An Alexander-type duality for valuations

We prove an Alexander-type duality for valuations for certain subcomplexes in the boundary of polyhedra. These strengthen and simplify results of Stanley (1974) and Miller-Reiner (2005). We give a generalization of Brion's theorem for this relative situation and we discuss the topology of the possible subcomplexes for which the duality relation holds.


Introduction
Let P ⊂ R d be a convex polytope with vertices in Z d and let q ∈ R d . Viewing q as a light source, let B ⊆ ∂P be the collection of points in the boundary of P visible from q -the bright side of P . That is, B is the set of points p ∈ ∂P for which the open segment (q, p) does not meet the relative interior of P . Sticking to these figurative terms, let D be the closure of the set of dark points ∂P \B. Stanley [14] showed that for integral n ≥ 1 the function E P,B (n) := |n · (P \B) ∩ Z d | is the restriction of a univariate polynomial (and, by abuse of notation, identified with that polynomial), and that (1) (−1) dim P E P,B (−n) = |n · (P \D) ∩ Z d | for all n ≥ 1.
By choosing q ∈ relint P , we have that (B, D) = (∅, ∂P ) and (1) reduces to the well-known Ehrhart-Macdonald reciprocity [8]; see [2] for details. The set B ⊆ ∂P is a particular case of what Ehrhart [5,6] calls a reciprocal domain, that is, a domain for which (1) holds.
For a subset S ⊂ R d+1 , the lattice point enumerator of S is the multivariate Laurent series d+1 . If we associate to P the pointed cone C(P ) := cone(P × {1}) ⊂ R d+1 , then F C(P ) (x) records the individual lattice points (a, n) ∈ Z d+1 for which a ∈ nP . Stanley [14,Prop. 8.3] actually proved the stronger result that (2) (−1) dim P F C(P \B) .
The relation (2) holds for general rational pointed polyhedral cones C but not for arbitrary subsets in the boundary of C. To see this, we can choose B as two non-adjacent triangles in the boundary of a 3-dimensional pyramid; one can check that B is not a reciprocal domain. The question which subsets in the boundary of C are reciprocal domains was investigated by Miller and Reiner [10]. They showed that the conditions giving rise to reciprocal domains are topological rather than geometric in nature. Let C ⊂ R d+1 be a rational, pointed polyhedral cone and let ∆ be a full-dimensional subcomplex of the boundary of C, i.e. ∆ is a polyhedral complex induced by a collection of facets of C. Let ∆ be the subcomplex generated by the facets F ∈ ∆. Their result is . If ∆ is a Cohen-Macaulay complex, then The proof of Theorem 1.1 in [10] is given in terms of combinatorial commutative algebra and relies on a connection between lattice point enumerators and Hilbert series of Z d -graded modules.
In this paper we give a simple proof of (1) and (3) that generalizes to a broader class of geometric objects and to valuations other than counting lattice points (see Theorem 3.1). Our proof relies on basic facts from topological combinatorics and, as a byproduct, gives a slightly more general class of complexes for which (1) holds. Like Theorem 1.1, our results are reminiscent of Alexander duality and we will emphasize this relation throughout.
The paper is organized as follows. In Section 2, we recall the notions of Λ-polytopes and valuations as well as (weakly) Cohen-Macaulay complexes. In Section 3 we state and prove an Alexander-duality type relation which contains Thm. 1.1 as a special case. In Section 4, we give a relative version of Brion's theorem which is interesting in its own right and highlights the role played by weakly Cohen-Macaulay complexes. In Section 5 we focus on the topology of full-dimensional (weakly) Cohen-Macaulay complexes in the boundary of spheres. The bright side B of P is homeomorphic to a ball of dimension dim P − 1 and thus Cohen-Macaulay. A natural question, which was answered affirmatively in [10], is if there exist full-dimensional Cohen-Macaulay complexes in the boundary of polytopes that are not balls. We will extend this result and we discuss possibly counterintuitive instances for which (1) and (3) apply.

Λ-polytopes, valuations, and weakly Cohen-Macaulay complexes
We start by setting the stage for the use of more general geometric objects and valuations, following McMullen [9]. Throughout, let Λ ⊂ R d be a fixed, full-dimensional discrete lattice or a vector space over some subfield of R. We denote by P = P(Λ) the collection of polytopes in R d with vertices in Λ. A Λ-valuation is a map ϕ from P into some abelian group such that ϕ(P ∪ Q) = ϕ(P ) + ϕ(Q) − ϕ(P ∩ Q) whenever P ∪ Q ∈ P (and hence P ∩ Q ∈ P) and such that ϕ(t + P ) = ϕ(P ) for all t ∈ Λ. We can extend ϕ to half-open polytopes as follows. If B ⊂ ∂P is a the union of facets where F J := {F j : j ∈ J}. In particular, if B = ∂P , we get where the sum is over all non-empty faces F of P . The following is the basis for our considerations.
