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Symmetry, oriented matroids and two conjectures of Michel Las Vergnas
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Symmetry, oriented matroids and two conjectures of Michel Las Vergnas Ilda P.F. da Silva Faculdade de Ciencias da Universidade de Lisboa/CFCUL, Dep.to de Matemática, Campo Grande, Edificio C6-Piso 2, P-1749-016 Lisbon, Portugal
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Article history: Available online xxxx To the memory of Michel Las Vergnas
The paper has two parts. In the first part we survey the existing results on the cube conjecture of Las Vergnas. This conjecture claims that the orientation of the matroid of the cube is determined by the symmetries of the underlying matroid. The second part deals with Euclidean representations of matroids as geometric simplicial complexes defined by symmetry properties abstracting those of zonotopes. Both sections involve arguments concerning simplicial regions illustrating, once more, the fundamental importance of the simplex conjecture of Las Vergnas. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Two of the long standing conjectures of M. Las Vergnas are: the simplex conjecture and the cube conjecture. They say: Las Vergnas simplex conjecture ([22]). Every oriented matroid has an acyclic reorientation whose LVface lattice is a simplex. Equivalently, every arrangement of pseudospheres contains a region that is a simplex. Las Vergnas cube conjecture ([23]). The matroid of the affine dependencies of the vertices of a real cube has exactly one class of orientations. The cube conjecture relating symmetry and unicity of the class of orientations was the motivation for a systematic study of the number of orientation classes of regular polytopes starting in J. P.
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.ejc.2015.03.020 0195-6698/© 2015 Elsevier Ltd. All rights reserved.
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Roudneff’s thesis [29, Chapter VIII], and continuing with I. Salaün, in [23]. We survey in the next section the main results on the cube conjecture. Simplicial regions of an arrangement of pseudospheres are at the core of the definitions of local perturbations of oriented matroids introduced by J. Edmonds, K. Fukuda and A. Mandel, [14,25], and also by M. Las Vergnas in [19–21]. They provide practical ways of generating new oriented matroids from a given oriented matroid and have proved very useful in the construction of counterexamples to some expected generalizations to oriented matroids of known results from linear programming and real hyperplane arrangements, as well as expected generalizations of combinatorial properties of pseudoline arrangements, [14,25], and e.g. [2]. In Section 2 we relate some experiments of using these techniques to disprove the conjecture of orientability of cubes (Conjecture 2.2 Section 2) and how, in studying them, we rediscovered previous results of M. Las Vergnas from [18,3]. A well known result of G. Ringel [2] on arrangements of pseudolines states that: Two simple arrangements of n-pseudolines can be transformed one into the other by a sequence of local perturbations that consist in ‘‘switching a triangle’’. The result can be generalized to realizable oriented matroids: Two realizable orientations of the uniform matroid are obtained from each other by a sequence of realizable perturbations that consist in switching a simplex. For ranks higher then 3, in contrast with the rank 3 case, the case of pseudoline arrangements, if one does not require both orientations realizable such a sequence is not known to exist. One cannot guarantee in general the existence of a simplex to switch! One of the enumerative results of R. Cordovil’s thesis, [6] (Chapter I, Part 2), is an elegant and short proof, via Tutte polynomial, of the Las Vergnas simplex conjecture for rank 3 matroids. Relevant first computational and theoretical results concerning the generation of all possible orientations of uniform matroids via ‘‘simplex switchings’’ and realizability of matroids are very nicely presented in J. Bokowski and B. Sturmfels book [5]. See [2] for further developments. Our brief Section 2 is devoted to symmetry of the Euclidean representations of oriented matroids. We point out, in particular, how the results of J. Lawrence [24] and I. da Silva [7] combined with a refinement of Ringel’s theorem led to a first version and proof, in [8], of the first nontrivial case of Bohne-Dress theorem. 2. Cubes, symmetry and orientability M. Las Vergnas proved [18,3] that graphic and more generally unimodular matroids have exactly one class of orientations. In the middle 1980s, M. Las Vergnas with J. P. Roudneff and later I. Salaün, started a systematic study on enumeration of the orientation classes of regular polytopes, to understand how symmetry of a matroid also encoded its orientation. They proved the following theorem: Theorem 2.1 ([29,23]). Of the 3-dimensional regular polytopes – the five platonic solids – three have one class of orientations and two, the dodecahedron and the icosahedron, have two classes of orientations. Of the 4-dimensional regular polytopes, the following four: the cross-polytope, the n-cube, the simplex, and the 24-cell have exactly one class of orientations. For every d ∈ N, the d-dimensional simplex and the cross-polytope have exactly one class of orientations. The theorem is proven by studying the circuits of rank 3: trapezoids and parallelograms, combined in 3-dimensional prisms. It is conjectured [23] that the remaining two regular polytopes of dimension 4: the 120-cell and the 600-cell also have exactly one class of orientations. Notice that with the above result the Las Vergnas cube conjecture was proven for n ≤ 4. Computationally the Las Vergnas cube conjecture is difficult to test even for small dimensions, since an explicit description of the real cube is a well known hard problem. The first computational breakthrough was achieved by J. Bokowski et al. [4] who proved: Theorem 2.2 ([4]). The cube conjecture of Las Vergnas is true for n ≤ 7. The approach of J. Bokowski, A. Guedes de Oliveira, U. Thiemann and A. Veloso da Costa combines results on the symmetry group of the matroid of the real cube with Roudneff’s theorem, [29], saying
