Hypersimplicial subdivisions

Let $\pi:{\mathbb R}^n \to {\mathbb R}^d$ be any linear projection, let $A$ be the image of the standard basis. Motivated by Postnikov's study of postitive Grassmannians via plabic graphs and Galashin's connection of plabic graphs to slices of zonotopal tilings of 3-dimensional cyclic zonotopes, we study the poset of subdivisions induced by the restriction of $\pi$ to the $k$-th hypersimplex, for $k=1,\dots,n-1$. We show that: - For arbitrary $A$ and for $k\le d+1$, the corresponding fiber polytope $\mathcal F^{(k)}(A)$ is normally isomorphic to the Minkowski sum of the secondary polytopes of all subsets of $A$ of size $\max\{d+2,n-k+1\}$. - When $A={\mathbf P}_n$ is the vertex set of an $n$-gon, we answer the Baues question in the positive: the inclusion of the poset of $\pi$-coherent subdivisions into the poset of all $\pi$-induced subdivisions is a homotopy equivalence. - When $A=\mathbf{C}(n,d)$ is the vertex set of a cyclic $d$-polytope with $d$ odd and any $n \ge d+3$, there are non-lifting (and even more so, non-separated) $\pi$-induced subdivisions for $k=2$.


Introduction
The main object of study in this paper are hypersimplicial subdivisions, defined as follows. Let A be a set of n points affinely spanning R d . Let ∆ n be the standard (n − 1)-dimensional simplex in R n . Consider the linear projection π : R n → R d sending the vertices of ∆ n to the points in A. (We implicitly consider the points in A labelled by [n], so that π sends e i to the point labelled by i). Let ∆ (k) n := k∆ n ∩[0, 1] n be the standard hypersimplex and A (k) the image of the vertices of ∆ (k) n under π (so that points in A (k) are labelled by k-subsets of [n]). A hypersimplicial subdivision of A (k) is a polyhedral subdivision of conv(A (k) ) such that every face of the subdivision is the image of a face of ∆ (k) n under π. Put differently, we call hypersimplicial subdivisions the π-induced subdivisions of the projection π : ∆ (k) n → conv(A (k) ), as introduced in [BS92, BKS94] (see also [Rei99,DLRS10]). See more details in Section 2.
One reason to study such subdivisions comes from the case where A ⊂ R 2 are the vertices of a convex polygon. Galashin [Gal18] shows that in this case fine hypersimplicial subdivisions, which we call hypertriangulations, are in bijection with maximal collections of chord-separated k-sets. These, in turn, correspond to reduced plabic graphs, [OPS15] which are a fundamental tool in the study of the positive Grassmannian [Pos06, Pos19].
More generally, it is of interest the case where A are the vertices of a cyclic polytope C(n, d) ⊂ R d . (The n-gon is the case d = 2). In [Pos19,Problem 10.3] Postnikov asks the generalized Baues problem for this scenario; that is, he asks whether the poset of hypersimplicial subdivisions of C(n, d) (k) has the homotopy type of a (n − d − 2)-sphere. For k = 1 this was shown to have a positive answer by Rambau and Santos [RS00]. For d = 2, Balitskiy and Wellman show the poset to be simply connected and again ask the Baues question for it ([BW19, Theorem 6.4 and Question 6.1]). We here give the answer to this: Theorem 1.1. Let P n be the vertices of any convex n-gon. The poset of hypersimplicial subdivisions B(∆ (k) n → P (k) n ) retracts onto the poset of coherent hypersimplicial subdivisions. In particular, it has the homotopy type of an (n − 4)-sphere.
[Pos19, Problem 10.3] also asks for which values of the parameters can all hypersimplicial subdivisions of C(n, d) (k) be lifted to zonotopal tilings of the cyclic zonotope. This was already known to be false for d = 1 [Pos19, Example 10.4] and we generalize the counterexamples to every odd dimension: Theorem 1.2. Consider the cyclic polytope C(n, d) ⊂ R d for odd d and n ≥ d + 3. Then, for every k ∈ [2, n − 2] there exist hypersimplicial subdivisions of C(n, d) (k) that do not extend to zonotopal tilings of the cyclic zonotope Z(C(n, d)).
In contrast, Galashin [Gal18] showed that the answer to Postnikov's question is positive in dimension two for hypertriangulations, a result that was generalized to all hypersimplicial subdivisions by Balitskiy and Wellman [BW19, Lemma 6.3].
The poset of coherent hypersimplicial subdivisions of any A is isomorphic to the face poset of a polytope, a particular case of a fiber polytope. When k = 1 this is just the secondary polytope of A, so for k > 1 we call it the k-th hypersecondary polytope of A. We study hypersecondary polytopes for any A ⊂ R d and k ≤ d + 1. Specifically, we show that these polytopes are normally equivalent to the Minkowski sum of certain faces of the secondary polytope of A. By symmetry, an analogue statement holds for n − d − 1 ≤ k < n. Theorem 1.3. Let A ⊆ R d be a configuration of size n and k ∈ [d + 1]. Let s = max(n − k + 1, d + 2). The hypersecondary polytope F (k) (A) is normally equivalent to the Minkowski sum of the secondary polytopes of all subsets of A of size s.
The paper is organized as follows: Section 2 introduces notation and basic background on induced subdivisions in general, and hypersimplicial subdivisions in particular. In Section 3 we look at coherent hypersimplicial subdivisions and hypersecondary polytopes as Minkowski sums and prove Theorem 1.3, among other results.
