Question 6 in [, Section 6] asks whether the scheme is connected. In our example this is not the case. Baues posets appear also as important objects i

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1 A point set whose space of triangulations is disconnected. Francisco Santos Departamento de Matematicas, Estadstica y Computacion Universidad de Cantabria, E-3907, Santander, SPAIN. santos@matesco.unican.es August, 999 Abstract By the \space of triangulations" of a nite point conguration A we mean either of the following two objects: the partially order set (poset) of all polyhedral subdivisions of A (the so-called Baues poset of A) considered as an abstract simplicial complex in the standard way or the graph of triangulations of A, whose vertices are the triangulations of A and whose edges are the geometric bistellar operations. We construct a point conguration in dimension 6 and a triangulation of it which admits no geometric bistellar operations. This triangulation is an isolated point in both the Baues poset and the graph of triangulations of A, which proves for the rst time that these two objects can be disconnected. Our example answers in the negative the generalized Baues question for triangulations. Introduction In this paper we construct explicitly a triangulation of a 6-dimensional point conguration with 34 vertices which admits no geometric bistellar operations (or ips, for short). This triangulation is an isolated element both in the poset of subdivisions (the Baues poset) and in the graph of triangulations of the point conguration. It has been a central open question in polytope combinatorics in the last decade whether point congurations exist for which these objects are disconnected (see, e.g. [40, Challenge 3]). The generalized Baues conjecture of Billera et al. [7, Section 3] would have implied the Baues poset to be homotopy equivalent to a sphere of dimension 37. Our construction is likely to have an impact in algebraic geometry too, via the connections between lattice polytopes and toric varieties [9, 0, 36]. For example, in [, Section ] and [, Section 4] the dierent authors dene certain algebraic schemes based on Baues posets of integer point congurations. This research was partially supported by grant PB97{0358 of the Spanish Direccion General de Ense~nanza Superior e Investigacion Cientca

2 Question 6 in [, Section 6] asks whether the scheme is connected. In our example this is not the case. Baues posets appear also as important objects in oriented matroid theory, zonotopal tilings and hyperplane arrangements, so our result has implications in these areas. We will come back to this particular point at the end of the introduction. Let A be a nite point set in the real ane space R k. A triangulation of A is a geometric simplicial complex with vertices in A which covers the convex hull of A (Denition.). Also important for this introduction are the polyhedral subdivisions of A. We loosely dene them as polyhedral complexes with vertices contained in A which cover the convex hull of A, although a more combinatorial framework (see [6], [9, Chapter 7], [9], or [39, Chapter 9]) is convenient if A is not in convex position, i.e. if some element of A is not a vertex of its convex hull conv(a). There are the following two ways to give a structure to the collection of all triangulations of a point conguration A: (A) Geometric bistellar operations (or \ips"). Geometric bistellar operations are the minimal possible changes which can be made in a triangulation of A to produce a new one: see Denition.3 and Figure. Part (c) in the gure is the familiar diagonal edge ip in two-dimensional triangulations, of frequent use in computational geometry and geometric combinatorics. (Non-geometric) bistellar operations are used in combinatorial topology [7, 7]. In contrast to our results, Pachner [7] shows that any two PL-homeomorphic combinatorial manifolds can be connected by a sequence of them. The graph of triangulations of A has as vertices the triangulations of A and as edges the geometric bistellar operations between them. Graphs of triangulations in dimension two are known to be always connected since the early days of computational geometry [] and extensively used in that area. For the vertex set of a convex polygon, the graph is a classical object in combinatorics, rst studied by Stashe and Tamari [38, 35] and related to associativity structures and to binary trees (see [3, 34]). It is disturbing that in dimension three, and even assuming convex position, we do not know whether the graph is always connected or not. The graph of triangulations of A contains as an induced subgraph the - skeleton of the secondary polytope of A introduced by Gel'fand et al. [8]. This is a polytope of dimension jaj?dim(a)? whose vertices are in bijection with the regular (or coherent) triangulations of A; see [6, 9, 4, 39]. Triangulations with less than jaj? dim(a)? geometric bistellar operations are called ipdecient. Flip-deciency cannot occur either in dimension two or in convex position in dimension three [3]. In non-convex position in dimension three there are highly ip-decient triangulations, with more than n vertices and less than 4n ips for arbitrarily large n [3, Section ]. In dimension four there are triangulations with arbitrarily many vertices and bounded number of ips [3, Sections 3 and 4]. Connectivity of the graph of triangulations is an important question in com-

3 putational geometry, where ips are used to enumerate triangulations or to search for optimal ones [, 5]. But it is also relevant theoretically: if the graph is connected, properties which hold for a particular triangulation and are preserved under ips must hold for any other one. (B) The Baues complex. The polyhedral subdivisions of A form a partially ordered set (poset) whose minimal elements are the triangulations. It is usually referred to as the Baues poset of A and its study is the generalized Baues problem. To be precise, Baues posets were introduced implicitly in [8] and explicitly in [7] in a more general situation where one has an ane projection from the vertex set of a polytope P to A. The Baues poset of is the poset of subdivisions of A which are induced by in a certain sense. When P is a simplex (hence of dimension jaj? ) all subdivisions of A are -induced and this is the case of interest to us. When P is a cube its projection is a zonotope and the -induced subdivisions are its zonotopal tilings [39]. When dim(a) =, the -induced subdivisions are cellular strings [7] (a generalization of monotone paths) on the polytope P. See [30] for a very complete account of the dierent guises in which Baues posets have appeared in the literature and for a survey of what is known about them. We call Baues complex of A the order complex [9] of the Baues poset. That is to say, the abstract simplicial complex whose simplices are the chains in the Baues poset. Billera et al. [7] proved that if dim(a) =, then the Baues complex of any projection : vert(p )! A is homotopy equivalent to a (dim(p )? )- sphere. This resolved a conjecture of Baues [5]. Their conjecture that in general the Baues complex is homotopy equivalent to a (dim(p )? dim(a))? -sphere was since called the generalized Baues conjecture, or GBC. It was inspired by the fact that the ber polytope [8] of the projection (a generalization of the secondary polytope) has dimension dim(p )? dim(a) and its poset of proper faces is isomorphic to a subposet of the Baues poset. Rambau and Ziegler [9] found a counterexample to the GBC, but the particularly interesting cases of P being a simplex or a cube remained open. The case of a cube was shown to be a special case of the simplex case in [3]. In this paper we disprove the simplex case (but the cube case remains open; see below). Baues complexes for various polytope projections have been studied in the last few years. They have been proved to be homotopy equivalent to spheres of the appropriate dimensions in the cases when dim(p )? dim(a) [9], when dim(a) and P is a simplex [4] or a zonotope [], when jaj? dim(a) 4 [4], and when is the canonical projection between two cyclic polytopes [8, 3]. The Baues poset and the graph of triangulations are related in the following way: If two triangulations T and T of A dier by a geometric bistellar operation, then there is a polyhedral subdivision of A whose only proper renements are T and T. Less trivial is the following strong converse of this, proved in [33]: If all the proper renements of a subdivision S of A are triangulations, then S has exactly two such proper renements and they are geometric bistellar neighbors. In other words, geometric bistellar operations are exactly the next-to-minimal elements in the Baues poset. This is additional evidence that 3