Theorem 2.1 ( [9]). If ϕ is a Λ-valuation, then for all n ∈ Z ≥0 ϕ P (n) := ϕ(nP ) agrees with a univariate polynomial of degree ≤ dim P and A Λ-complex is a polyhedral complex K such that every face is a Λ-polytope. The complex is pure if all inclusion-maximal faces have the same dimension. For example, the collection of proper faces of a Λ-polytope P is a pure Λ-complex, called the boundary complex B(P ). The underlying set of K is denoted by |K| and, since this is the disjoint union of relatively open polytopes, we can extend ϕ to Λ-complexes by setting For a subcomplex ∆ ⊂ K, a face F ∈ ∆ is an interior face of ∆ if lk K (F ) ⊂ ∆ and a boundary face otherwise. The boundary of ∆ is the subcomplex ∂∆ of all boundary faces. Note that for F ∈ ∆, we have lk ∆ (F ) = ∅ = {∅} with reduced Euler characteristic for all non-empty faces F ∈ K. Thus K is Cohen-Macaulay if additionally H i (K) = 0 for all 0 ≤ i < dim K. This is a stronger condition as, for instance, weakly Cohen-Macaulay complexes are not necessarily connected. Since G ⊆ F implies lk K (F ) ⊆ lk K (G), we get that K is weakly Cohen-Macaulay if and only if every vertex link of K is Cohen-Macaulay. Munkres [12] proved that Cohen-Macaulayness of a complex K is a topological property of the underlying pointset |K| and hence K is weakly Cohen- Note that what we define is the notion of (weakly) Z-CM complexes as our ring of coefficients is Z throughout (cf. [3, Sect. 11]); however, most of our results hold for general rings of coefficients.
Finally, a pure Λ-complex K of dimension d is a homology manifold, if for every face F of K, the reduced homology of lk K (F ) is identically zero or if In particular, if |K| is a manifold, then K is a homology manifold, and every homology manifold is weakly CM.

An Alexander-type duality
In this section we prove Alexander-type duality relations for Λ-valuations that relates complementary complexes ∆ and ∆ inside Λ-complexes.
Theorem 3.1 (Alexander-type duality for valuations). Let K be a d-dimensional Λ-complex such that K is a homology manifold with boundary and let B ⊂ ∂K be a full-dimensional, weakly Cohen-Macaualay subcomplex. Let D be the closure of ∂P \B. If ϕ is a Λ-valuation, then for all n ≥ 1 and For the proof of the theorem we need to relate the combinatorics of inclusion-exclusion for the valuation ϕ to the topology of ∆. The main observation, captured in the following lemma, is that weakly Cohen-Macaulay (d − 1)-complexes which are embedded into the boundary of a d-dimensional homology manifold are rather restricted.
In other words, a full-dimensional, weakly Cohen-Macaulay subcomplex of a homology manifold is again a homology manifold.
Proof. The link lk ∆ (F ) is a subcomplex of L = lk R (F ), which has the homology of a k-sphere. Thus, if F is an interior face of ∆, then lk ∆ (F ) = L and H * (lk ∆ (F )) = H * (S k ).
If lk ∆ (F ) L is a proper subcomplex, it is sufficient to show that H k (lk ∆ (F )) = 0 for k = dim lk ∆ (F ), as ∆ is weakly Cohen-Macaulay. For this observe that |L|\| lk ∆ (F )| is non-empty. By Alexander duality for homology spheres [11, § 72], we get that Alternatively, it is sufficient to show that lk ∆ (F ) is homotopic to a subcomplex of dimension k − 1. To see this, note that lk ∆ (F ) is a full-dimensional subcomplex of the k-dimensional homology manifold L. Thus, lk ∆ (F ) has a free face and, using Whitehead's language of cellular collapses [15], lk ∆ (F ) collapses to a subcomplex of its (k − 1)-skeleton. Since a collapse in particular provides a certificate for deformation retraction, this finishes the proof.
Proof of Theorem 3.1. As a subset of R d , |K| is partitioned by the relative interiors of faces G ∈ K and thus ϕ |K|\|B| (n) = G∈K\B ϕ relint G (n), For the case n = 0: as ϕ |K| (n) = ϕ n|K| (1), is it is sufficient to prove the claim for n = −1. From Theorem 2.1 and (4), we get where for a face σ ∈ K It follows from Lemma 3.2, that W σ = 1 if σ ∈ K\D which proves the claim. The proof of the case n = 0 is analogous.