In Section 4 we study the connection of hypersimplicial subdivisions with zonotopal tilings. In particular, we extend to tiles of positive dimension the concept of A-separated sets introduced in [GP17]. With this machinery we show that if all hypertriangulations of A are separated then all hypersubdivisions are separated too (Corollary 4.11). In Section 5 and Section 6 we prove Theorem 1.2 and Theorem 1.1 respectively. Finally, we briefly discuss the enumeration of hypersimplicial subdivisions of P (2) n in Section 7. Acknowledgements. We thank Alexander Postnikov for inspiring us to work on this and Alexey Balitskiy, Pavel Galashin and Julian Wellman for comments on a first version of this paper.

Preliminaries and notation
2.1. Fiber polytopes. We here briefly recall the main concepts and results on fiber polytopes. See [BS92] or [Rei99] for more details.
Let π : R n → R d be a linear projection map. Let Q ⊂ R n be a polytope and let A = π(vertices(Q)). A π-induced subdivision of A is a polyhedral subdivision S (in the sense of, for example, [DLRS10]), such that every face of S is the image under π of a face F of Q.
Given a vector w ∈ (R n ) * the face Q w of Q selected by w is the convex hull of all vertices of Q which minimize w. A π-coherent subdivison is a π-induced subdivision in which the faces of Q are chosen "coherently" via a vector w ∈ (R n ) * . More precisely, we define the π-coherent subdivision of A given by w to be S(Q π → A, w) := π(F ) : ∃w ∈ (R n ) * s.t.w| ker(π) = w| ker(π) and Qw = F .
The fiber fan of the projection Q π → A is the stratification of (R n ) * according to what π-coherent subdivision is produced. It is a polyhedral fan with linearity space equal to {w ∈ (R n ) * : ker(π) ⊂ ker(w)} + {w ∈ (R n ) * : w| Q = constant}.
As we will see below, it is the normal fan of a certain polytope To define F(Q π → A), we look at fine π-induced subdivisions. A π-induced subdivision S is fine if dim(F ) = dim(π(F )) for each of the faces F ≤ Q whose images are cells in S Put differently, a fine π-induced subdivision is the image of a subcomplex of Q that is a section of π : Q → conv(A). To each fine π-induced subdivision S we associate the following point: where c(F ) denotes the centroid of F .
Definition 2.1. The fiber polytope of the projection π : Q → conv(A) is the convex hull of the vectors GKZ(S) for all fine π-induced subdivisions. We denote it F(Q → A).
The main property of the fiber polytope is the following result of Billera and Sturmfels. In fact, for the purposes of this paper this theorem can be taken as a definition of the fiber polytope, since our results are mostly not about the polytope but about its normal fan (see, eg Section 3).
In particular, the face lattice of F(Q → A) is isomorphic to the poset of πcoherent subdivisions ordered by refinement. For example, vertices of F(Q → A) correspond bijectively to fine π-coherent subdivisions.
Two cases of this construction are of particular importance. Let A = {a 1 , . . . , a n } ⊂ R d be a configuration of n points. Then: (1) If we let π : ∆ n → conv(A) be the affine map e i → a i bijecting vertices of ∆ n to A, then all the polyhedral subdivisions of A are π-induced, and the coherent ones are usually called regular subdivisions of A. The corresponding fiber polytope is the secondary polytope of A and we denote it be the zonotope generated by the vector configuration A × {1} ⊂ R d+1 . The π in the previous case extends to a linear map π : [0, 1] n → Z(A) still sending e i → a i . Then the π-induced subdivisions are precisely the zonotopal tilings of Z(A). The corresponding fiber polytope is the fiber zonotope of Z(A) (or of A) and we denote it F Z (A).
2.2. The Baues problem. The poset of all π-induced subdivisions (excluding the trivial subdivision for technical reasons) is called the Baues poset of the projection and we denote it B(Q → A). The subposet of π-coherent subdivisions is denoted B coh (Q → A). The Baues problem is, loosely speaking, the question of how similar are B(Q → A) and B coh (Q → A), formalized as follows: To every poset P one can associate a simplicial complex called the order complex of P by using the elements of P as elements and chains in the poset as simplices. In particular, one can speak of the homotopy type of P meaning that of its order complex. Similarly, an order preserving map of posets f : P 1 → P 2 induces a simplicial map between the corresponding order complexes, and one can speak of the homotopy type of f .
The prototypical example is the following: if P is the face poset of a polyhedral complex C, then the order complex of P is (isomorphic to) the barycentric subdivision of C. In particular, since B coh (Q → A) is the face poset of the polytope F(Q → A), it is homotopy equivalent (in fact, homeomorphic) to a sphere of dimension dim(Q) − dim(A) − 1. Rei99] for a (not-so-recent) survey about this question, and [San06, Liu17] for examples where the answer is no and having Q a simplex and a cube, respectively.

Question 2.3 (Baues Problem). Under what conditions is the inclusion
2.3. Cyclic polytopes. Cyclic polytopes are a family of polytopes of particular interest for this manuscript and are defined as follows. The trigonometric moment curve (also known as the Carathéodory curve), is parametrized by Let t 1 , . . . , t n be n cyclically equidistant numbers in [0, 2π), for example, The cyclic polytope C(n, d) is the convex hull of φ(t 1 ), . . . , φ(t n ). The combinatorics of the cyclic polytope can be nicely described in terms of the circuits of the corresponding oriented matroid. Namely, all circuits are of the form ({a 1 , a 3 , . . . }, {a 2 , a 4 , . . . }) such that a 1 < a 2 < · · · < a d+2 and their opposites (giving the label i to the vertex φ(t i )).