4 they provide the right concept of vicinity between triangulations. In particular, the triangulation without geometric bistellar neighbors constructed in this paper does not rene any other proper subdivision of the point conguration in question and, hence, is an isolated element in the Baues poset. The structure of the paper is as follows. Section contains preliminaries on triangulations and geometric bistellar operations. The rest of the paper is divided into Sections and 3, which can be read independently. In Section. we study the relation between the ips in triangulations T and T and the ips in any triangulation which renes the product T T. In Section. we introduce a particular way of rening such a product based on the wellknown staircase triangulation of the product of two simplices. In Section.3 we prove that this renement has no ips when T is any triangulation satisfying certain hypotheses and T is a very simple triangulation on four elements in dimension two (Theorem.6). In Section 3. we describe a triangulation of a four-dimensional point conguration with 8 elements. Then, we compute its ips (Section 3.) and introduce a locally acyclic orientation of its -skeleton (Section 3.3). This triangulation is based on the construction in [3, Section 4] and satises the requirements needed in Theorem.6. In Section 3.4 we modify the coordinates of the point conguration so that they become integer, an essential feature for the applications to algebraic geometry. Hence: Theorem There are two point congurations A R 4 and A R with integer coordinates and with 8 and four elements respectively and a triangulation of the product A A without geometric bistellar neighbors. This triangulation has ? 6 maximal simplices. It would be desirable to improve our construction in any of the following ways, in case this is possible: Smaller dimension: This is open starting in dimension three. Theorem.6 cannot provide a triangulation in dimension less than ve, because the graph of a two dimensional triangulation cannot be oriented locally acyclicly and without reversible edges [3, Remark 9]. In dimension three, we know triangulations which admit locally acyclic orientations without reversible edges, sinks or sources (Remark 3.4). But we have not obtained one whose ippable circuits are \sandwich circuits", as required in Theorem.6. Convex position: Our triangulation has 34 vertices, of which only 96 are extremal, i.e. vertices in the convex hull. The general procedure described in [3, Section 4.4] and [3, Proposition 3.3] substitutes any element of a point conguration by two extremal points, increasing the dimension by one. Applying it 8 times to our triangulation one obtains a triangulation without ips and in convex position, with dimension 34 and 55 vertices. Flip-deciency occurs in convex position starting in dimension four [3]. 4

5 General position: A point conguration of dimension d is said to be in general position if every d points are anely independent. If this happens, then every circuit anely spans A and the Denition.3 of geometric bistellar operation is much simpler: part (ii) is superuous, since the link mentioned in it is empty. Flip-decient triangulations in general position exist starting in dimension three [3]. The techniques in this paper cannot be adapted to general position in any obvious way, since they are based on taking products. Unimodular congurations: Unimodularity is interesting in the connections to algebraic geometry. A point conguration is unimodular if all its ane bases have the same determinant, up to a sign. This implies convex position. Let A (Z 0 ) k be a nonnegative, integer point conguration. Sturmfels [36, Chapter 0] has shown that the radical of every monomial A-graded ideal (mono-a-gi for short) equals the Stanley ideal of some triangulation of A. If A is unimodular, then the converse is true: the Stanley ideal of every triangulation equals the radical of some mono-a-gi (and every mono-a-gi is radical). Using a notion of ips between monomial A-graded ideals devised by Maclagan and Thomas [5], a triangulation without ips whose Stanley ideal equals the radical of some mono-a-gi would imply that the parameter space of A- graded ideals constructed by Sturmfels in [37, Section 5] is not connected. We do not know whether our triangulation without ips has this property. Lifting triangulations and Lawrence polytopes; relation to zonotopal tilings and oriented matroids: If a point conguration A has a centrally symmetric Gale transform, then its convex hull is called a Lawrence polytope [0, 39]. This implies that A is in convex position. Let A be a point conguration, let A denote a Gale transform of A and let M(A ) be the oriented matroid of ane dependences in A (roughly speaking, its set of circuits). Every generic one-element extension of M(A ) induces a triangulation of A. The triangulations which can be obtained in this way are called lifting triangulations (see [0, Section 9.6] or [3, Section 4]). It can be easily proved that our triangulation is not lifting. On the other hand, Lawrence polytopes only have lifting triangulations [3, Section 4.3]. A lifting triangulation without ips would imply that the extension space of the realizable oriented matroid M(A ) is not connected, thus disproving the extension space conjecture of oriented matroid theory [0, pp. 95{96]. This conjecture is equivalent to the following two: via the Bohne-Dress Theorem [0, 39], to the generalized Baues conjecture for projections from cubes to zonotopes; with the results in [3, Section 4.3], to the generalized Baues conjecture for projections from simplices to Lawrence polytopes. Finally, the extension space conjecture is the case k = d? of the following far-reaching conjecture by MacPherson, Mnev and Ziegler [30, Conjecture ]: that the poset of all strong images of rank k of any realizable oriented matroid M of rank d (the OM-Grassmannian of rank k of M) is homotopy equivalent to the usual real Grassmannian G k (R d ). This conjecture is relevant in the combinatorial dierential geometry introduced by MacPherson [6]. 5