Since the boundary of every Λ-polytope is a sphere, we can extend the validity of (1) to general Λ-valuations.  The boundary of the 4-cube contains a 2-dimensional torus T = ∂P 1 × ∂P 2 , which decomposes ∂P into two solid tori S 1 = P 1 × ∂P 2 and S 2 = ∂P 1 × P 2 . As these are 3-manifolds with boundary, both S 1 and S 2 are pure 3-dimensional weakly Cohen-Macaulay subcomplexes. The Ehrhart function for a k-cube is E [0,1] k (n) = (n + 1) k . Thus the relative Ehrhart function is E P,S 1 (n) = (n + 1) 4 − 4(n + 1) 3 + 4(n + 1) 2 = n 4 − 2n 2 + 1 Towards a proof for Theorem 1.1, let us record the following general lemma. For a polyhedral cone C and a point a ∈ C, let σ a ⊂ C be the unique face with a ∈ relint σ a .
Lemma 3.5. Let C ⊂ R d+1 be a rational (d + 1)-dimensional cone and ∆ ⊂ B(C) an arbitrary subcomplex. Then Notice that lk ∆ (σ) ⊆ lk B(C) (σ) ∼ = S d−dim σ . Thus, the coefficient of x a in the equation above is the Euler characteristic of the Alexander dual of | lk ∆ (σ a )| ⊂ S d−dim σa .
Proof. From Ehrhart theory (cf. [14,Prop. 7.1]), we have for a rational cone G which shows that the right-hand side is supported on relint(C)∪|∆|. Now for a ∈ |∆|∩Z d+1 , the coefficient of x a on the right-hand side is which proves the claim.
Proof of Theorem 1.1. If ∆ is Cohen-Macaulay, then for every face F ∈ ∆, the link lk ∆ (F ) has the reduced Euler characteristic of a (d − 1 − dim F )-sphere if F is interior and the reduced Euler characteristic of a point otherwise. Together with Lemma 3.5 this gives us x a

A relative Brion theorem
In this section we give a version of Brion's theorem [4] (see also [1]) suitable in the presence of a forbidden subcomplex. To make our results more transparent, let us start with the Let C = {x ∈ R d+1 : a i , x ≤ 0 for i = 1, 2, . . . , m} be a polyhedral cone. For a non-empty face F ⊆ C let I(F ) = {i ∈ [m] : a i , x = 0 for all x ∈ F } and define the tangent cone of C at F as is the product of a linear space and a pointed polyhedral cone and thus has Euler characteristic χ(C J ) = 0. Moreover, by sending the point p ∈ C J to infinity, the faces of C J are exactly those faces F ⊆ C for which p ∈ T C (F ) and the left-hand side of the stated equation computes the Euler characteristic of C J .
From that, we can deduce the usual Brianchon-Gram relation. If P ⊂ R d is polytope and F a face, then the tangent cone of P at F is defined analogously as above and, equivalently, T P (F ) = q F + cone(P − q F ), where q F ∈ relint F . In particular, we have T P (P ) = R d and T P (∅) = P .
Corollary 4.2. If P ⊂ R d is a polytope, then Proof. Let C = C(P ) ⊂ R d+1 be cone associated to P . Let H = R d ×{1}.
which proves the claim.
We also get an interesting complementary version as follows. For every face F ⊆ P , the tangent cone is of the form T P (F ) = aff(F ) + C P (F ) where C P (F ) is the unique cone contained in aff(F ) ⊥ . Let us define the inverted tangent cone as T −1 P (F ) = aff(F ) − C P (F ). Corollary 4.3. Let P ⊂ R d be a full-dimensional polytope. Then For a non-empty face F ⊆ P , the inverted tangent cone is given by Then, with appropriate identifications, relint(−C) ∩ H = relint(P ) and T C (F ) ∩ H = T −1 P (F ). Lemma 4.1 now yields the result.
For dealing with forbidden subcomplexes, we will also need the following relative versions of the two Brianchon-Gram relations. If ∆ ⊆ B(P ) is a full-dimensional subcomplex of the boundary, then this induces a subcomplex ∆ F ⊆ B(T P (F )) in the tangent cone of every face F P . This subcomplex is pure of dimension d − 1 or empty. We write T P,∆ (F ) = T P (F )\|∆ F | for the tangent cone minus the faces induced by ∆, and T −1 P,∆ (F ) for the analogously defined relative inverted tangent cone. and where the sums are over all non-empty faces F ⊆ P .