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that an oriented matroid can be reconstructed from its lower rank labeled contractions. The proof of the unicity of the orientation class consists in the computational reconstruction of the pseudosphere arrangement of the n-dimensional cube (n ≤ 7) from its rank 3 contractions. Using results of [10,11] we could prove (by hand) a stronger version of Theorem 2.2: For n ≤ 7 the orientation of the real affine cube as well as the underlying matroid can be reconstructed from its signed circuits of rank 3 (rectangles) and orientability. To the cube conjecture of Las Vergnas, suggesting that the unicity of the orientation of the cube is encoded in the symmetry of the underlying matroid, we added the question of understanding at what extent the symmetry of the underlying matroid is a consequence of orientability. We precise the main ideas in the next paragraph. 2.1. Orientability and the Las Vergnas Cube Conjecture We recall from [11] that a cube or cubic matroid is a matroid M = M (C n ), over the set C n = {0, 1}n , which satisfies the following two conditions: (i) Every circuit of length 4 (rectangle) of the real cube is a circuit of M. (ii) Every hyperplane with 2n−1 points of the real cube (a facet or a skew-facet) is a hyperplane of M. The class of cubes is a large class. It contains, in particular, all affine cubes, the matroids AffF (C n ) of the affine dependencies of C n over an arbitrary field F . The real cube is AffR (C n ). Along with the cube conjecture of Las Vergnas, re-enunciated as Conjecture 2.1 we consider also Conjecture 2.2. on the orientability of cubes (both true for n ≤ 7): Conjecture 2.1 (Las Vergnas cube conjecture). The real cube has a unique class of orientations. Conjecture 2.2 (Conjecture of orientability of cubes [11]). The real cube is the unique orientable cube (or cubic matroid). The next theorem summarizes some nontrivial properties of orientable cubes: Theorem 2.3 (Proposition 3 and Corollary 1 of [11]). Every orientable cubic matroid M = M (C n ) satisfies the following four properties: (i) M is a matroid of rank n + 1. (ii) The circuits of M with smallest number of elements have exactly 4 elements and are the circuits of length 4 of the real cube. (iii) The cocircuits of M with smallest number of elements have exactly 2n−1 elements and are the cocircuits of length 2n−1 of the real cube. (iv) Every class of orientations of M contains a unique orientation whose signed length 4 circuits and signed length 2n−1 cocircuits are signed as in the canonical orientation of the real cube. Condition (iv) of the above theorem allows a reformulation of both cube conjectures as the following reconstruction problems about the real cube: Conjecture 2.1′ (Las Vergnas Cube Conjecture). The canonical orientation Aff (C n ) of the real cube can be reconstructed from its underlying matroid AffR (C n ) and its signed rectangles or/and its positive cocircuits. Conjecture 2.2′ (Conjecture of Orientability of Cubes). The real affine cube Aff (C n ) can be reconstructed from orientability and its rectangles and its facets and skew-facets. Notice that, stated in this form, the conjecture of Las Vergnas says that the orientation of the real cube is the only way of packing the net of its rectangles along the flats of its highly symmetric underlying matroid. The orientability conjecture says that the symmetric underlying matroid of the real cube is the only way of packing in an oriented way (in an oriented matroid) the net of rectangles of a cube along its facets and skew facets. We point out that if both conjectures are true they provide a purely combinatorial characterization of the affine and linear dependencies of C n over the reals.
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Notice also that in this form the conjecture of orientability of cubes is a deterministic version of the following well known probabilistic conjecture: Conjecture 2.3 ([27,31]). The asymptotic behavior of the probability of a random n × n Bernoulli matrix Mn being singular is: limn→∞ Prob(det(Mn ) = 0) = ( 21 + o(1))n . 2.2. Pushing elements onto hyperplanes and orientability Pushing an element onto a hyperplane is a perturbation of a matroid that leaves the class of cubic matroids invariant and therefore can be used to produce new cubes. We recall the definition: Definition 2.1 (Pushing an Element onto a Hyperplane in a Matroid). Let M = M (E ) be a matroid of rank r over a set E with family of hyperplanes H ⊆ 2E . Assume that M has no coloops. For every pair (G, e) ∈ H × E in general position in M (i.e. for every hyperline L ⊂ G the set L ∪ e is a hyperplane of M) define G := {H ∈ H : H ⊂ G ∪ e}. Then HG,e := H \ G ∪ {G ∪ e} is the family of hyperplanes of a new matroid MG,e of rank r over E, the matroid obtained from M by pushing the element e onto the hyperplane G. Theorem 2.4 (Proposition 5 of [12]). If M = M (C n ), n ≥ 3, is a cubic matroid then every matroid obtained from M by pushing an element onto a hyperplane is also a cubic matroid. Starting with the real cube we can use this operation to obtain non-representable and non-orientable cubes, see examples in [11]. Theorem 2.5 ([11], implicit in [18,3]). If F is a finite field of prime characteristic p then for every n ≥ p + 1 the affine cube AffF (C n ) is not orientable. The operation of pushing an element onto a hyperplane defined above when applied to an oriented matroid does not, in general, preserve orientability. K. Fukuda and A. Tamura proved [15] that orientability is preserved when the element and the hyperplane are not only in general position in the matroid but also near each other in the oriented matroid. Roughly speaking a pair (G, e) of a hyperplane and an element are near each other in an oriented matroid M (E ) if the pair (G, e) is in general position in the underlying matroid and all the hyperplanes of M contained in G ∪ e determine the same partition of the complement, E \ (G ∪ e), in the oriented matroid, in other words, there is an acyclic reorientation whose LV-face lattice is a pyramid with base M (G) and vertex e. Notice that when |G| = r (G) the pyramid is a simplex. If an oriented matroid M contains a pair (G, e) of a hyperplane and an element near each other then M induces not only a canonical orientation MG,e of the matroid MG,e obtained by pushing e onto G, but also a new orientation MeGe of the matroid M, obtained by pushing the element e across the hyperplane G (see [15] or [2]). From this point of view Conjectures 2.1 and 2.2 imply the next, apparently weaker conjecture: Conjecture 2.4. The oriented real affine cube Aff (C n ) contains no pair (G, e) of a hyperplane and an element near each other. If such a pair (G, e) exists both cube Conjectures 2.1 and 2.2 are false. 2.3. Bland–Las Vergnas family Mn and extensions of the cross-polytope As pointed out above, Theorem 2.5. stating the non-orientability of affine cubes over finite fields, although obtained with a very short proof in [11], is in fact implicit in [3] once we notice that all those cubes contain Bland–Las Vergnas minor-minimal non-orientable matroids Mn . In [12] the matroids Mn are proved to be a particular case of a larger family of minor-minimal non-orientable matroids with 2n elements and rank n all constructed by performing on a crosspolytope the same sequence of operations. The cube Conjectures 2.1 and 2.2 actually suggest, that such a sequence of operations must always produce non-orientable matroids.