6 Triangulations and ips Let A be a nite subset of the real ane space R k and let d denote the dimension of the ane subspace a(a) spanned by A. This is what we call a point conguration of dimension d. Denition. A triangulation of A is any collection T of anely independent subsets of A with the following properties: (i) if S is in T, then every subset of S is in T, i.e. T is an abstract simplicial complex; (ii) if S and S 0 are in T, then conv(s)\conv(s 0 ) is a face of both conv(s) and conv(s 0 ), i.e. T induces a geometric simplicial complex in R k ; (iii) [ ST conv(s) = conv(a), i.e. T covers the convex hull of A. We will call faces or simplices of T all its elements, maximal faces or maximal simplices of T the elements of dimension d and facets of T the facets of the maximal simplices, i.e. the simplices of dimension d?. This deviates from standard use in simplicial complexes, where the facets are the maximal simplices. With our conventions, every facet of conv(a) is triangulated by facets of T. A facet is interior if its convex hull intersects the interior of conv(a) and is a boundary facet otherwise. It has some combinatorial advantages to consider elements of a triangulation only the maximal simplices, as is done in recent literature such as [6], [] or [9]. Here we prefer to use the convention that lower dimensional ones are also elements, in order to work more easily with the links and stars of simplices, dened as follows. If S T, then star T (S) = fs 0 T : S [ S 0 T g and link T (S) = fs 0 T : S [ S 0 T ; S \ S 0 = ;g: The following result appears, for example, in []: Lemma. Let A be a point conguration and let T be an abstract simplicial complex of pure dimension dim(a) with vertices contained in A and containing only anely independent subsets. T is a triangulation of A if and only if: (i) The link of every interior facet S of T has exactly two elements, which lie on opposite sides of the hyperplane of a(a) spanned by S. (ii) There exists a point in conv(a) which lies in the convex hull of exactly one maximal simplex of T. Geometric bistellar operations arose in the following terms in the work of Gel'fand, Kapranov and Zelevinsky [8] [9, Chapter 7], who called them modications. Following the terminology of matroid theory, we call a minimal anely dependent subset of A a circuit (see [0] or [39] for details). The Radon partition of a circuit Z is the unique partition Z = Z + [ Z? such that the convex hulls of the two parts intersect. Equivalently, Z + and Z? contain respectively the elements with positive and negative coecient in the unique (up to a scalar multiple) ane dependence equation on Z. The pair (Z + ; Z? ) is called an oriented circuit; of course, if (Z + ; Z? ) is an oriented circuit, then so is (Z? ; Z + ), 6

7 and the two of them are the only orientations of Z = Z + [Z?. Since our interest will always be in oriented circuits we will use the word circuits for the pairs (Z + ; Z? ) and will call the underlying unoriented circuit the support of (Z + ; Z? ). The support Z of a circuit (Z + ; Z? ) admits exactly two triangulations T + (Z) and T? (Z), which have Z + and Z? as their unique minimal non-faces. I.e: T + (Z) := fs Z : Z + 6 Sg; T? (Z) := fs Z : Z? 6 Sg: Denition.3 Let T be a triangulation of A and let (Z + ; Z? ) A be a circuit of A. Suppose that the following conditions are satised: (i) The triangulation T + (Z) is a subcomplex of T. (ii) All the maximal simplices of T + (Z) have the same link L in T. In particular, T + (Z)L is a subcomplex of T. Here AB := fs [T : S A; T Bg is the join of two simplicial complexes A and B. Then, we can obtain a new triangulation T 0 of A by replacing the subcomplex T + (Z) L of T by the complex T? (Z) L. This operation of changing the triangulation is called a geometric bistellar operation, geometric bistellar ip (or a ip, for short) supported on the circuit (Z + ; Z? ). We call (Z + ; Z? ) a ippable circuit and we say that T and T 0 are geometric bistellar neighbors. Figure shows the three types of geometric bistellar ips possible in dimension two (parts (a), (b) and (c)) and the two types in general position in dimension three (parts (d) and (e)). See [33] for a generalization of the concept of ip in the general framework of Baues posets and ber polytopes. (a) (d) (b) (e) (c) Figure : Examples of geometric bistellar operations. Our denition of ip supported on a circuit explicitly assumes the circuit oriented so that the star of the negative part is \ipped out" and the star of the positive part is \ipped in". This convention is not made by other authors (see [9, page 3]) and will be relevant in our exposition. It is sometimes more convenient to focus on ippable facets, which we now introduce, instead of ippable circuits. Let S be the common facet of two 7

8 maximal simplices S [ fa g and S [ fa g of a triangulation T. The d + elements of S [ fa ; a g anely span a(a). Hence, there is a unique circuit (Z + ; Z? ) with support contained in them. This circuit can be oriented so that a ; a Z +. We call (Z + ; Z? ) the circuit supported on S. We say that S is a ippable facet of T if there is a ip supported on (Z + ; Z? ). In the following statement, if (Z + [ Z? ) is full-dimensional, then the link L of T + (Z) is considered to have one maximal simplex (the empty set). Lemma.4 Let T be a triangulation of a point conguration A of dimension d. Let (Z + ; Z? ) be a ippable circuit, with jz + j = k. Let l be the number of maximal simplices in the common link L of T + (Z) in T. Then, (Z + ; Z? ) is supported on l? k facets of T. In particular: (i) If T uses all the elements of A as vertices, then every ippable circuit is supported on at least one ippable facet. (ii) If conv(z + [ Z? ) has dimension d 0 and intersects the interior of conv(a), then (Z + ; Z? ) is supported on at least (d +? d 0 )? k ippable facets. Proof: The ippable facets of T on which the circuit (Z + ; Z? ) is supported are those of the form S T where S is an interior facet of T + (Z) and T is a maximal simplex in L. The triangulation T + (Z) has k maximal simplices and every pair of them are adjacent. Hence it has? k interior facets. Every proper subset of Z + is a simplex in T + (Z) T. If every element of A is a vertex in T, then Z + has at least two elements, which implies part (i). If conv(z + [Z? ) intersects the interior of conv(a), then L is a triangulation of a (d? d 0 )-sphere and, hence, it has at least d +? d 0 maximal simplices. Rening the product of two triangulations Throughout this section, let T and T be triangulations using all the elements of respective point congurations A of dimension d and A of dimension d. Their product T T is the polyhedral subdivision of A A into products S S of simplices S T and S T. Clearly, T T uses all the elements of A A.. Flippable circuits in renements of T T Here we let T be any triangulation of A A rening T T. We are going to study the relation between the ips in T and the ips in T and T. We consider the two natural projections : A A! A and : A A! A and will often use the following properties. Lemma. (i) T = f (S) : S T g and T = f (S) : S T g. (ii) For every simplex S T and for i f; g: star Ti ( i (S)) = i (star T (S)) and link Ti ( i (S)) i (link T (S)): 8