Cyclic polytopes can also be defined by using the polynomial moment curve t → (t, t 2 , . . . , t d ) instead of the trigonometric moment curve and the combinatorial type remains the same. However, the coherence of subdivisions and hence fiber polytopes depend also on the embedding (see Example 3.12). When using the trigonometric moment curve in even dimension the cyclic polytope has more symmetry. That is, it is invariant under the cyclic group action on the vertices. When d = 2 the cyclic polytope C(n, 2) is a is a regular polygon and we abbreviate it by P n .
The Baues problem is known to have positive answer for cyclic polytopes in the following two cases: SZ93]). Let n > d ∈ N. Then, the following two cases of the Baues question have a positive answer: • When Q = ∆ n and A = C(n, d) is the cyclic polytope of dimension d with n vertices [RS00]. 2.4. Hypersecondary polytopes. Let A = {a 1 , . . . , a n } ∈ R d be a point configuration. For each k = 1, . . . , n−1 we consider the following k-th deleted (Minkowski) sum of A with itself, which we denote A (k) : The k-th deleted sum of the standard (n − 1)-simplex ∆ n := conv(e 1 , . . . , e n ) equals the k-th hypersimplex of dimension n − 1: (Observe that the notation ∆ (k) n here is an abbreviation of conv(vertices(∆ n ) (k) )). As mentioned above, the projection R n → R d × {1} that sends the vertices of ∆ n to A extends to a linear map R n → R d+1 that sends the unit cube [0, 1] n to the zonotope Z(A). In turn, this linear map restricts to an affine map sending each ∆ (k) n ⊂ R n to A (k) ⊂ R d × {k}. We use the same letter π for all these projections. • For a cyclic polytope C(n, d), all triangulations in the standard sense (that is, all hypertriangulations of C(n, d) (1) ) are lifting [RS00]. The same is not known for non-simplicial subdivisions. • For arbitrary k and a convex n-gon P n , all hypertriangulations of P (k) n are lifting [Gal18]. The same result for all hypersimplicial subdivisions has recently been provedin [BW19]. Non-lifting triangulations of A (1) are not known in dimension two but easy to construct in dimension three or higher. For example, if a subdivision S of A has the property that its restriction to some subset B of A cannot be extended to a subdivision of B, then S is non-lifting. Such subdivisions (and triangulations) exist when A is the vertex set of a triangular prism together with any point in the interior of it, the vertex set of a 4-cube, or the vertex set of ∆ 4 × ∆ 4 , among other cases (see, e.g., Geometrically, we think of [X, Y ] as the zonotope X + Z(Y \ X), but we prefer the combinatorial notation where the tile is described as the set of vertices of [0, 1] n of which it is the projection.
Every tile is a cell in a coherent zonotopal tiling of Z(A), by letting w(j) be −1, 0 or 1 depending on whether j is in X, Y \ X, or none of them. Indeed, this w gives value at least −|X| to every point in Z(A), with equality if and only if the point belongs to [X, Y ].
Turning our attention to hypersimplices, observe that every face of the hypersimplex ∆ (k) n is the intersection of a face of [0, 1] n with the hyperplane {x : Therefore we can denote the projection under π of any face of ∆ (1) Intersection of zonotopal tilings with the hyperplane at level k induces an order-preserving map (2) The normal fan of F Z (A) refines the normal fan of F (k) (A).
Proof. For the first claim, notice that the intersection of a zonotopal tiling S = , which clearly is hypersimplicial. We denote r (k) (S) as S (k) for simplicity. The second claim follows from the fact that Example 2.8. Consider the regular hexagon P 6 . Figure 1 shows a hypersimplicial subdivision of P  This subdivision is not coherent. To see this, suppose there is a lifting vector w ∈ (R * ) 6 whose regular subdivision is this. Then notice that the presence of the edge [1, 136] (2) implies w 3 +w 6 < w 2 +w 5 , the presence of the edge [3, 235] (2) implies w 2 +w 5 < w 1 +w 4 and the presence of the edge [5, 145] (2) implies w 1 +w 4 < w 3 +w 6 , together forming a contradiction. This contrasts the fact that every subdivision of a convex polygon is regular.
2.6. Lifting subdivisions via Gale transforms. The Bohne-Dress Theorem. As a general reference for the contents of this section we recommend the book [DLRS10], more specifically Chapters 4, 5 and 9.
A Gale transform of a point configuration A = {a 1 , . . . , a n } is a vector configuration G A = {a * 1 , . . . , a * n } with the property that a vector (λ 1 , . . . , λ n ) ∈ R n is the coefficient vector of an affine dependence in A if and only if it is the vector of values of a linear functional on G A . The definition implicitly assumes a bijection between A and G A given by the labels 1, . . . , n.
Gale duality is an involution: the Gale duals of a Gale dual of A are linearly isomorphic to A when considering A as a vector configuration via homogenization, by which we mean looking at affine geometry on the points a 1 , . . . , a n as linear algebra on the vectors (a 1 , 1), . . . , (a n , 1). In fact, if A and B are Gale duals to one another then their oriented matroids are dual, which implies that their ranks add up to n. In our setting where A has affine dimension d and hence rank d + 1, its Gale duals have rank n − d − 1.