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Conjecture 2.5 ([12]). Let Sn = Aff (E ) be a cross-polytope over a 2n-element set E ⊆ Rn (rank(Sn ) = n + 1). Let N = N (E ∪ p) be a single element extension of Sn with p lying on n-facets of Sn . Assume that (G, p) is a pair hyperplane/element in general position in N. Then the matroid N(G,p) obtained from N by pushing p into G is not orientable. To conclude this section we comment on a last theorem from [11], now using the Folkman–Lawrence representation Theorem [13]: Theorem 2.6 (Theorem 2 of [11]). Consider an orientable cube M = M (C n ). Every orientation class O (M ) satisfies the following condition: Let ∆(M ) denote the cell decomposition of the unit sphere S n of Rn+1 determined by the signed arrangement of pseudohyperplanes representing an orientation M ∈ O (M ). ∆(M ) has exactly 2(n + 1) topes, ±T0 , ±T1 , . . . ± Tn whose face lattices are isomorphic to the face lattice of the n-cross polytope. Moreover, given two topes T , T ′ ∈ {±T0 , ±T1 , . . . ± Tn } such that T ̸= ±T ′ , T and T ′ have exactly one vertex (signed cocircuit) in commun. We believe that by appealing to this Theorem it may be possible to translate both Conjectures 2.1 and 2.2 in terms of the topology of the extension space of the cross-polytope. However we do not know how to do it. A final remark on symmetry: notice that Theorems 2.3 and 2.6 actually say that every orientable cube contains structures – the signed circuits of rank 3, the signed cocircuits complementary to the facets and skew-facets, the topes of the above theorem – that remain invariant under the group of symmetries of the real oriented cube Aff (C n ). Everything that we can deduce from these structures using just matroid properties and orientability – and we can actually deduce few things more than those stated in [11] – must have the symmetry of the real cube. Is the real oriented cube what we end up with? 3. Euclidean representations of oriented matroids, symmetry and simplices Oriented matroids of rank r over an n element set are simplicial complexes, homeomorphic to r dimensional balls, that have a canonical geometric realization inside the cube [−1, 1]n of Rn . These geometric simplicial complexes work as combinatorial abstractions of r dimensional linear subspaces of Rn . Using the notation of [2] the simplicial complex we are considering is the complex ∆(V ⊥ ), the complex of the chains of the poset of covectors V ⊥ = (V ⊥ , ≼) of the matroid. In what follows W ⊥ = Max(V ⊥ ) denotes the set of maximal covectors or topes of the oriented matroid. V and W = MaxV are the sets respectively of vectors and maximal vectors. Representing every covector X = (X + , X − ) by the corresponding sign vector X ∈ {−1, 0, 1}n and every maximal chain 0 ≼ X1 ≼ . . . ≼ Xr −1 ≼ T , T ∈ W ⊥ by the corresponding geometric r-simplex, conv(0, X1 , . . . , Xr −1 , T ), contained in [−1, 1]n , we obtain a canonical geometric realization of the cell complex ∆(V ⊥ ) that is homeomorphic to an r-ball [13,25], or [2]. We call this geometric realization of ∆(V ⊥ ), as in [26], the crinkled zonotope C (M ) of the oriented matroid. When an oriented matroid M of rank r over [n] is realizable and we consider a linear r-dimensional subspace V of Rn realizing M then, projecting the cube [−1, 1]n orthogonally into V , we obtain a zonotope Z (V ). The crinkled zonotope of M is the lifting of the first barycentric subdivision of the r dimensional zonotope Z (V ) to the simplicial complex of Rn contained in the cube [−1, 1]n , whose vertices are precisely the centers of faces of the cube [−1, 1]n projecting onto the barycenters of the faces of the zonotope Z (V ). See the next Fig. 1. Like zonotopes, crinkled zonotopes are Euclidean objects characterized by (central) symmetry properties. A first characterization of oriented matroids in terms of symmetry was established in [7]. In [9] that first characterization was considerably improved and we provided several sets of axioms for the maximal (co)vectors of an oriented matroid generalizing in a satisfactory way the results of J. Lawrence [24] for uniform matroids, and improving previous results of W. Bienia and R. Cordovil [1] and K. Handa [16].
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The first step in the approach through symmetry to oriented matroids was the generalization of the notion of lopsidedness introduced by J. Lawrence together with a direct and useful way of recovering the covectors of the matroid from its maximal covectors – see (V) in the next Theorem 3.1 – obtained in [7]. Notation. To simplify the writing in what follows we use 2E := {−1, 1}E and 3E := {−1, 0, 1}E . Recall that given X , Y ∈ 3E , the support of X is the set X := {i ∈ E : X (i) = ̸ 0}, and that the sign vector X ◦ Y ∈ 3E is defined by X ◦ Y (i) = X (i), if i ∈ X and X ◦ Y (i) = Y (i), if i ̸∈ X . Theorem 3.1 (Proposition 2.1, Theorem 4.1, Corollary 4.5, Lemma 4.6 of [9]). W ⊆ 3E is the set of maximal vectors of an oriented matroid if and only if the following three conditions are satisfied: (Z1) ∀ W , W ′ ∈ W , W = W ′ . The set E \ W , W ∈ W is the set of loops of the matroid. (Z2) W = −W . (Central symmetry with respect to 0 ∈ RE .) (Z3) If the set W ∩ FX of points of W contained in the face FX of the cube [−1, 1]n centered at X , is centrally symmetric with respect to X then the orthogonal projection of W onto FX is precisely W ∩ FX :
[X ◦ W ∈ W H⇒ X ◦ (−W ) ∈ W ] H⇒ X ◦ W ⊆ W . Moreover, if W satisfies (Z 1), (Z 2), (Z 3) then: (V) the set V of vectors of the oriented matroid is defined by:
V := {X ∈ 3E : X ◦ W ⊆ W } = {X ∈ 3E : X ◦ W ∈ W H⇒ X ◦ (−W ) ∈ W }. (B) A cobase of the oriented matroid is a maximal subset B ⊆ E satisfying the condition: W (B) = 2B . (C) A signed cocircuit of the oriented matroid is a signed subset X ∈ 3E such that W (C ) = 2C − {±X }. The next proposition is the generalization to crinkled zonotopes, of a classical result of G.C. Shephard [30] on associated pairs of zonotopes. Shephard’s theorem says, in our terminology, the following: Given a linear subspace V of Rn , denoting by Z = Z (V ) the zonotope obtained projecting the cube [−1, 1]n orthogonally onto V and by W the set of maximal vectors of the oriented matroid canonically associated with V , there is a partition 2E = W ⊎ W ⊥ ⊎ B of the vertices of the cube [−1, 1]n that project orthogonally in V into respectively: interior points of Z , vertices of Z and boundary points of Z . Proposition 3.1 (Maximal Covectors, Maximal Vectors and Boundary Vectors). Let W ⊆ 3E be the set of maximal vectors of an oriented matroid over a set E. Then: (1)
W ⊥ := Max{3E \ (∪X ∈V X ◦ 2E )}. (2) Set B := 2E \ (W ⊎ W ⊥ ). Every element Z ∈ B has a unique decomposition Z = X ◦ Y with X ∩ Y = ∅, X ∈ V and Y ∈ V ⊥ . Note that in the uniform case one has trivially W ⊥ = 2E \ W . Fig. 1 represents all the pairs of orthogonal crinkled zonotopes of matroids on a 3-element set, one in black the other in grey. Only one simplex of the black complex assumed as ∆(V ⊥ ) is also represented in all the cases. Although the number of elements is too small, the black complex in the lower left cube is clearly the lifting of the barycentric subdivision of a centrally symmetric hexagon, the zonotope representing the black oriented matroid. In the two right figures the orthogonal combinatorial projections as in Proposition 3.1 of the vertex T = (1, −1, 1) of the cube in the boundary of both complexes are also represented. Another Euclidean representation of oriented matroids whose contraction by one element is realizable is given by Bohne-Dress theorem. Theorem 3.2 (Bohne-Dress Theorem [2]). Let M be a realizable oriented matroid. Denote by Z (M ) a zonotope representing M. There is a bijection M ′ −→ Z(M ) between the one element liftings of M and the zonotopal tilings of the zonotope Z (M ).