9 By an ane equality on a point conguration A we mean a valid expression of the form P az + a a = P bz? b b, in which Z + and Z? are disjoint subsets of A, all the coecients a and b are strictly positive reals, and P a a = P b b. For example, there is a unique ane equality (up to a scalar multiple) associated to any circuit (Z + ; Z? ) of A. Any ane equality C on A A is projected to ane equalities on A i, that we denote i (C) (i f; g), in the following natural way: substitute every point (a ; a ) A A which appears in the equality by its projection a i and reduce the coecients of points which at the end appear more than once. One or both of the projected equalities might result in a trivial equality 0 = 0. This happens, for example, for the equality (a; a 0 ) + (b; b 0 ) = (a; b 0 ) + (b; a 0 ) for any given points a; b A and a 0 ; b 0 A. We are interested in the ane equalities on A A associated to the ippable circuits of T. It is more convenient to look at ippable facets of T instead, which can be done by part (i) of Lemma.4. Interior facets of T fall into the following three categories: (A) Facets which are interior to a maximal cell S S of T T. The points involved in the equality are contained in S S, whose two projections are anely independent. Hence, the projected equalities are trivial. (B ) Facets common to two maximal cells S S and S 0 S of T T with the same second factor. Now the points involved in the equality are contained in (S [ S 0 ) S. The second projection has to be a trivial equality as in the previous case. The rst projection can in principle be trivial or be the equality associated to the circuit contained in S [ S 0. Lemmas. and.3 below show that the second is always the case and that the circuit in question is ippable in T, if the original facet is ippable in T. (B ) Facets common to two maximal cells S S and S S 0 of T T with the same rst factor. This case is analogous to (B ). Lemma. Let S be an interior facet common to two maximal cells S S and S 0 S of T T with the same second factor. Let (Z + ; Z? ) be the circuit on that facet and let C denote the ane equality associated to the circuit. Then, (C) is the ane equality on a circuit (Z 0 + ; Z0?) of A, with Z 0 + (Z + ) and Z 0? (Z? ). Proof: Let Z 0 + and Z 0? be the sets of points on one and the other side of the projected equality (C). The conditions Z 0 + (Z + ) and Z 0? (Z? ) are obvious. Let (a; b) and (a 0 ; b 0 ) be the two points in link T (S). Clearly a 6= a 0 and they are not in (S). Hence, a and a 0 appear in (C) and (C) is not trivial. Since Z 0 + [ Z 0? (Z + [ Z? ) is contained in the union of two adjacent simplices of T, (Z 0 + ; Z0?) must be a circuit. Lemma.3 In the conditions of Lemma., if (Z + ; Z? ) supports a ip of T, then: 9

10 (i) For any a (Z + ) there is a unique b A such that (a; b) Z + and there is no b A such that (a; b) Z?. (ii) Z 0 + = (Z + ) and Z 0? = (Z? ). (iii) (Z 0 + ; Z0?) supports a ip of T. (iv) There is no pair of elements a Z 0 + and b A such that (a; b) is in the common link in T of the positive triangulation of (Z + ; Z? ). (v) (Z + ) = (Z? ). Proof: We denote Z = Z + [ Z? and Z 0 = Z 0 + [ Z 0?. Let a (Z + ). Let b A with (a; b) Z +. Since (Z + ; Z? ) is ippable, Z n (a; b) is a simplex of T and hence projects to a simplex of T. On the other hand, Z projects to a dependent set. This implies that no other point (a; b 00 ) is contained in Z because otherwise (Z) = (Z n (a; b)). This nishes part (i). The containments Z 0 + (Z + ) and Z 0? (Z? ) appear in the statement of Lemma.. Their converses hold because any element in Z 0 + n (Z + ) or in Z 0? n (Z? ) must appear as the rst coordinate of points both in Z + and Z?, contradicting part (i). This proves part (ii). Let us now see part (iii). By parts (i) and (ii), for any a Z 0 + there is a b A with (a; b) Z + and with Z 0 n a = (Z n (a; b)). Since (Z + ; Z? ) is ippable in T, Z 0 n a is in T. Hence, T contains as a subcomplex the positive triangulation of Z 0. Let a and a 0 be two dierent elements in Z 0 +. We have to prove that link T (Z 0 n a) = link T (Z 0 n a 0 ), for which we check only one of the two containments. Let b; b 0 A with (a; b); (a 0 ; b 0 ) Z +, as before. We have link T (Z 0 n a) = link T ( (Z n(a; b))) (link T (Z n(a; b))) = (link T (Z n(a 0 ; b 0 ))), where the rst equality holds by part (ii) of the statement, the containment in the middle by part (ii) of Lemma. and the last equality because (Z + ; Z? ) is ippable. With similar arguments (link T (Z n(a 0 ; b 0 ))) star T ( (Z n(a 0 ; b 0 ))) = star T (Z 0 n a 0 ). In conclusion, link T (Z 0 n a) star T (Z 0 n a 0 ). Since no element of Z 0 n a 0 can appear in link T (Z 0 n a), we have link T (Z 0 n a) link T (Z 0 n a 0 ). For part (iv), if (a; b 0 ) is a point in the common link of the positive triangulation of (Z + ; Z? ) with a Z 0 + = (Z + ), then let (a; b) Z + as before. We have that Z [(a; b 0 )n(a; b) is a simplex in T. But Z 0 = (Z) = (Z [(a; b 0 )n(a; b)). This is impossible since Z 0 is dependent in A. Part (v) is trivial: the second projection (C) of the ane equality on the circuit (Z + ; Z? ) is the trivial equality. This implies that every element in (Z) is the second coordinate of points in both Z + and Z?.. The staircase renement of T T We rst recall what the staircase triangulation of the product of two simplices is. Let S and S be two simplices of dimensions d and d, and suppose that their vertices are given in a specic order. The total order in the vertices of S and S induces the following partial order in the vertices of the product S S : (a; b) (a 0 ; b 0 ) if and only if a a 0 and b b 0. Every chain in this partial order 0

