplicial complex). For more on convex polytopes we refer the reader to [1]. The f -and h-polynomials associated to a finite simplicial complex, as well as their many-variable analogues, were defined in Section 1. Similarly, for an n-dimensional simple convex polytope the associated f -and h-polynomial are defined as those associated to the dual of its boundary complex.
Let P n be an n-dimensional simple convex polytope. Let F 1 , F 2 , . . . , F m be the codimension-one faces of P n (also called facets). Let F denote the set of all faces of P n and t = (t 1 , t 2 , . . . , t m ). If F ∈ F, then I(F ) = {i|F ⊂ F i }. We define: Similarly, we define: We now introduce a different formula for the h-polynomial in several variables. The vertices and edges of a polytope form in an obvious way a nonoriented graph. Following [1, pp. 93-96] we introduce an orientation on the edges of P n using an admissible vector. A vector w ∈ R n is called admissible for P n if x, w = y, w for any two vertices x and y of P n . Geometrically, this means that no hyperplane in R n with w as a normal contains more than one vertex of P n . It is shown in [1, Theorem 15.1, p. 93] that the set of admissible vectors is dense in R n . Any vector w which is admissible for P n induces an orientation of the edges of P n according to the following rule: An edge determined by vertices x and y is oriented towards x (and away from y) if x, w ≤ y, w .
For each vertex v ∈ P n , denote by F in v ∈ F the face determined by the inward-pointing edges at v, and by F out v ∈ F the face determined by the outward-pointing edges at v. We have I(v) = {i|v ∈ F i }. Moreover: . Using an admissible vector w we now define: .
For example, if P 2 is a 4-gon, then which can be simplified to .
To prove the next theorem we need the following combinatorial lemma. Proof: The following identity is well-known: . . , x). Let u = 1 + t where 1 = (1, . . . , 1) and evaluate the above identity when x = 1.
Theorem 3.2. Let P n be an n-dimensional simple convex polytope and denote by L the dual of its boundary complex. Suppose L is a finite simplicial complex and w is an admissible vector for P n . Then Proof: To prove this formula we use a different "cell" structure on Σ(t, L) obtained with the help of an admissible vector. We refer the reader to [4] for more details concerning this construction. There is one orbit of (open) "cells" for each vertex v ∈ P n . The dimension of C v is dim(F in v ). C v is constructed as follows. Let F in v denote the union of the relative interiors of those faces F ′ ∈ F which are contained in F in v and contain v.
To prove our formula we have to show that the contribution c v of C v to the l 2 -Euler characteristic is exactly For a face F we denote by G F its stabilizer. |G F | denotes the order of the finite group G F . We have: where |G I(v)−I(F ′ ) | denotes i∈I(v)−I(F ′ ) (t i + 1). Since |G v | = (1 + t) v the proof is complete if Written explicitly, the above formula coincides with the identity proved in the previous lemma. Summing over vertices of P n , the proof is completed. Remark 3.3. It follows from the previous theorem that H w (t) does not depend on the choice of the admissible vector w.
The next corollary is a reciprocity statement.