The normal fan of the secondary polytope F (1) (A) of A lives naturally in the ambient space of G A : it equals the common refinement of all the complete fans with rays taken from G A . Put differently, vectors w ∈ span(G A ) are in natural bijection to lifting functions A → R (where the latter, which forms a linear space isomorphic to R n , is considered modulo the linear subspace of affine functions restricted to A). Under this identification, w 1 and w 2 define the same coherent subdivision of A if and only if they lie in exactly the same family of cones among the finitely many cones spanned by subsets of B. The precise combinatorial rule to construct the . Any vector w ∈ span(G A ) induces such an extension, but the definition is more general since M needs not be realizable, or it may be realizable but not extend the given realization G A of M * (A). Yet, any such extension w allows to define a subdivision S(w) of A as follows.
Proposition 2.9. With the notation above, the following rules define, respectively, a polyhedral subdivision S (1) (A, w) of A and a zonotopal tiling In particular, lifting subdivisions of A (1) are precisely the ones that can be obtained by the construction in Proposition 2.9(1).

Normal fans of hypersecondary polytopes
The goal of this section is to study hypersecondary polytopes, and the relations between them and the secondary zonotope. Most of such relations say that the normal fan of one of the polytopes refines that of another one. We introduce the following definition to this effect: Definition 3.1. Let P, Q ∈ R d be two polytopes. We say that Q is a Minkowski summand of P , and write Q ≤ P , if any of the following equivalent conditions holds: (1) The normal fan of P refines that of Q.
(2) P + Q is combinatorially isomorphic to P . If P and Q are Minkowski summands of one another then they are normally equivalent and we write P ∼ = Q.
Remark 3.2. The equivalence of these two conditions follows from the fact that the normal fan of P + Q is the common refinement of the normal fans of P and Q. It can be shown Q ≤ P is also equivalent to the existence of a polytope Q and an ε > 0 such that P = Q + εQ, hence the name "Minkowski summand".
Throughout this section we will assume that A ⊆ R d is a point configuration that spans affinely R d . As a first example, it follows from Proposition 2.7 that: (2) Let k 0 = 0 < k 1 < · · · < k p = n be a sequence of integers with k i+1 − k i ≤ d + 1 for all i. Then, In particular: (2) If n ≥ 2d + 2 then Lemma 3.5. Let S be coherent zonotopal subdivision of A and let B ⊆ A be a spanning subset. Then there is at most one X ⊆ A\B, such that [X, X ∪ B] ∈ S.
Proof. Let w ∈ (R * ) n such that S = S(Z(A), w). Since B is of maximal dimension, there is at most onew such thatw| ker(π) = w| ker(π) and w · b = 0 for every b ∈ B.
If suchw exists then the only tile of the form [X, X ∪ B] that is in S is the one where X = {x ∈ A |w · x < 0}. If no suchw exists then there is no tile of that form in the subdivision.
In the following result and in the rest of this section we denote by A J the subset of A labelled by J, for any J ⊂ [n].
Lemma 3.6. Fix k ≥ 1 and a lifting vector w ∈ (R n ) * , for a point configuration A of size n. For each tile [X, Y ] ⊂ 2 [n] such that Y \X a basis of A, the following are equivalent: ( . If, moreover, k > 1, then they are also equivalent to: (4) There are is equivalent to saying that if, for a given w we know the subdivisions that w induces in A (k) and in A\x (k) for every x then we also know the subdivision induced in A (k+1) . For a cell [X, Y ] (k+1) with |X| = k, Lemma 3.6 says that its presence in S(A (k+1) , w) is determined by its presence in S(A (k) , w) and S(A\x (k) , w).
The converse is only true for small k: Proof. One direction is Proposition 3.7. For the other direction we have that by Lemma 3.6 then S(A (k+1) , w) determines S(A (k+1) Proposition 3.9. For every configuration A ⊆ R d of size n > d + 2 and every k ∈ [d] we have that Proof. We need to prove that for every w ∈ R d , knowing S(A (k) [n]\i , w) for every i determines S(A (k) , w). It is enough to prove it for a generic w, so we can assume the subdivisions are fine. Let [X, Y ] be a tile such that Y \X is an affine basis.
There is exactly onew that agrees with w in ker(π) and such thatw · x = 0 for every [n]\i , w) if and only ifw · x < 0 for every x ∈ X andw · x > 0 for every x ∈ [n]\Y . Notice that as n > d + 2, [n]\i , w) if and only ifw · x < 0 for every x ∈ X\i for all x ∈ X\i andw · x > 0 for every x ∈ [n]\(Y ∪ i). As |[n]\(Y \X)| > 2, we can do this for two different elements in [n]\(Y \X) so we can verify the sign ofw · i for every i ∈ [n]\(Y \X).
A consequence of this is that Proposition 3.8 can be strengthened as follows: Proposition 3.10. For every configuration A ⊆ R d of size n > d + 2 and every k ∈ [d] we have that Notice that if n = d + 1 then the fiber polytopes are just points and if n = d + 2 they are just segments and in particular F (k+1) (A) ∼ = F (k) (A). Now we are ready to prove the main result of this section: Theorem 3.11. Let A ⊆ R d be a configuration of size n and k ∈ [d + 1]. Let s = max(n − k + 1, d + 2). Then Proof. We prove this by iterating Proposition 3.10 several times. At each iteration, The iteration stops at level 1 with the desired result (notice that Minkowski sum is idempotent with respect to normal equivalence).