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Fig. 1. All pairs of orthogonal crinkled zonotopes of [−1, 1]3 .
The first nontrivial case of this theorem, when the rank r (M ) = 2, was established in [8] with a proof that is constructive and combines essentially the first version of Theorem 3.1 of [7] with a generalized version of the classical Theorem of Ringel on arrangements of pseudolines. As we pointed out then, the generalization of that proof to higher dimensions required the Las Vergnas simplex conjecture that remains unsolved. In all proofs of Bohne-Dress theorem, [28,17], there is a volume argument involved which cannot be used when the matroid M is not realizable. Proposition 3.1-(2) provides a notion of combinatorial orthogonal projection onto a crinkled zonotope that could be the starting point to define a convenient notion of crinkled zonotopal tiling of a crinkled zonotope, generalizing the Bohne-Dress Theorem to all oriented matroids. Actually [26] was a first step in this direction. 4. Final remark As a final remark connecting both sections: observe that if Conjecture 2.1 or Conjecture 2.2 fails, i.e. if there are oriented cubes other than the real cube, Theorem 2.5 guarantees that in all of them we will have the same notion of crinkled zonotope, since all of them will have a similar subset of acyclic reorientations whose face lattice is exactly the same as the face lattice of the real cube. References [1] W. Bienia, R. Cordovil, An axiomatic of non-Radon partitions, European J. Combin. 8 (1987) 1–4. [2] A. Björner, M. Las Vergnas, B. Sturmfels, N. White, G. Ziegler, Oriented Matroids, second ed., in: Encycl. of Maths and Appl., vol. 46, Cambridge University Press, 1999. [3] R. Bland, M. Las Vergnas, Orientability of matroids, J. Combin. Theory Ser. B 24 (1978) 94–123. [4] J. Bokowski, A. Guedes de Oliveira, U. Thiemann, A. Veloso da Costa, On the cube problem of Las Vergnas, Geom. Dedicata 63 (1) (1996) 25–43. [5] J. Bokowski, B. Sturmfels, Computational Synthetic Geometry, in: Lecture Notes in Mathematics, Springer, 1989. [6] R. Cordovil, Quelques proprietés algébriques des matroïdes (Thèse de doctorat d’État es Sciences), Université de Paris VI, 1981. [7] I.P.F. da Silva, Quelques proprietés des matroïdes orientés (Thèse de Doctorat), de L’Université de Paris VI, Paris, 1987. [8] I.P.F. da Silva, Fillings of 2n-gons with rhombi, Discrete Math. 111 (1993) 137–144. [9] I.P.F. da Silva, Axioms for maximal vectors of an oriented matroid: a combinatorial characterization fo the regions determined by an arrangement of pseudohyperplanes, European J. Combin. 16 (1995) 125–145. [10] I.P.F. da Silva, Recursivity and geometry of the hypercube, Linear Algebra Appl. 397 (2005) 223–233. [11] I.P.F. da Silva, Cubes and orientability, Discrete Math. 308 (2008) 3574–3585. [12] I.P.F. da Silva, On minimal non-orientable matroids with 2n-elements and rank n, European J. Combin. 30 (2009) 1825–1832. [13] J. Folkman, J. Lawrence, Oriented matroids, J. Combin. Theory Ser. B 25 (1978) 199–236. [14] K. Fukuda, Oriented matroid programming, Ph.D. thesis, Waterloo, Ontario, 1981. [15] K. Fukuda, A. Tamura, Local deformation and orientation transformation in oriented matroids, Ars Combin. 25A (1988) 243–258.
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[16] K. Handa, Characterization of oriented matroids in terms of topes, European J. Combin. 11 (1990) 41–45. [17] B. Huber, J. Rambau, F. Santos, The Cayley trick, lifting subdivisions and Bohne-Dress theorem, J. Eur. Math. Soc. 2 (2) (2000) 179–198. [18] M. Las Vergnas, Matroïdes orientables, 1974, p. 77. [19] M. Las Vergnas, Matroïdes orientables, C. R. Acad. Sci. Paris Ser. A 280 (1975) 61–64. [20] M. Las Vergnas, Sur les extensions principales d’un matroïde, C. R. Acad. Sci. Paris Ser. A 280 (1975) 187–190. [21] M. Las Vergnas, Extensions pontuelles d’une géometrie combinatoire orienté, in: Problèmes Combinatoires et Theorie des Graphes, Actes Coll. Orsay 1976, Colloques Internationaux, C.N.R.S, 260, 1976, pp. 265–270. [22] M. Las Vergnas, Convexity in oriented matroids, J. Combin. Theory Ser. B 29 (1980) 231–243. [23] M. Las Vergnas, J.P. Roudneff, I. Saläun, Regular polytopes, preprint 1989, p. 12 (unpublished). [24] J. Lawrence, Lopsided sets and orthant intersection by convex sets, Pacific J. Math. 104 (1) (1983) 155–173. [25] A. Mandel, Topology of oriented matroids, Ph.D. thesis, Waterloo, 1981. [26] V. Moulton, I. da Silva, Crinkled Zonotopes, Preprint 1998. online at: http://webpages.fc.ul.pt/~ipsilva/publications.html. [27] A.M. Odlyzco, On subspaces spanned by random selections of ±1 vectors, J. Combin. Theory Ser. A 47 (1988) 124–133. [28] J. Richter-Gebert, G. Ziegler, Zonotopal tilings and the Bohne-Dress theorem, in: Proc. Jerusalem Combinatorics 93 H. Barcelo, G. Kalai (Eds.), Contemporary Math. 178, Amer. Math. Soc., 1993, pp. 211–232. [29] J.P. Roudneff, Matroïdes orientés et arrangements de pseudodroites (Thèse de 3 ème cycle), Université de Paris VI, 1986. [30] G.C. Shephard, Combinatorial properties of the associated zonotope, Canad. J. Math. 26 (1974) 302–321. [31] T. Tao, V. Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603–628.