11 is an anely independent subset of S S. The collection of all such chains is a triangulation of S S, described for example in [9, Section 7.3.D] and [4, p. 8] and used also in algebraic topology [6, page 69]. All the maximal simplices contain the vertices (a min ; b min ) and (a max ; b max ), where a min and a max (respectively b min and b max ) are the minimum and maximum vertices of S (respectively of S ). More visually, if we arrange the vertices of S S in a (d +)(d +) rectangular grid, then the maximal simplices of our triangulation are the monotone staircases from (a min ; b min ) to (a max ; b max ). For this reason it is called the staircase triangulation of S S and we denote it stair(s S ). We will encounter an example of a staircase triangulation in Remark 3.4. Any pair of adjacent maximal simplices in stair(s S ) dier in the replacement of a point (a; b 0 ) by a point (a 0 ; b), for consecutive pairs of vertices a < a 0 and b < b 0 in the total orderings of S and S. In particular, every ip of stair(s S ) is supported on a circuit (f(a; b 0 ); (a 0 ; b)g; f(a; b); (a 0 ; b 0 )g), for consecutive pairs of vertices a < a 0 and b < b 0. The converse is also true although we do not need it: stair(s S ) has exactly d d ips, supported on the circuits of that form. For the sequel, it is convenient to reformulate the construction of the staircase triangulation as follows: We will express the ordering of the vertices of S and S by giving an acyclic orientation to their respective -skeletons. Then, the circuits which produce ips are those of the form (f(a; b 0 ); (a 0 ; b)g; f(a; b); (a 0 ; b 0 )g) where fa; a 0 g and fb; b 0 g are edges of S and S whose reversal produces no directed cycle. We call such edges reversible edges. Suppose now that we have two triangulations T and T of two point congurations A and A as in Section., and that we are given respective locally acyclic orientations of the -skeletons of T and T. By this we mean orientations of the edges which are acyclic on every simplex. Then, rening each product S S T T in its staircase way (according to the given locally acyclic orientations) produces a triangulation of A A. That this is indeed a triangulation follows from the fact that in the staircase triangulation of the product of two simplices S S, each face F F is triangulated according to the staircase triangulation corresponding to the restrictions of the orderings of S to F and of S to F. We call this triangulation the staircase renement of T T and denote it stair(t T ). Observe that the staircase renement depends on the orientations of the -skeletons of T and T. In our notation, T and T implicitly denote not only the triangulations but also specic locally acyclic orientations of their -skeletons. The analysis of ippable circuits carried out in Section. gives the following: Lemma.4 If the locally acyclic orientation of at least one of T and T does not have any reversible edge, then every ippable circuit of stair(t T ) projects, in the sense of Lemma., either to a ippable circuit of T or to a ippable circuit of T. Proof: Recall the three types (A), (B ) and (B ) of possible interior facets in

12 a renement of T T that we mentioned before Lemma.. Those of type (B ) are in the conditions of Lemma. and by Lemma.3 they project to a ippable circuit of T. Analogue lemmas for the second instead of the rst projection deal with facets of type (B ). Hence, we only need to deal with facets of type (A), i.e. facets interior to a cell S S of T T. A ip in one of these facets will be in particular a ip in the staircase triangulation of S S. Such a ip is supported in a circuit of the form (f(a; b 0 ); (a 0 ; b)g; f(a; b); (a 0 ; b 0 )g) where fa; a 0 g is a reversible edge in the orientation of S and fb; b 0 g a reversible edge in the orientation of S. For this circuit to be ippable it is clearly necessary (and sucient, although we do not need it) that both edges fa; a 0 g and fb; b 0 g be reversible in in every simplex of T and T in which they appear. Our hypotheses forbid the existence of at least one of these edges..3 Towards a triangulation without ips Let A = fo; p; q; rg R be a point conguration whose unique circuit is (fp; q; rg; fog). In other words, A consists of four points in general position with o in the interior of the triangle conv(fp; q; rg). Let T be the triangulation whose maximal simplices are fo; p; qg, fo; p; rg, and fo; q; rg. We consider its -skeleton oriented in the following locally acyclic way: o is a global source and there is a cycle p! q! r! p. The statement of Theorem.6 below needs an explanation. We have already said what we mean by a locally acyclic orientation with no reversible edges. Having no sinks means, as in graph theory, that every vertex is the source of at least one edge in the orientation. Let us see what we mean by \all its ippable circuits are sandwich circuits". Let T be a triangulation of a point conguration A and let (Z + ; Z? ) be a ippable circuit such that Z + has at least two elements. Every ippable circuit has this property when all the points of A are vertices in T, as in the proof of part (i) of Lemma.4. Then, every pair fa + ; a? g with a + Z + and a? Z? is an edge in T. Denition.5 In the above conditions, we say that the ippable circuit (Z + ; Z? ) is a sandwich circuit if: (i) every a + of Z + is either the source in all the edges fa + ; a? g with a? Z? or the sink in all those edges; and (ii) there is at least one a + Z + which is the source and at least one which is the sink. The intuitive meaning of this is that Z + [ Z? is ordered in such a way that all the negative elements are consecutive and there are some positive elements at the beginning and some at the end. Theorem.6 Let T be a triangulation using all the points of a point con- guration A and whose -skeleton has been oriented in a locally acyclic way, without global sinks or reversible edges, and with the property that all its ippable circuits are sandwich circuits. Then, the staircase renement stair(t T ) of T T does not have any ips, where T is the -dimensional triangulation described above.