Example 3.12. Consider the regular hexagon P 6 . The secondary polytope F (1) (P 6 ) is the 3-dimensional associahedron, as seen in Figure 2. Its border consists of 6 pentagons and 3 squares. By Theorem 3.11, the hypersecondary polytope F (2) (P 6 ) is normally equivalent to the Minkowski sum of those 6 pentagons, see Figure 3. It has 66 vertices and the facets consist of 27 quadrilaterals (18 rectangles, 6 rhombi and 3 squares), 6 pentagons, 2 hexagons and 6 decagons. The short edges correspond to flips which do not change the set of vertices of the triangulation and the long edges correspond to those flips that do change the set of vertices.  The GKZ vector corresponding to the triangulation from Example 2.8 is in the center of one of the hexagons. There are 4 non-coherent hypertriangulations of P (2) 6 , which come in pairs with the same GKZ-vector, each in the center of one of the two hexagons. If instead of a regular hexagon we had a hexagon where the three long diagonals do not intersect in the same point, two of those subdivisions would become coherent and the hypersecondary polytope would have instead of each hexagon a triple of rhombi around the new vertex.
The order complex of the Baues poset B (2) (P 6 ) is the (barycentric subdivision of the border of the) hyperassociahedron F (2) (P 6 ) where the hexagons are replaced by cubes. In particular it satisfies the Baues problem, that is, B (2) (P 6 ) retracts onto F (2) (P 6 ). We will generalize this in Section 6.

Throughout this section let A ⊂ R d be a point configuration labelled by [n], and let Z(A) ⊂ R d+1 be the zonotope generated by the vector configuration
Following [GP17], we say that two points X 1 , X 2 ⊂ [n] are separated with respect to A or A-separated for short if there is an affine functional positive on A X1\X2 and negative on A X2\X1 . Equivalently, if there is no oriented circuit (C + , C − ) in A with C + ⊂ X 1 \X 2 and C − ⊂ X 2 \X 1 . Their motivation is that the notions of strongly separated and chord separated that were introduced in [LZ98] and [Gal18, OPS15] are equivalent to "C(n, 1)-separated" and "C(n, 2)-separated" respectively ([GP17, Lemmas 3.7 and 3.10]). 1 One of their main results is as follows (their statement is a bit more general, since it is stated for arbitrary oriented matroids, rather then "point configurations"): We here extend their definition to separation of tiles. In the rest of the paper we omit A and write "separated" instead of A-separated: Definition 4.2. Let [X 1 , Y 1 ] and [X 2 , Y 2 ] be two tiles. We say they are separated if there is no circuit The following diagram illustrates the circuits forbidden by the first two conditions in this definition. The third condition forbids circuits with support fully contained in the middle cell: OPS15] uses the expression "weakly separated" for "chord separated", but "weakly separated" had a different meaning in [LZ98] By the orthogonality between circuits and covectors in an oriented matroid [BLVS + 99, Proposition 3.7.12], and the fact that covectors of a realized oriented matroid are the sign vectors of affine functionals this definition is equivalent to: The following diagram illustrates the sign-patterns of covectors witnessing that two tiles are separated: is a covector in A if and only if a circuit as in the definition of separation does not exist. The following result clarifies the relation between separation of points and tiles. In it, we say that a tile [X, Y ] is fine if Y \X is an independent set. Fine tiles are the ones that can be used in fine zonotopal tilings of Z(A).
The converse holds if the tiles are fine.
Proof. For the first direction, by induction on |Y 1 \X 1 |+|Y 2 \X 2 |, we can assume that [X 1 , Y 1 ] is not a singleton and that every tile properly contained in it is separated from [X 2 , Y 2 ]. In particular, taking any element i ∈ Y 1 \X 1 we have that both [X 1 ∪ i, Y 1 ] and [X 1 , Y 1 \i] are separated from [X 2 , Y 2 ]. By Proposition 4.3, that implies the following two covectors: \Y 2 then the first or the second covector, respectively, show that [X 1 , Y 1 ] and [X 2 , Y 2 ] are separated. If i ∈ Y 2 \X 2 then elimination of i in these two covectors gives a covector with values which again shows that [X 1 , Y 1 ] and [X 2 , Y 2 ] are separated. For the converse, suppose first that [X 1 , Y 1 ] and [X 2 , Y 2 ] are separated and let V be the covector showing it. Let B 1 and B 2 be points in them. Since the set C := (Y 1 \ X 1 ) ∩ (Y 2 \ X 1 ) is independent and is contained in the zero-set of V , no matter what signs we prescribe for its elements there is a covector V that agrees with V where V is not zero and has the prescribed signs on C. This implies the points B 1 and B 2 are separated.
(2) There is a zonotopal tiling of Z(A) using both.
(3) There is a coherent zonotopal tiling of Z(A) using both. (4) There is a polyhedral subdivision of A using Y 1 \X 2 and Y 2 \X 1 as cells.
(5) There is a coherent polyhedral subdivision of A using Y 1 \X 2 and Y 2 \X 1 as cells.
Proof. Throughout the proof, let A = {a 1 , . . . , a n } and denoteã i = (a i , 1) the corresponding generator of Z(A).