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Symmetry, oriented matroids and two conjectures of Michel Las Vergnas Ilda P.F. da Silva Faculdade de Ciencias da Universidade de Lisboa/CFCUL, Dep.to de Matemática, Campo Grande, Edificio C6-Piso 2, P-1749-016 Lisbon, Portugal
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Article history: Available online xxxx To the memory of Michel Las Vergnas
The paper has two parts. In the first part we survey the existing results on the cube conjecture of Las Vergnas. This conjecture claims that the orientation of the matroid of the cube is determined by the symmetries of the underlying matroid. The second part deals with Euclidean representations of matroids as geometric simplicial complexes defined by symmetry properties abstracting those of zonotopes. Both sections involve arguments concerning simplicial regions illustrating, once more, the fundamental importance of the simplex conjecture of Las Vergnas. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Two of the long standing conjectures of M. Las Vergnas are: the simplex conjecture and the cube conjecture. They say: Las Vergnas simplex conjecture ([22]). Every oriented matroid has an acyclic reorientation whose LVface lattice is a simplex. Equivalently, every arrangement of pseudospheres contains a region that is a simplex. Las Vergnas cube conjecture ([23]). The matroid of the affine dependencies of the vertices of a real cube has exactly one class of orientations. The cube conjecture relating symmetry and unicity of the class of orientations was the motivation for a systematic study of the number of orientation classes of regular polytopes starting in J. P.
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.ejc.2015.03.020 0195-6698/© 2015 Elsevier Ltd. All rights reserved.
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Roudneff’s thesis [29, Chapter VIII], and continuing with I. Salaün, in [23]. We survey in the next section the main results on the cube conjecture. Simplicial regions of an arrangement of pseudospheres are at the core of the definitions of local perturbations of oriented matroids introduced by J. Edmonds, K. Fukuda and A. Mandel, [14,25], and also by M. Las Vergnas in [19–21]. They provide practical ways of generating new oriented matroids from a given oriented matroid and have proved very useful in the construction of counterexamples to some expected generalizations to oriented matroids of known results from linear programming and real hyperplane arrangements, as well as expected generalizations of combinatorial properties of pseudoline arrangements, [14,25], and e.g. [2]. In Section 2 we relate some experiments of using these techniques to disprove the conjecture of orientability of cubes (Conjecture 2.2 Section 2) and how, in studying them, we rediscovered previous results of M. Las Vergnas from [18,3]. A well known result of G. Ringel [2] on arrangements of pseudolines states that: Two simple arrangements of n-pseudolines can be transformed one into the other by a sequence of local perturbations that consist in ‘‘switching a triangle’’. The result can be generalized to realizable oriented matroids: Two realizable orientations of the uniform matroid are obtained from each other by a sequence of realizable perturbations that consist in switching a simplex. For ranks higher then 3, in contrast with the rank 3 case, the case of pseudoline arrangements, if one does not require both orientations realizable such a sequence is not known to exist. One cannot guarantee in general the existence of a simplex to switch! One of the enumerative results of R. Cordovil’s thesis, [6] (Chapter I, Part 2), is an elegant and short proof, via Tutte polynomial, of the Las Vergnas simplex conjecture for rank 3 matroids. Relevant first computational and theoretical results concerning the generation of all possible orientations of uniform matroids via ‘‘simplex switchings’’ and realizability of matroids are very nicely presented in J. Bokowski and B. Sturmfels book [5]. See [2] for further developments. Our brief Section 2 is devoted to symmetry of the Euclidean representations of oriented matroids. We point out, in particular, how the results of J. Lawrence [24] and I. da Silva [7] combined with a refinement of Ringel’s theorem led to a first version and proof, in [8], of the first nontrivial case of Bohne-Dress theorem. 2. Cubes, symmetry and orientability M. Las Vergnas proved [18,3] that graphic and more generally unimodular matroids have exactly one class of orientations. In the middle 1980s, M. Las Vergnas with J. P. Roudneff and later I. Salaün, started a systematic study on enumeration of the orientation classes of regular polytopes, to understand how symmetry of a matroid also encoded its orientation. They proved the following theorem: Theorem 2.1 ([29,23]). Of the 3-dimensional regular polytopes – the five platonic solids – three have one class of orientations and two, the dodecahedron and the icosahedron, have two classes of orientations. Of the 4-dimensional regular polytopes, the following four: the cross-polytope, the n-cube, the simplex, and the 24-cell have exactly one class of orientations. For every d ∈ N, the d-dimensional simplex and the cross-polytope have exactly one class of orientations. The theorem is proven by studying the circuits of rank 3: trapezoids and parallelograms, combined in 3-dimensional prisms. It is conjectured [23] that the remaining two regular polytopes of dimension 4: the 120-cell and the 600-cell also have exactly one class of orientations. Notice that with the above result the Las Vergnas cube conjecture was proven for n ≤ 4. Computationally the Las Vergnas cube conjecture is difficult to test even for small dimensions, since an explicit description of the real cube is a well known hard problem. The first computational breakthrough was achieved by J. Bokowski et al. [4] who proved: Theorem 2.2 ([4]). The cube conjecture of Las Vergnas is true for n ≤ 7. The approach of J. Bokowski, A. Guedes de Oliveira, U. Thiemann and A. Veloso da Costa combines results on the symmetry group of the matroid of the real cube with Roudneff’s theorem, [29], saying