13 Proof: By Lemma.4, every ippable circuit of stair(t T ) projects (in the sense of Lemma.) either to a ippable circuit of T or to the unique ippable circuit (fp; q; rg; fog) of T. We discard these two cases in the following two lemmas. Lemma.7 In the conditions of Theorem.6, no ippable circuit of stair(t T ) projects via to a ippable circuit of T. Proof: The proof will be based in the fact that all ippable circuits of T are sandwich circuits. Let (Z + ; Z? ) be a ippable circuit of T T with support Z = Z + [ Z?. Suppose that it projects in the sense of Lemma. to a ippable circuit (Z 0 + ; Z0?) of T with support Z 0. By part (v) of Lemma.3, (Z) = (Z? ) is contained in a simplex S of T, say S = fo; p; qg. By parts (i) and (ii) of Lemma.3, for every a Z 0 + there is a unique b A such that (a; b) Z and, moreover, (a; b) Z +. We now claim that: If a is a sink in the edges fa; a? g with a? Z? 0, then b is the last element of S (i.e., b = q): Let a 0 be a point which is a source in all the edges fa; a? g with a? Z?, 0 which exists because Z 0 is a sandwich circuit. Let b 0 A be the unique element with (a 0 ; b 0 ) Z +. Let S be a simplex of stair(t T ) containing Z n (a 0 ; b 0 ) and with (S) = S, which exists because Z is ippable and (Z) = S. Let us look at S as a staircase in the rectangular grid representing the vertices of (S) S. Parts (i) and (iv) of Lemma.3 can be rephrased saying that in every column corresponding to a positive element of Z 0 + there is a unique element of S, and it belongs to Z +. The fact that S is a staircase implies that the element used in a column of the last part of the sandwich is the last one in the ordering of that column, i.e. the last element of S. If a is a source in the edges fa; a? g with a? element of S (i.e., b = o): Same proof. Z 0?, then b is the rst From the above we conclude that (Z + ) = fo; qg. By part (v) of Lemma.3 (Z) = fo; qg and, hence, Z is not only contained in simplices of stair(t T ) which project to fo; p; qg but also in simplices which project to fo; q; rg. This is impossible since the same arguments applied to these simplices would prove that (Z) = fo; rg. Lemma.8 In the conditions of Theorem.6, no ippable circuit of stair(t T ) projects via to the ippable circuit of T. Proof: Here we use that the triangulation T has no sinks, and Lemmas.3 and. applied to the second projection. Let (Z + ; Z? ) be a ippable circuit in stair(t T ) with support Z and suppose that it projects, in the sense of Lemma., to the circuit (fp; q; rg; fog) of A. 3

14 Let S be any simplex in T such that (Z) S, which exists since (Z? ) = (Z) and Z is ippable. Let a, b and c be the unique elements of A such that (a; p); (b; q); (c; r) Z +, as given by part (ii) of Lemma.3. We will prove below that c is the last element in the total order on the vertices of S. In the same way one proves that a and b are the last element of S. In particular, a = b = c. This implies that (Z + ) = (Z? ) is a single element, and that this element is the sink of every simplex of T containing it. This nishes the proof since T has no global sinks. Let us prove the claim that c is the last element in S. Let S be a simplex of stair(t T ) containing Z n (a; p) and with (S) = S, which exists since Z is ippable and (Z) S. By part (i) of Lemma.3 p 6 (S), i.e. (S) fo; q; rg. Moreover, when we look at S as a staircase in the rectangular grid formed with S and fo; q; rg we have a unique element in the row corresponding to r. Since this is the last row, this element has to be the last element in the row. 3 A construction in dimension four This section is devoted to describing a triangulation in dimension four which satises the hypotheses of the T of Theorem Description of the triangulation Throughout Section 3, let A be the following point conguration in R 4, with 8 elements. Indices are regarded modulo 8: (i) The origin O = (0; 0; 0; 0). (ii) The 8 points v j :=? 0; 0; cos? 4 j ; sin? 4 j, for j = 0; : : : ; 7. (iii) The 8 points h i :=? cos? 4 i ; sin? 4 i ; 0; 0, for i = 0; : : : ; 7. (iv) The 3 points t i;j := h i + v j? O =? cos? 4 i ; sin? 4 i ; cos? 4 j ; sin? 4 j, for all values of i; j f0; : : : ; 7g with i + j odd. (v) The 3 points s i+ with i + j even. = (t i +t i+;j )=, for all values of i; j f0; : : : ; 7g The convex hull of A is a 4-polytope whose vertices are the 3 points t i;j and with the following 48 facets: Eight square anti-prisms P k+ points t i;j with j fk; k + g, (k = 0; : : : ; 7) whose eight vertices are the Another eight square anti-prisms P 0 k+ (k = 0; : : : ; 7) whose eight vertices are the points t i;j with i fk; k + g, and 3 tetrahedra T i;j (i; j f0; : : : ; 7g; i + j even) having as vertices t i?;j, t i+;j, t i and t i. 4

15 The two series of 8 anti-prisms form two solid tori in the three-dimensional topological which are glued to one another along edges, leaving some spaces for the 3 tetrahedra T i;j. The point s i+ lies in an edge incident to the tetrahedra T i;j and T i+ and to the anti-prisms P j+ and P 0. The point i+ v j (respectively h i ) lies in the square between the two anti-prisms P j? and P j+ (respectively P 0 i? and P 0 ). i+ The ane symmetry group of A has 8 elements, since any of the 6 antiprisms can be sent to any other one in eight ways. We will be interested in the following subgroup G with 64 elements. Let g h and g v denote the rotations of angle =4 on the rst two and on the last two coordinates respectively, and let g t be the exchange of the rst two and last two coordinates. Then, G is generated by g := g h, g := g v, g 3 := g h g v = g v g h, and g 4 := g t. Either of the rst two is redundant. All these isometries x the origin O. Table shows the permutations induced on the other points of A. g : (h i 7! h i+ ; v j 7! v j ; t i;j 7! t i+;j ; s i+ 7! s i+ 5 ) g : (h i 7! h i ; v j 7! v j+ ; t i;j 7! t i ; s i+ 7! s i+ 5 ) g 3 : (h i 7! h i+ ; v j 7! v j+ ; t i;j 7! t i+ ; s i+ 7! s i+ 3 3 ) g 4 : (h i 7! v i ; v j 7! h j ; t i;j 7! t j;i ; s i+ 7! s j+ ;i+ ) Table : The ane symmetries of the triangulation T. We now dene a triangulation T of A which has G as its symmetry group. For this, we give in Table a representative of each of the nine orbits of maximal simplices in T. We name the orbits with the rst nine letters of the capital Greek alphabet. The indices i and j in Table range over all the possibilities modulo 8 with i + j even. This produces 3 simplices in each orbit. The other 3 are obtained by applying the transformation g 4 and we name them as follows: A 0 j;i := g 4 (A i;j ), B 0 j;i := g 4 (B i;j ),? 0 j;i := g 4(? i;j ), 0 j;i := g 4( i;j ), E 0 j;i := g 4 (E i;j ), Z 0 j;i := g 4 (Z i;j ), H 0 j;i := g 4 (H i;j ), 0 j;i := g 4( i;j ), and I 0 j;i := g 4 (I i;j ). That these 9 64 simplices of dimension four are indeed the maximal simplices of a triangulation of A will be proved afterwards. Remark 3.4 at the end of Section 3.3 may help to clarify this construction and its genesis. A i;j := fo ; v j+ ; t i ; s i? ; s i+ g B i;j := fo ; v j+ ; t i+;j ; s i? ; s i+ g? i;j := fo ; v j+ ; t i+;j ; t i+ ; s i+ g i;j := fo ; v j+ ; t i?;j ; t i ; s i? g E i;j := fo ; v j+ ; v j ; t i+;j ; s i? g Z i;j := fo ; v j+ ; v j ; t i?;j ; s i? g H i;j := fv j+ ; v j ; t i?;j ; t i+;j ; s i? g i;j := fv j+ ; t i?;j ; t i+;j ; s i? ; s i+ g I i;j := fv j+ ; t i ; t i?;j ; s i? ; s i+ g Table : Representatives for the nine orbits of maximal simplices in T. 5