• 1 ⇒ 3. Suppose the tiles are separated. By Proposition 4.3 this implies there is a linear functional v ∈ (R d+1 ) * such that v ·ã i takes the following values on the generators of Z(A): Let w ∈ (R n ) * be defined as follows on each i ∈ [n]: where N is a very large positive number. Since w is negative in X 1 , positive in [n]\Y 1 , and zero in Y 1 \X 1 , the tile selected by w in the subdivision Similarly, the vector w ∈ (R n ) * defined by w i = w i + 2v ·ã i has the following values which shows that [X 2 , Y 2 ] is also in S(Z(A), w), since the difference between w and w is a linear function. That is, our hypothesis is that there is a lift M of A that contains the covectors Elimination of the element n + 1 among these covectors gives us a covector of Proposition 4.3. • 1 ⇒ 5. Let v as in the proof of 1 ⇒ 3, and define w ∈ (R n ) * as follows: Then w and the w defined by w i = w i + v ·ã i show that Y 1 \X 2 and Y 2 \X 1 are cells in S(A, w). • 4 ⇒ 1 For C 1 := Y 1 \X 2 and C 2 := Y 2 \X 1 to be cells in a subdivision it is necessary that their convex hulls intersect in a common face. That is, there must be a covector in A that is zero in C 1 ∩ C 2 , negative on C 1 \ C 2 , and positive on C 2 \ C 1 . These are precisely the same conditions as required in Proposition 4.3. • 3 ⇒ 2 and 5 ⇒ 4 are obvious.
Remark 4.7. With this theorem, it is now easy to see that Lemma 3.5 also holds for non coherent subdivisions. If Y 1 \X 1 = Y 2 \X 2 is a spanning set then there can not be a linear functional vanishing on it, so [X 1 , Y 1 ] and [X 2 , Y 2 ] are not separated (unless X 1 = X 2 , in which case they are the same cell).
Remark 4.8. The definition of separated points and tiles makes sense for an arbitrary oriented matroid M, since it uses only the notion of circuits, and Proposition 4.3 still holds in tis more general setting.
The notions of zonotopal tiling and of subdivision also make sense for arbitrary oriented matroids: the former is interpreted as "extension of the dual oriented matroid" via Theorem 2.10 and the latter is studied in detail in [San02]. In this setting the implications (2) ⇒ (4) ⇒ (1) of Theorem 4.6 still hold, the first one as a consequence of the oriented matroid analogue of Proposition 2.9 and the second one because our proof above works at the level of oriented matroids. Yet: (1) The notion of coherent subdivisions needs a realization of the oriented matroid be given. Not only the notion does not make sense for nonrealizable oriented matroids. Also, different realizations of the same oriented matroid may have different sets of coherent subdivisions, and non-isomorphic secondary polytopes/zonotopes. (2) The implication (4) ⇒ (2) fails in the example of [San02, Section 5.2] (see Proposition 5.6(i) in that section), and the implication (1) ⇒ (4) fails in the Lawrence polytope that one can construct from that example. Proof. By induction on the dependence rank of the tiles we only need to show that if [X 1 , Y 1 ] is dependent then there is a tile [X 1 , Y 1 ] properly contained in [X 1 , Y 1 ], covering level k, and non-separated from [X 2 , Y 2 ].
Let (C + , C − ) be a circuit showing that [X 1 , Y 1 ] and [X 2 , Y 2 ] are not-separated. Let C = C + ∪ C − be its support.
If there is an element a ∈ (Y 1 \X 1 )\C then both [X 1 ∪ a, Y 1 ] and [X 1 , Y 1 \a] are not separated from [X 2 , Y 2 ], and one of them still covers level k, since dependent sets are of size at least 3.
If there is no such an a, then Y 1 \X 1 ⊂ C. Since C is a circuit we conclude that Y 1 \X 1 = C. By definition, we have that C − ⊂ Y 2 and C + ⊂ [n]\X 2 . Again, we take as new tile [X 1 ∪ a, Y 1 ] or [X 1 , Y 1 \b], depending on which of the two still covers level k, where a ∈ C + and b ∈ C − . Corollary 4.11. Let A be a point configuration in general position ("uniform") and let k ∈ [n − 1]. If no hypertriangulation of A (k) contains two non-separated tiles, then no hypersimplicial subdivision of A (k) contains them either.
Proof. Suppose that a subdivision S contains two non-separated tiles [X 1 , Y 1 ] and [X 2 , Y 2 ]. Let [X 1 , Y 1 ] and [X 2 , Y 2 ] be the tiles guaranteed by Proposition 4.10. Then, we can refine [X 1 , Y 1 ] and [X 2 , Y 2 ] to fine subdivisions using [X 1 , Y 1 ] and [X 2 , Y 2 ]. By general position this extends to a hypertriangulation refining S and with two non-separated tiles.

Non-separated subdivisions
We call a subdivision S of A (k) non-separated if it contains two non-separated cells. Non-separated subdivisions are certainly non-lifting.
Proof. A hypertriangulation of a configuration A with n = d + 3 has all its fulldimensional cells of one of the following forms, where a < b ∈ [n] and we omit the superscript (k), which will be clear from the context: To simplify notation, we denote these four cells simply as ab, ab, ab and ab, respectively (observe that we always write the indices a and b in increasing order). For example, in this notation the subdivision S of C(4, 1) (2) mentioned above becomes S = {14, 12, 34, 14} (2) . One reason for this notation is that via the correspondence in Proposition 2.9 the tile [X, Y ] corresponds in G A to the cone spanned by X ∪ [n] \ Y , where we use B to denote the set of vectors opposite to B, for B ⊂ [n].