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that an oriented matroid can be reconstructed from its lower rank labeled contractions. The proof of the unicity of the orientation class consists in the computational reconstruction of the pseudosphere arrangement of the n-dimensional cube (n ≤ 7) from its rank 3 contractions. Using results of [10,11] we could prove (by hand) a stronger version of Theorem 2.2: For n ≤ 7 the orientation of the real affine cube as well as the underlying matroid can be reconstructed from its signed circuits of rank 3 (rectangles) and orientability. To the cube conjecture of Las Vergnas, suggesting that the unicity of the orientation of the cube is encoded in the symmetry of the underlying matroid, we added the question of understanding at what extent the symmetry of the underlying matroid is a consequence of orientability. We precise the main ideas in the next paragraph. 2.1. Orientability and the Las Vergnas Cube Conjecture We recall from [11] that a cube or cubic matroid is a matroid M = M (C n ), over the set C n = {0, 1}n , which satisfies the following two conditions: (i) Every circuit of length 4 (rectangle) of the real cube is a circuit of M. (ii) Every hyperplane with 2n−1 points of the real cube (a facet or a skew-facet) is a hyperplane of M. The class of cubes is a large class. It contains, in particular, all affine cubes, the matroids AffF (C n ) of the affine dependencies of C n over an arbitrary field F . The real cube is AffR (C n ). Along with the cube conjecture of Las Vergnas, re-enunciated as Conjecture 2.1 we consider also Conjecture 2.2. on the orientability of cubes (both true for n ≤ 7): Conjecture 2.1 (Las Vergnas cube conjecture). The real cube has a unique class of orientations. Conjecture 2.2 (Conjecture of orientability of cubes [11]). The real cube is the unique orientable cube (or cubic matroid). The next theorem summarizes some nontrivial properties of orientable cubes: Theorem 2.3 (Proposition 3 and Corollary 1 of [11]). Every orientable cubic matroid M = M (C n ) satisfies the following four properties: (i) M is a matroid of rank n + 1. (ii) The circuits of M with smallest number of elements have exactly 4 elements and are the circuits of length 4 of the real cube. (iii) The cocircuits of M with smallest number of elements have exactly 2n−1 elements and are the cocircuits of length 2n−1 of the real cube. (iv) Every class of orientations of M contains a unique orientation whose signed length 4 circuits and signed length 2n−1 cocircuits are signed as in the canonical orientation of the real cube. Condition (iv) of the above theorem allows a reformulation of both cube conjectures as the following reconstruction problems about the real cube: Conjecture 2.1′ (Las Vergnas Cube Conjecture). The canonical orientation Aff (C n ) of the real cube can be reconstructed from its underlying matroid AffR (C n ) and its signed rectangles or/and its positive cocircuits. Conjecture 2.2′ (Conjecture of Orientability of Cubes). The real affine cube Aff (C n ) can be reconstructed from orientability and its rectangles and its facets and skew-facets. Notice that, stated in this form, the conjecture of Las Vergnas says that the orientation of the real cube is the only way of packing the net of its rectangles along the flats of its highly symmetric underlying matroid. The orientability conjecture says that the symmetric underlying matroid of the real cube is the only way of packing in an oriented way (in an oriented matroid) the net of rectangles of a cube along its facets and skew facets. We point out that if both conjectures are true they provide a purely combinatorial characterization of the affine and linear dependencies of C n over the reals.
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Notice also that in this form the conjecture of orientability of cubes is a deterministic version of the following well known probabilistic conjecture: Conjecture 2.3 ([27,31]). The asymptotic behavior of the probability of a random n × n Bernoulli matrix Mn being singular is: limn→∞ Prob(det(Mn ) = 0) = ( 21 + o(1))n . 2.2. Pushing elements onto hyperplanes and orientability Pushing an element onto a hyperplane is a perturbation of a matroid that leaves the class of cubic matroids invariant and therefore can be used to produce new cubes. We recall the definition: Definition 2.1 (Pushing an Element onto a Hyperplane in a Matroid). Let M = M (E ) be a matroid of rank r over a set E with family of hyperplanes H ⊆ 2E . Assume that M has no coloops. For every pair (G, e) ∈ H × E in general position in M (i.e. for every hyperline L ⊂ G the set L ∪ e is a hyperplane of M) define G := {H ∈ H : H ⊂ G ∪ e}. Then HG,e := H \ G ∪ {G ∪ e} is the family of hyperplanes of a new matroid MG,e of rank r over E, the matroid obtained from M by pushing the element e onto the hyperplane G. Theorem 2.4 (Proposition 5 of [12]). If M = M (C n ), n ≥ 3, is a cubic matroid then every matroid obtained from M by pushing an element onto a hyperplane is also a cubic matroid. Starting with the real cube we can use this operation to obtain non-representable and non-orientable cubes, see examples in [11]. Theorem 2.5 ([11], implicit in [18,3]). If F is a finite field of prime characteristic p then for every n ≥ p + 1 the affine cube AffF (C n ) is not orientable. The operation of pushing an element onto a hyperplane defined above when applied to an oriented matroid does not, in general, preserve orientability. K. Fukuda and A. Tamura proved [15] that orientability is preserved when the element and the hyperplane are not only in general position in the matroid but also near each other in the oriented matroid. Roughly speaking a pair (G, e) of a hyperplane and an element are near each other in an oriented matroid M (E ) if the pair (G, e) is in general position in the underlying matroid and all the hyperplanes of M contained in G ∪ e determine the same partition of the complement, E \ (G ∪ e), in the oriented matroid, in other words, there is an acyclic reorientation whose LV-face lattice is a pyramid with base M (G) and vertex e. Notice that when |G| = r (G) the pyramid is a simplex. If an oriented matroid M contains a pair (G, e) of a hyperplane and an element near each other then M induces not only a canonical orientation MG,e of the matroid MG,e obtained by pushing e onto G, but also a new orientation MeGe of the matroid M, obtained by pushing the element e across the hyperplane G (see [15] or [2]). From this point of view Conjectures 2.1 and 2.2 imply the next, apparently weaker conjecture: Conjecture 2.4. The oriented real affine cube Aff (C n ) contains no pair (G, e) of a hyperplane and an element near each other. If such a pair (G, e) exists both cube Conjectures 2.1 and 2.2 are false. 2.3. Bland–Las Vergnas family Mn and extensions of the cross-polytope As pointed out above, Theorem 2.5. stating the non-orientability of affine cubes over finite fields, although obtained with a very short proof in [11], is in fact implicit in [3] once we notice that all those cubes contain Bland–Las Vergnas minor-minimal non-orientable matroids Mn . In [12] the matroids Mn are proved to be a particular case of a larger family of minor-minimal non-orientable matroids with 2n elements and rank n all constructed by performing on a crosspolytope the same sequence of operations. The cube Conjectures 2.1 and 2.2 actually suggest, that such a sequence of operations must always produce non-orientable matroids.