Proof. We prove only the first statement as the proof of the second relation is analogous. Let p ∈ R d be an arbitrary point. If p is not contained in the affine span of any face of ∆, then [T P,∆ (F )](p) = [T P (F )](p) for all non-empty faces F ⊆ P and the identity is Corollary 4.2. Thus, suppose that p is contained in some hyperplane spanned by a facet in ∆.
If p ∈ P , then the unique face F ⊆ P containing p in the relative interior is a face of ∆. In this case p ∈ T P,∆ (G) if and only if G and F are contained in a common face of ∆. That is, if D is contained in the closed star st ∆ (F ) : The right-hand side of the stated equation evaluated at p can be written as This is the difference of the unreduced Euler characteristics of two contractible complexes and therefore 0 = 1 − 1.
If p ∈ R d \P , let F 1 , . . . , F k ⊆ P be the (d − 1)-dimensional faces of ∆ for which p is contained in the affine hyperplane H i := aff(F i ) spanned by F i . We have to show that We can rewrite the left-hand side of (5) as But for a fixed I, we have that s I is equal to the left-hand side of the Brianchon-Gram relation applied to F I and a point p ∈ F I inside aff(F I ). Thus s I = 0.
We can now state our generalization of Brion's theorem. and Proof. The first statement follows from the same consideration as in [1]: Observe that for S ⊂ R d , we have F S (x) = a∈Z d [S](a)x a and from Lemma 4.4 we get where the sum is over all non-empty faces F ⊆ P . Now if F is not a vertex, the relative tangent cone T P,∆ (F ) is not pointed, that is, t + T P,∆ (F ) = T P,∆ (F ) for some t = 0. On the level of lattice point enumerators, this means x t F T P,∆ (F ) (x) = F T P,∆ (F ) (x) and thus F T P,∆ (F ) (x) = 0. This proves the first statement.
By the same token, we get from Lemma 4.4 and thus Since ∆ is weakly Cohen-Macaulay, we have that ∆ i is Cohen-Macaulay and by Theorem 1.1 For the finishing touch, we calculate

Topology of reciprocal domains
Theorem 1.1 and Theorem 3.1 apply to full-dimensional (weakly) Cohen-Macaulay complexes in the boundaries of polytopes. In this section we discuss what forms these complexes can take. In [10], Miller and Reiner gave an example of a full-dimensional Cohen-Macaulay subcomplex in the boundary of a polytope that is not contractible and hence not a ball; they argued that, for instance, the Mazur manifold can occur. The purpose of this section is to generalize this remark. We refer to [13] for the basic notions of PL topology. Then there exists a (d + 1)-polytope P and a subcomplex B ⊆ B(P ) such that B is PLhomeomorphic to B. In particular, B is a full-dimensional weakly Cohen-Macaulay subcomplex of ∂P .
Any homology manifold B satisfying assumptions (a) and (b) is a homology ball.
Proof. By [7,Thm. 3], there is a contractible PL manifold M for which ∂M is PL homeomorphic to ∂B. Then the gluing of M and B along their boundaries is a PL-sphere S, since it is PL (because B and M are PL), simply connected (by property (a) of B and the fact that M is contractible) and has the homology of a sphere (since both M and B have the homology of a sphere); consequently, it is a PL sphere by the generalized Poincaré conjecture [16]. In particular, we have that there exists a subdivision S of S that is combinatorially equivalent to the boundary complex S of a (d + 1)-polytope P . The subcomplex of S corresponding to B is the desired complex B.
Corollary 5.2. Every contractible PL d-manifold B, d ≥ 5 can be realized, up to PL homeomorphism, as a full-dimensional weakly Cohen-Macaulay subcomplex in the boundary of a (d + 1)-polytope.
This suggests that every PL manifold satisfying (a) and (b) of Theorem 5.1 is contractible. This is not the case: Example 5.3. Let S denote a PL homology sphere that is not S d , such as Poincaré's homology sphere, and let ∆ denote any facet of S. Then B := (S − ∆) × [0, 1] is a homology ball, but homotopy equivalent to S − ∆, which has π 1 (S) = π 1 (S − ∆) = 0 and is consequently not contractible.
Theorem 3.1 applies more generally to subcomplexes in the boundary of homology manifolds; in this case, we are surprisingly flexible: Theorem 5.4. Let M denote any homology manifold with vanishing reduced homology. Then there exists a homology ball that contains M as a full-dimensional subcomplex of its boundary.
Proof. Let D(M, ∂M ) denote the double of M (that is, the result of gluing two manifolds PL homeomorphic to M along their isomorphic boundaries). By excision, the complex D(M, ∂M ) is a homology manifold without boundary which is homologically equivalent to a sphere. Thus, the cone over D(M, ∂M ) is a homology ball, as desired.