With this notation, Proposition 2.9(2) gives us that the following is a (coherent) zonotopal tiling of Z(C(d + 3, d)) ( Figure 5 shows the case of C(6, 3)): : a odd, b odd}∪{ab : a odd, b even}∪{ab : a even, b odd}∪{ab : a even {an : a < n even} ∪ {an : a odd}. These flips transform S 0 into two new coherent tilings S 1 and S 2 , also shown in Figure 5. The two flips are not compatible, since both want to remove the tile 1n from S 0 , and we can only remove it once. But 1n only affects level 1 of the tiling, which means that in any S (k) 0 with k ≥ 2 we can do these two flips one after the other. After performing them we get a subdivision that contains (for k ∈ [2, d − 2]) the non-separated cells 12 and n − 1n.
To further generalize this construction we need the following easy lemma:  Corollary 5.4. For every odd d, every n ≥ d + 3, and every k ∈ [2, n − 2], there is a non-separated hypertriangulation of C(n, d) (k) .
Question 5.5. Are there non separated hypertriangulations of C(n, d) (k) for d ≥ 4 even? The case of C(n, 2) suggests that the answer is no.
6. Baues posets for A = P n In this section we will restrict ourselves to the case when A is a convex polygon P n .
Remark 6.4. The maps U and D are well defined thanks to the fact that all hypersimplicial subdivisions of P n are lifting ([ BW19]). For more general configurations the definitions above would only make sense restricted to lifting subdivisions.
Example 6.5. Consider the subdivision T ∈ B (2) (P 6 ) in Figure 6   As seen in the right part of the figure, the cells in D(T ) are precisely the ones that have a full-dimensional intersection with the first level.
The main result in this section is that U and D induce homotopy equivalences of the corresponding order complexes (Corollary 6.13). To prove this we use the following criterion, originally proved by Babson [Bab93]. Another proof can be found in [SZ93] and some generalizations appear in [BWW05]: Lemma 6.6 (Babson's Lemma). Let f : P → Q be an order preserving map between two posets. Suppose that for every q ∈ Q we have that (1) f −1 (q) is contractible, and (2) f −1 (q) ∩ P ≤p is contractible, for every p ∈ f −1 (Q ≥q ). Then f is a homotopy equivalence. Suppose there exists X ∈ vertices (k) (S) and [I, J] ∈ S such that [X, ∪ uh S (X)] and [I, J] are not separated. Since d = 2 we may assume that |J \ I| ≤ 2 and there is Y ∈ [X, uh S (X)] (k+2) such that [X, Y ] is not separated from [I, J]. So we have a circuit (C + , C − ) such that C + ∈ Y \ I and C − ∈ J \ X Further, since S ∈ B (k+ 1 2 ) (P n ) we can also assume |I| ≤ k − 1. Let y ∈ Y \ X. Since y ∈ uh S (X) \ X we have that there is x ∈ X such that [X \ x, X ∪ y] is a face of a cell in S. Then by Corollary 4.9 and the fact that S is pairwise separated we have that [X \ x, X ∪ y] is separated from [I, J]. So C + can not be contained in X ∪ y.
This means that C + = Y \ X. Notice that for every i ∈ [n] \ C there is y ∈ C + such that (C + \ y ∪ i, C − ) is a circuit. So if there is an i ∈ X \ I, this circuit would imply that [X, Y \ y] is not separated from [I, J], which can not be as [X, Y \ y] is a face of some cell in S. But this means X \ I = ∅ which is a contradiction since |X| = k > k − 1 = |I|.
Corollary 6.8. Let S ∈ B (k+ 1 2 ) (P n ). Then together with all their faces, form the unique coarsest subdivision in the fibre D −1 (S).
Proof. We need to show that for X 1 , X 2 ∈ vertices (k) (S), [X 1 , uh S (X 1 )] and [X 2 , uh S (X 2 )] are separated. If not, we can again assume there are subsets Y 1 ⊆ uh S (X 1 ) and Y 2 ⊆ uh S (X 2 ) of cardinality k + 2 such that [X 1 , Y 1 ] and [X 2 , Y 2 ] are separated. As any subtile of them are faces of S, we have that there is a circuit C + = Y 1 \ X 1 and C − = Y 2 \ X 2 . Similarly as the proof of 6.7, this implies that X 1 = X 2 . The corollary follows from the fact that every cell in a subdivision in D −1 (S) not coming from S is of type 1 and hence it is contained in [X, uh S (X)] for some X. are also inŜ, but they are not maximal: they are edges.
Proof. LetŜ be the maximal element of D −1 (S), as described in Corollary 6.8.
Let T ∈ D −1 (S), which is a refinement ofŜ. If a cell [X, Y ] (k+1) ∈ T is such that |X| < k, then [X, Y ] ∈ S which implies that it is contained in a cell of D(T ). Then, [X, Y ] (k+1) is contained in a cell of T . Thus, for T to be a refinement of T , it is enough that [X, Y ] (k+1) ∈ T is contained in a cell of T for every [X, Y ] ∈ T with |X| = k.   (1) and (2) in Babson's Lemma follow from Corollary 6.8 and Lemma 6.10, respectively, since a poset with a unique maximal element is clearly contractible. For U the proof is completely symmetric.