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Conjecture 2.5 ([12]). Let Sn = Aff (E ) be a cross-polytope over a 2n-element set E ⊆ Rn (rank(Sn ) = n + 1). Let N = N (E ∪ p) be a single element extension of Sn with p lying on n-facets of Sn . Assume that (G, p) is a pair hyperplane/element in general position in N. Then the matroid N(G,p) obtained from N by pushing p into G is not orientable. To conclude this section we comment on a last theorem from [11], now using the Folkman–Lawrence representation Theorem [13]: Theorem 2.6 (Theorem 2 of [11]). Consider an orientable cube M = M (C n ). Every orientation class O (M ) satisfies the following condition: Let ∆(M ) denote the cell decomposition of the unit sphere S n of Rn+1 determined by the signed arrangement of pseudohyperplanes representing an orientation M ∈ O (M ). ∆(M ) has exactly 2(n + 1) topes, ±T0 , ±T1 , . . . ± Tn whose face lattices are isomorphic to the face lattice of the n-cross polytope. Moreover, given two topes T , T ′ ∈ {±T0 , ±T1 , . . . ± Tn } such that T ̸= ±T ′ , T and T ′ have exactly one vertex (signed cocircuit) in commun. We believe that by appealing to this Theorem it may be possible to translate both Conjectures 2.1 and 2.2 in terms of the topology of the extension space of the cross-polytope. However we do not know how to do it. A final remark on symmetry: notice that Theorems 2.3 and 2.6 actually say that every orientable cube contains structures – the signed circuits of rank 3, the signed cocircuits complementary to the facets and skew-facets, the topes of the above theorem – that remain invariant under the group of symmetries of the real oriented cube Aff (C n ). Everything that we can deduce from these structures using just matroid properties and orientability – and we can actually deduce few things more than those stated in [11] – must have the symmetry of the real cube. Is the real oriented cube what we end up with? 3. Euclidean representations of oriented matroids, symmetry and simplices Oriented matroids of rank r over an n element set are simplicial complexes, homeomorphic to r dimensional balls, that have a canonical geometric realization inside the cube [−1, 1]n of Rn . These geometric simplicial complexes work as combinatorial abstractions of r dimensional linear subspaces of Rn . Using the notation of [2] the simplicial complex we are considering is the complex ∆(V ⊥ ), the complex of the chains of the poset of covectors V ⊥ = (V ⊥ , ≼) of the matroid. In what follows W ⊥ = Max(V ⊥ ) denotes the set of maximal covectors or topes of the oriented matroid. V and W = MaxV are the sets respectively of vectors and maximal vectors. Representing every covector X = (X + , X − ) by the corresponding sign vector X ∈ {−1, 0, 1}n and every maximal chain 0 ≼ X1 ≼ . . . ≼ Xr −1 ≼ T , T ∈ W ⊥ by the corresponding geometric r-simplex, conv(0, X1 , . . . , Xr −1 , T ), contained in [−1, 1]n , we obtain a canonical geometric realization of the cell complex ∆(V ⊥ ) that is homeomorphic to an r-ball [13,25], or [2]. We call this geometric realization of ∆(V ⊥ ), as in [26], the crinkled zonotope C (M ) of the oriented matroid. When an oriented matroid M of rank r over [n] is realizable and we consider a linear r-dimensional subspace V of Rn realizing M then, projecting the cube [−1, 1]n orthogonally into V , we obtain a zonotope Z (V ). The crinkled zonotope of M is the lifting of the first barycentric subdivision of the r dimensional zonotope Z (V ) to the simplicial complex of Rn contained in the cube [−1, 1]n , whose vertices are precisely the centers of faces of the cube [−1, 1]n projecting onto the barycenters of the faces of the zonotope Z (V ). See the next Fig. 1. Like zonotopes, crinkled zonotopes are Euclidean objects characterized by (central) symmetry properties. A first characterization of oriented matroids in terms of symmetry was established in [7]. In [9] that first characterization was considerably improved and we provided several sets of axioms for the maximal (co)vectors of an oriented matroid generalizing in a satisfactory way the results of J. Lawrence [24] for uniform matroids, and improving previous results of W. Bienia and R. Cordovil [1] and K. Handa [16].
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The first step in the approach through symmetry to oriented matroids was the generalization of the notion of lopsidedness introduced by J. Lawrence together with a direct and useful way of recovering the covectors of the matroid from its maximal covectors – see (V) in the next Theorem 3.1 – obtained in [7]. Notation. To simplify the writing in what follows we use 2E := {−1, 1}E and 3E := {−1, 0, 1}E . Recall that given X , Y ∈ 3E , the support of X is the set X := {i ∈ E : X (i) = ̸ 0}, and that the sign vector X ◦ Y ∈ 3E is defined by X ◦ Y (i) = X (i), if i ∈ X and X ◦ Y (i) = Y (i), if i ̸∈ X . Theorem 3.1 (Proposition 2.1, Theorem 4.1, Corollary 4.5, Lemma 4.6 of [9]). W ⊆ 3E is the set of maximal vectors of an oriented matroid if and only if the following three conditions are satisfied: (Z1) ∀ W , W ′ ∈ W , W = W ′ . The set E \ W , W ∈ W is the set of loops of the matroid. (Z2) W = −W . (Central symmetry with respect to 0 ∈ RE .) (Z3) If the set W ∩ FX of points of W contained in the face FX of the cube [−1, 1]n centered at X , is centrally symmetric with respect to X then the orthogonal projection of W onto FX is precisely W ∩ FX :
[X ◦ W ∈ W H⇒ X ◦ (−W ) ∈ W ] H⇒ X ◦ W ⊆ W . Moreover, if W satisfies (Z 1), (Z 2), (Z 3) then: (V) the set V of vectors of the oriented matroid is defined by:
V := {X ∈ 3E : X ◦ W ⊆ W } = {X ∈ 3E : X ◦ W ∈ W H⇒ X ◦ (−W ) ∈ W }. (B) A cobase of the oriented matroid is a maximal subset B ⊆ E satisfying the condition: W (B) = 2B . (C) A signed cocircuit of the oriented matroid is a signed subset X ∈ 3E such that W (C ) = 2C − {±X }. The next proposition is the generalization to crinkled zonotopes, of a classical result of G.C. Shephard [30] on associated pairs of zonotopes. Shephard’s theorem says, in our terminology, the following: Given a linear subspace V of Rn , denoting by Z = Z (V ) the zonotope obtained projecting the cube [−1, 1]n orthogonally onto V and by W the set of maximal vectors of the oriented matroid canonically associated with V , there is a partition 2E = W ⊎ W ⊥ ⊎ B of the vertices of the cube [−1, 1]n that project orthogonally in V into respectively: interior points of Z , vertices of Z and boundary points of Z . Proposition 3.1 (Maximal Covectors, Maximal Vectors and Boundary Vectors). Let W ⊆ 3E be the set of maximal vectors of an oriented matroid over a set E. Then: (1)
W ⊥ := Max{3E \ (∪X ∈V X ◦ 2E )}. (2) Set B := 2E \ (W ⊎ W ⊥ ). Every element Z ∈ B has a unique decomposition Z = X ◦ Y with X ∩ Y = ∅, X ∈ V and Y ∈ V ⊥ . Note that in the uniform case one has trivially W ⊥ = 2E \ W . Fig. 1 represents all the pairs of orthogonal crinkled zonotopes of matroids on a 3-element set, one in black the other in grey. Only one simplex of the black complex assumed as ∆(V ⊥ ) is also represented in all the cases. Although the number of elements is too small, the black complex in the lower left cube is clearly the lifting of the barycentric subdivision of a centrally symmetric hexagon, the zonotope representing the black oriented matroid. In the two right figures the orthogonal combinatorial projections as in Proposition 3.1 of the vertex T = (1, −1, 1) of the cube in the boundary of both complexes are also represented. Another Euclidean representation of oriented matroids whose contraction by one element is realizable is given by Bohne-Dress theorem. Theorem 3.2 (Bohne-Dress Theorem [2]). Let M be a realizable oriented matroid. Denote by Z (M ) a zonotope representing M. There is a bijection M ′ −→ Z(M ) between the one element liftings of M and the zonotopal tilings of the zonotope Z (M ).