16 There are 7 orbits of boundary facets in T, with representatives given in Table 3. The rst ve orbits triangulate the 6 square anti-prisms. The last two orbits triangulate the 3 tetrahedra T i;j. For example, T 0;0 is triangulated with 0;0 n v, 0 0;0 n h, I 0;0 n v and I 0 0;0 n h. Each of the four 3-simplices joins the two points s? and ;? s i+, which lie in two opposite edges of T 0;0 to one of the remaining four edges of T 0;0.? i;j n O = fv j+ ; t i+;j ; t i+ ; s i+ g P j+ H i;j n s i? = fv j+ ; v j ; t i?;j ; t i+;j g P j+ H i+ n v j+ = fv j+ ; t i ; t i+ ; s i+ g P j+ i;j n s i? I i;j n s i? = fv j+ ; t i?;j ; t i+;j ; s i+ g P j+ = fv j+ ; t i?;j ; t i ; s i+ g P j+ i;j n v j+ = ft i?;j ; t i+;j ; s i? ; s i+ g T i;j I i;j n v j+ = ft i ; t i?;j ; s i? ; s i+ g T i;j Table 3: Representatives for the seven orbits of boundary facets in T. There are 9 orbits of interior facets in the triangulation T, with representatives given in Table 4. For each one we write its two representations as the complement of a vertex in a maximal simplex of T followed by its four vertices. As before, the indices i and j range over all the possibilities modulo 8 and with i + j even. This gives 3 elements of each orbit and the other 3 are obtained applying the transformation g 4 to them. i;j := A i;j n O = I i;j n t i?;j = fv j+ ; t i ; s i+ ; s i? g i;j := A i;j n v j+ = B 0 j;i n h i+ = fo; t i ; s i? ; s i+ g i;j := A i;j n t i = B i;j n t i+;j = fo; v j+ ; s i? ; s i+ g i;j := A i;j n s i? = Z i+ n v j+ = fo; v j+ ; t i ; s i+ g i;j := A i;j n s i+ = i;j n t i?;j = fo; v j+ ; t i ; s i? g i;j := B i;j n O = i;j n t i?;j = fv j+ ; t i+;j ; s i? ; s i+ g i;j := B i;j n s i? =? i;j n t i+ = fo; v j+ ; t i+;j ; s i+ g i;j := B i;j n s i+ = E i;j n v j = fo; v j+ ; t i+;j ; s i? g i;j :=? i? n v j = 0 n j;i h i+ = fo; t i ; t i+;j ; s i? g i;j :=? i? n t i = E i;j n v j+ = fo; v j ; t i+;j ; s i? g i;j :=? i? n s i? = i+ n s i+ 3 = fo; v j ; t i ; t i+;j g i;j := i;j n O = I i;j n s i+ = fv j+; t i?;j ; t i ; s i? g i;j := i;j n t i = Z i;j n v j = fo; v j+ ; t i?;j ; s i? g i;j := E i;j n O = H i;j n t i?;j = fv j+ ; v j ; t i+;j ; s i? g o i;j := E i;j n t i+;j = Z i;j n t i?;j = fo; v j+ ; v j ; s i? g i;j := E i;j n s i? = Z i+;j n s i+ 3 = fo; v j+; v j ; t i+;j g i;j := Z i;j n O = H i;j n t i+;j = fv j+ ; v j ; t i?;j ; s i? g i;j := H i;j n v j = i;j n s i+ = fv j+; t i?;j ; t i+;j ; s i? g i;j := i;j n t i+;j = I i;j n t i = fv j+ ; t i?;j ; s i+ ; s i? g Table 4: Representatives for the nineteen orbits of interior facets in T. 6

17 C( 0;0 ); C( 0;0 ); C( 0;0 ) : ( p? )O + +p t?;0 + s ; = p v + t 0; + p s? ;? C( 0;0 ) : ( p + )v + ( p + )h + s? ;? = O + ( + p )s ; C( 0;0 ) = C( 0 0;0); C( 0;0 ) = C( 0 0;0) : t ;0 + t 0; = s ; C( 0;0 ) : s? ;? + ( + p )v + s ; = (? p )O + p v + ( + p )t 0; C( 0;0 ); C( 0;0 ); C(o 0;0 ); C( 0;0 ) : (? p )O + p t ;0 + ( + p )t?;0 = v + s? ;? C( 0;0 ) : s? ;? + ( + p )t ; = (? p )O + p v + ( + p )t ;0 C( 0;0 ) : p s ; + v 0 + (3? p )O = v + t ;0 + (? p )s? ;? C( 0;0 ) = C( 0 ;?) : v 0 + h = O + t ;0 C( 0;0 ) : ( + p )t 0;? + p v = ( p? )O + t ;0 + s? ;? C( 0;0 ) = C( 0 ;?) : p s? ;? + p s ;? 3 = (p? )O + ( p + )t 0;? C( 0;0 ) : t 0; + p v 0 + (? p )O + t?;0 = v + s? ;? C( 0;0 ) : s? ;? + ( + p )s 3 ;? + ( + p )v = O + ( + p )v 0 + ( + p )t ;0 C( 0;0 ) : p v0 + s ; = v + p t?;0 + +p t ;0 Table 5: Ane equalities for the circuits supported on interior facets of T. 3. T is a triangulation, with two orbits of ips In order to prove that T is a triangulation and compute its ips, in Table 5 we display a list of ane equalities valid on A. Each equality involves only the points of A in or joined to one representative of one of the 9 orbits of interior facets in T, although dierent orbits may produce the same equality. In other words, the points on the left and right part of each equality are the positive and negative parts of the circuit supported on the corresponding interior facet of T. The group G acts freely over the circuits supported on facets except for those of types (same circuit as ), and, in which the stabilizer has exactly two elements, as indicated in Table 5. The reader can check the equalities with a symbolic computation program or can verify them by hand in the following way: rst check that in every equation the sum of coecients is equal on both sides; second, forget the appearances of the point O and perform the following substitutions: t k;l = h k + v l, s k+ ;l+ (h k + h k+ + v l + v l+ )=, h k+ + h k? = p h k, and v l+ + v l? = p v l. After this is done all terms should cancel out in all the equations. Lemma 3. T is a triangulation of A. Proof: We rst check that A i;j is anely independent. For this we compute the determinant of the points O, v 0, t 0;0, s? ;? 3, and s ;? (we are taking i = 0 = 7