Theorem 6.14. Let A be the vertex set of a convex n-gon. The inclusion B Proof. The proof is by induction on k. The base case, k = 1, is the main result of Rambau and Santos in [RS00]. Now let us suppose that B (k) coh (A) → B (k) (P n ) is a homotopy equivalence and we will prove that B (k+1) coh (P n ) → B (k+1) (P n ) is also a homotopy equivalence. Consider the following diagram, which commutes by Proposition 6.2: The maps i (k) and i (k+1) are the inclusions of coherent subdivisions into all subdivisions. The maps r (k) and r (k+1) are the restriction of each zonotopal tiling to its k and k + 1 levels; that is, S → S (k) and S → S (k+1) respectively. They are homotopy equivalences since they can be geometrically realized as the identity maps among the normal fans of F Z (P n ), F (k) (P n ) and F (k+1) (P n ). Since D and U are homotopy equivalences by Corollary 6.13, and i (k) is a homotopy equivalence by inductive hypothesis, the dotted arrow i (k+1) must also be a homotopy equivalence.
Corollary 6.15. The restriction map r (k) : B Z (P n ) → B (k) (P n ) is a homotopy equivalence.
Proof. We now use the following commutative diagram: The top arrow is a homotopy equivalence by [SZ93] and the bottom arrow by Theorem 6.14. The left arrow is also a homotopy equivalence, as mentioned in the proof of Theorem 6.14, so the right arrow is a homotopy equivalence too.

Hypercatalan numbers
Let C (k) n be the number of hypertriangulations of P (k) n , which we will call hypercatalan number. When k = 1 these are the usual Catalan numbers C n . In this section we look at the case k = 2. For a triangulation T of P n and a vertex i ∈ [n] we write deg T (i) for the number of diagonals (edges excluding the sides of P n ) in T incident to i and we call it the degree of i. Proof. Let T be a triangulation of P n . To get a hypertriangulation of P (2) n that agrees with T we need to triangulate [i, U T (i)] (2) for every i. As [i, U T (i)] (2) is a polygon with deg T (i)+2 vertices, the number of ways to triangulate it is C deg T (i) . So for each triangulation T there are  The computation for n = 6 is as follows. Triangulations of the hexagon fall into three symmetry classes: • Two triangulations with degree sequence 020202, each contributing 1 · 2 · 1 · 2 · 1 · 2 = 8 to the sum. • Six triangulations with degree sequence 012012, each contributing 1 · 1 · 2 · 1 · 1 · 2 = 4 to the sum. • Six triangulations with degree sequence 011103, each contributing 1 · 1 · 1 · 1 · 1 · 5 = 5 to the sum. This gives a total of 2 · 8 + 6 · 4 + 6 · 5 = 70 fine subdivisions in B (2) (P 6 ). Proof. Let k 1 . . . k j be the sequence of the degrees of the vertices of T which are positive. The terms of this sequence add up to 2n − 2. The contribution of T to the sum is j i=1 C ki . Observe that the number (n + 2) − j is the number of ears in T , which lies between 2 and n 2 + 1. Thus, j lies between n 2 + 1 and n. For the lower bound, take into account that for every k ≥ 1 one has 2 k−1 ≤ C k , we deduce the contribution of T to be at least 2 2n−2−j . Plugging in that j ≤ n, we get the desired lower bound.
For the upper bound, let l be number of degree 1 vertices. Reorder the k i so that the last l are equal to 1. We have that j−l i=1 k i = 2n − 2 − l. Now take into account that for k ≥ 2 we have that C k ≤ 2 2k−3 , so C kj ≤ 2 2(2n−2−l)−3(j−l) = 2 n−10+3e+l where e = n + 2 − j is the number of ears. So to prove the upper bound we need to show that 3e + l ≤ 3n+6 2 . Suppose T is the triangulation that maximizes 3e + l. If there was a vertex of inner degree 1 such that it is not adjacent to an ear, flipping this edge would not decrease the number 3e + l. So we can assume every degree 1 vertex is next to an ear. But then the vertex of degree 1 can not be neighbour to two ears, otherwise n = 2, and it can not be neighbour to another vertex of degree 1, otherwise n = 3. Also, an ear can not be neighbour to two degree 1 vertices, otherwise n = 2. So the other neighbours of a pair of consecutive vertices (ear,degree 1) must have degree at least 2. Let e the number of ears not adjacent to any degree 1 vertex. Then e − e = l is the number of pairs (ear,degree 1) and we have: l + 2e = 3l + 2e ≤ n + 2 l + 3e ≤ n + 2 + e ≤ 3(n + 2) 2 Corollary 7.4. For n ≥ 6, 2 n−2 ≤ C (2) n C n ≤ 2 5 2 n−7 .
Remark 7.5. The lower bound of 2 n−2 of Lemma 7.3 for the contribution of a single triangulation T is attained by a zigzag triangulation, in which all degrees are 2 except for two 1s and two 0s. When T is a star triangulation in which a vertex is joined to all others, the contribution of T is C n−1 ∼ 4 n (neglecting a polynomial factor). A higher contribution is obtained by the following procedure: start with any triangulation T 0 (e.g. a zig-zag or a star). Let T 1 be obtained by adding an ear at each boundary edge of T 0 , let T 2 be obtained from T 1 in the same way, etcetera. This method produces triangulations that contribute about 4.133 n (according to our computations) for n large.
Remark 7.6. By [Gal18, Theorem 1.2], hypercatalan numbers are bounded from above by the number of fine zonotopal tilings of Z(P n ), which is sequence A060595 in the Online Encyclopedia of Integer Sequences. The known terms are n 3 4 5 6 7 1 2 10 148 7686