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Fig. 1. All pairs of orthogonal crinkled zonotopes of [−1, 1]3 .
The first nontrivial case of this theorem, when the rank r (M ) = 2, was established in [8] with a proof that is constructive and combines essentially the first version of Theorem 3.1 of [7] with a generalized version of the classical Theorem of Ringel on arrangements of pseudolines. As we pointed out then, the generalization of that proof to higher dimensions required the Las Vergnas simplex conjecture that remains unsolved. In all proofs of Bohne-Dress theorem, [28,17], there is a volume argument involved which cannot be used when the matroid M is not realizable. Proposition 3.1-(2) provides a notion of combinatorial orthogonal projection onto a crinkled zonotope that could be the starting point to define a convenient notion of crinkled zonotopal tiling of a crinkled zonotope, generalizing the Bohne-Dress Theorem to all oriented matroids. Actually [26] was a first step in this direction. 4. Final remark As a final remark connecting both sections: observe that if Conjecture 2.1 or Conjecture 2.2 fails, i.e. if there are oriented cubes other than the real cube, Theorem 2.5 guarantees that in all of them we will have the same notion of crinkled zonotope, since all of them will have a similar subset of acyclic reorientations whose face lattice is exactly the same as the face lattice of the real cube. References [1] W. Bienia, R. Cordovil, An axiomatic of non-Radon partitions, European J. Combin. 8 (1987) 1–4. [2] A. Björner, M. Las Vergnas, B. Sturmfels, N. White, G. Ziegler, Oriented Matroids, second ed., in: Encycl. of Maths and Appl., vol. 46, Cambridge University Press, 1999. [3] R. Bland, M. Las Vergnas, Orientability of matroids, J. Combin. Theory Ser. B 24 (1978) 94–123. [4] J. Bokowski, A. Guedes de Oliveira, U. Thiemann, A. Veloso da Costa, On the cube problem of Las Vergnas, Geom. Dedicata 63 (1) (1996) 25–43. [5] J. Bokowski, B. Sturmfels, Computational Synthetic Geometry, in: Lecture Notes in Mathematics, Springer, 1989. [6] R. Cordovil, Quelques proprietés algébriques des matroïdes (Thèse de doctorat d’État es Sciences), Université de Paris VI, 1981. [7] I.P.F. da Silva, Quelques proprietés des matroïdes orientés (Thèse de Doctorat), de L’Université de Paris VI, Paris, 1987. [8] I.P.F. da Silva, Fillings of 2n-gons with rhombi, Discrete Math. 111 (1993) 137–144. [9] I.P.F. da Silva, Axioms for maximal vectors of an oriented matroid: a combinatorial characterization fo the regions determined by an arrangement of pseudohyperplanes, European J. Combin. 16 (1995) 125–145. [10] I.P.F. da Silva, Recursivity and geometry of the hypercube, Linear Algebra Appl. 397 (2005) 223–233. [11] I.P.F. da Silva, Cubes and orientability, Discrete Math. 308 (2008) 3574–3585. [12] I.P.F. da Silva, On minimal non-orientable matroids with 2n-elements and rank n, European J. Combin. 30 (2009) 1825–1832. [13] J. Folkman, J. Lawrence, Oriented matroids, J. Combin. Theory Ser. B 25 (1978) 199–236. [14] K. Fukuda, Oriented matroid programming, Ph.D. thesis, Waterloo, Ontario, 1981. [15] K. Fukuda, A. Tamura, Local deformation and orientation transformation in oriented matroids, Ars Combin. 25A (1988) 243–258.
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[16] K. Handa, Characterization of oriented matroids in terms of topes, European J. Combin. 11 (1990) 41–45. [17] B. Huber, J. Rambau, F. Santos, The Cayley trick, lifting subdivisions and Bohne-Dress theorem, J. Eur. Math. Soc. 2 (2) (2000) 179–198. [18] M. Las Vergnas, Matroïdes orientables, 1974, p. 77. [19] M. Las Vergnas, Matroïdes orientables, C. R. Acad. Sci. Paris Ser. A 280 (1975) 61–64. [20] M. Las Vergnas, Sur les extensions principales d’un matroïde, C. R. Acad. Sci. Paris Ser. A 280 (1975) 187–190. [21] M. Las Vergnas, Extensions pontuelles d’une géometrie combinatoire orienté, in: Problèmes Combinatoires et Theorie des Graphes, Actes Coll. Orsay 1976, Colloques Internationaux, C.N.R.S, 260, 1976, pp. 265–270. [22] M. Las Vergnas, Convexity in oriented matroids, J. Combin. Theory Ser. B 29 (1980) 231–243. [23] M. Las Vergnas, J.P. Roudneff, I. Saläun, Regular polytopes, preprint 1989, p. 12 (unpublished). [24] J. Lawrence, Lopsided sets and orthant intersection by convex sets, Pacific J. Math. 104 (1) (1983) 155–173. [25] A. Mandel, Topology of oriented matroids, Ph.D. thesis, Waterloo, 1981. [26] V. Moulton, I. da Silva, Crinkled Zonotopes, Preprint 1998. online at: http://webpages.fc.ul.pt/~ipsilva/publications.html. [27] A.M. Odlyzco, On subspaces spanned by random selections of ±1 vectors, J. Combin. Theory Ser. A 47 (1988) 124–133. [28] J. Richter-Gebert, G. Ziegler, Zonotopal tilings and the Bohne-Dress theorem, in: Proc. Jerusalem Combinatorics 93 H. Barcelo, G. Kalai (Eds.), Contemporary Math. 178, Amer. Math. Soc., 1993, pp. 211–232. [29] J.P. Roudneff, Matroïdes orientés et arrangements de pseudodroites (Thèse de 3 ème cycle), Université de Paris VI, 1986. [30] G.C. Shephard, Combinatorial properties of the associated zonotope, Canad. J. Math. 26 (1974) 302–321. [31] T. Tao, V. Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603–628.