18 and j =?): p ? p p p p p ? + p? p 4 4 =? p 4 p 4? + p? p 4 4 p = 8 : Since A i;j is an ane basis, for any point a A n A i;j and b A i;j, if a and b have non-zero coecient in the unique ane equality supported on A i;j [fag, then A i;j [fagnfbg is again a basis. Hence, the ane equalities C( i;j ), C( i;j ), C( i;j ), and C( i;j ) imply, respectively, that I i;j, B i;j, Z i+, and i;j are also bases. Using the same argument and the other equalities we conclude that? i;j, E i;j, H i;j, and i;j are bases too. Thus, T consists only of independent subsets of A and we can apply Lemma. to it. The lists of boundary and interior facets of T given in Tables 3 and 4 are complete and non-redundant in the sense that exactly one representative of each of the ve facets of each of the nine orbits of 4-simplices appears exactly once in them. Moreover, in each of C( i;j ); : : : ; C( i;j ) the two vertices joined to the interior facet in question appear both on the left-hand side of the equality. This implies that these two points are in opposite sides of the interior facet. With this and Lemma. we only have to check that some point of conv(a) lies in the convex hull of exactly one simplex of T. For a generic point in the interior of conv(a) but suciently close to a facet of conv(a), this follows from the fact that T induces a triangulation on that facet. Lemma 3. T has two orbits of ips under the action of its symmetry group G, with representatives supported on the circuits (fo; t i?;j ; s i+ g; fv j+ ; t i ; s i? g) and (fv j ; s i+ g; fv j+ ; t i?;j ; t i+;j g): Proof: That (fo; t i?;j ; s i+ g; fv j+ ; t i ; s i? g) and (fv j ; s i+ g; fv j+ ; t i?;j ; t i+;j g) are circuits follows respectively from the equalities C( 0;0 ) and C( 0;0 ) of Table 5. The rst circuit spans conv(a) and the maximal simplices in its positive triangulation are A i;j, i;j, and I i;j. The second circuit spans the anti-prism facet P j+ of conv(a). are H i;j n s i? The maximal simplices in its positive triangulation and i;j n s i?. In other words, T contains its positive triangulation and all the maximal simplices in it have the same link. Hence, both circuits are ippable. We now check that the other 5 orbits of interior facets are not ippable, using Lemma.4 for this. In the rst place, if a ippable circuit has more than two positive elements, then it must be supported on at least? 3 facets of T. This implies that the circuits supported on the facets of types,,,, and 8

19 are not ippable, because they are supported only on one facet of T each and their positive parts have at least three elements. The circuits supported on facets of types (same circuit as, o, and ),,,, and all contain the element O and hence we can apply part (ii) of Lemma.4 to them. According to it we would need at least 3,, 3,, and 3 ippable facets for them respectively, but they are supported in only 4,,,, and facets. Only the facets of types and remain to be checked. 0;0 and 0;0 are both supported on the circuit (ft ;0 ; t 0; g; fs ; since the link of ft ;0 ; s ; g).this circuit is not ippable g in T contains fo; v ; v g (simplex Z ; ), the link of ft 0; ; s g ; contains fo; v ; t ; g (simplex? 0;0 ) and the points v and t ; g lie both on the same side of the hyperplane spanned by fo; v ; t ;0 ; t 0; ; s ; (equation C( ; )). 3.3 A locally acyclic orientation of the edges of T We now introduce an orientation of the -skeleton of the triangulation T. There are 4 orbits of edges. In Table 6 below we show a representative for each orbit, and a particular maximal simplex containing it. We orient the representatives from the vertex which appears rst to the second and let each orbit be oriented by the action of the symmetry group G. fo; v j g A i fo; t i g A i;j fv j ; t i+;j g? i+ fv j ; s i? ; g A i? ft i?;j ; t i+;j g H i;j fs i? ; ; t ig A i;j ft i?;j ; s i+ ; g i;j fv j ; v j+ g E i;j fo; s i+ ; g A i;j ft i ; v j g i+ fs i+ ; 3 ; v jg E i+ ft i ; t i+;j g? i? fs i? ; ; s i+ g A i;j ; ft i ; s i? ; g B i? Table 6: The fourteen orbits of edges in T. That this orientation is locally acyclic (i.e. acyclic on every simplex) can be seen in Table 7, where the total order induced on the vertices of each maximal representative simplex is shown. It has no reversible edges since the reversal of any edge produces a cycle in the simplex included after it in Table 6. The only vertices which are sinks of some maximal simplex in Table 7 are v j+, t i and s i+, but the rst two are not sinks in A i;j and the last one is not a sink in? i;j. Hence, the orientation has no sinks. Finally, let us see that the two types of ippable circuits of T (see Lemma 3.) are sandwich circuits, in the terms of Denition.5. This holds since the orientation of the -skeleton of T is compatible only with the following total order on the supports of the circuits: + O! t + i?;j! s? i?! v? j+!? t i! s + i+ ; 9