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62
Combinatorial generation of small point configurations and hyperplane arrangements
, 2003
"... A recent progress on the complete enumeration of oriented matroids enables us to generate all combinatorial types of small point configurations and hyperplane arrangements in general dimension, including degenerate ones. This extends a number of former works which concentrated on the nondegenerate ..."
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A recent progress on the complete enumeration of oriented matroids enables us to generate all combinatorial types of small point configurations and hyperplane arrangements in general dimension, including degenerate ones. This extends a number of former works which concentrated on the nondegenerate case and are usually limited to dimension 2 or 3. Our initial study on the complete list for small cases has shown its potential in resolving geometric conjectures. 1
Mod 2 Cohomology of Combinatorial Grassmannians
"... this paper we explore the relationship between combinatorial vector bundles and real vector bundles. As a consequence of our results we get theorems relating the topology of the combinatorial Grassmannians to that of their real analogs. The theory of oriented matroids gives a combinatorial abstract ..."
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Cited by 7 (3 self)
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this paper we explore the relationship between combinatorial vector bundles and real vector bundles. As a consequence of our results we get theorems relating the topology of the combinatorial Grassmannians to that of their real analogs. The theory of oriented matroids gives a combinatorial abstraction of linear algebra; a kdimensional subspace of R
A characterization of cocircuit graphs of uniform oriented matroids
 JOURNAL OF COMBINATORIAL THEORY, SERIES B
, 2006
"... ..."
Topological representations of matroids
 J. Amer. Math. Soc
, 2002
"... Abstract. There is a onetoone correspondence between geometric lattices and the intersection lattices of arrangements of homotopy spheres. When the arrangements are essential and fully partitioned, Zaslavsky’s enumeration of the cells of the arrangement still holds. An application of the theory sh ..."
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Cited by 6 (0 self)
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Abstract. There is a onetoone correspondence between geometric lattices and the intersection lattices of arrangements of homotopy spheres. When the arrangements are essential and fully partitioned, Zaslavsky’s enumeration of the cells of the arrangement still holds. An application of the theory shows that all minimal cellular resolutions of matroid Steiner ideals are bounded subcomplexes of homotopy sphere arrangements of the given matroid. As a result the Betti numbers of the ideal are computed and seen to be equivalent to Stanley’s formula in the special case of face ideals of independence complexes of matroids. 1.
Topology of Combinatorial Differential Manifolds
 Topology
"... We prove that all combinatorial differential manifolds involving only Euclidean oriented matroids are PL manifolds. In doing so we introduce a new notion of triangulations of oriented matroids, and show that any triangulation of a Euclidean oriented matroid is a PL sphere. In Section 5 we adapt thes ..."
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Cited by 6 (2 self)
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We prove that all combinatorial differential manifolds involving only Euclidean oriented matroids are PL manifolds. In doing so we introduce a new notion of triangulations of oriented matroids, and show that any triangulation of a Euclidean oriented matroid is a PL sphere. In Section 5 we adapt these results to get a new definition of triangulations of oriented matroid polytopes, and show that any triangulation of a Euclidean oriented matroid polytope is a PL ball. 1 Introduction In [M], MacPherson introduced a new class of combinatorial objects called combinatorial differential manifolds, or CD manifolds. The idea of a CD manifold is to give a simplicial complex a combinatorial analog to a differential structure. The role of "tangent spaces" in this theory is played by oriented matroids, objects about which we will have much more to say later. If M is a real differential manifold and j : jjXjj !M is a smooth triangulation of M , then we shall see in Section 2.2 that the differentia...
VapnikChervonenkis dimension and (pseudo)hyperplane arrangements
, 1997
"... An arrangement of oriented pseudohyperplanes in affine dspace defines on its set X of pseudohyperplanes a set system (or range space) (X, R), R ⊆ 2 X of VCdimension d in a natural way: to every cell c in the arrangement assign the subset of pseudohyperplanes having c on their positive side, and let ..."
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Cited by 4 (1 self)
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An arrangement of oriented pseudohyperplanes in affine dspace defines on its set X of pseudohyperplanes a set system (or range space) (X, R), R ⊆ 2 X of VCdimension d in a natural way: to every cell c in the arrangement assign the subset of pseudohyperplanes having c on their positive side, and let R be the collection of all these subsets. We investigate and characterize the range spaces corresponding to simple arrangements of pseudohyperplanes in this way; such range spaces are called pseudogeometric, and they have the property that the cardinality of R is maximum for the given VCdimension. In general, such range spaces are called maximum, and we show that the number of ranges R ∈ R for which also X −R ∈ R, determines whether a maximum range space is pseudogeometric. Two other characterizations go via a simple duality concept and ‘small ’ subspaces. The correspondence to arrangements is obtained indirectly via a new characterization of uniform oriented matroids: a range space (X, R) naturally corresponds to a uniform oriented matroid of rank X  − d if and only if its VCdimension is d, R ∈ R implies X − R ∈ R and R  is maximum under these conditions.
Balanced Signings of Oriented Matroids and Chromatic Number
 FLOW LATTICE OF ORIENTED MATROIDS 19
, 2003
"... This paper regards an optimization problem which is an oriented matroid analogue of the graph chromatic number. There are several ways in which a `chromatic number' might be defined for more general matroids. One such formulation, introduced by Goddyn, Tarsi and Zhang [7], depends only on the s ..."
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Cited by 4 (3 self)
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This paper regards an optimization problem which is an oriented matroid analogue of the graph chromatic number. There are several ways in which a `chromatic number' might be defined for more general matroids. One such formulation, introduced by Goddyn, Tarsi and Zhang [7], depends only on the sign patterns of (signed) circuits (or cocircuits). The result is a natural invariant of an oriented matroid. The invariant can be viewed as a `discrepancy in ratio' of a pseudohyperplane arrangement, and thus should be of interest to geometers. The main theorem answers a question raised in [7]. We first state the result and some consequences, using a minimal set of definitions. Detailed definitions appear in Section 2. It is convenient to view an oriented matroid to be a matroid in which every circuit C (and cocircuit B) has been partitioned C = C C  , (and B = B subject to a standard orthogonality condition. We regard each bipartition as an unordered pair }, where one of the parts may be empty. For E(O), the reorientation I of is the new oriented matroid obtained from by repartitioning each circuit C (and cocircuit B) according to the rules } ## {C C), C  C)} and } ## {B B), B  B)}
On the topology and geometric construction of oriented matroids and convex polytopes
 Transactions Amer. Math. Soc
, 1991
"... Abstract. This paper develops new combinatorial and geometric techniques for studying the topology of the real semialgebraic variety 31 (M) of all realizations of an oriented matroid M. We focus our attention on point configurations in general position, and as the main result we prove that the reali ..."
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Abstract. This paper develops new combinatorial and geometric techniques for studying the topology of the real semialgebraic variety 31 (M) of all realizations of an oriented matroid M. We focus our attention on point configurations in general position, and as the main result we prove that the realization space of every uniform rank 3 oriented matroid with up to eight points is contractible. For these special classes our theorem implies the isotopy property which states the spaces 32(M) are pathconnected. We further apply our methods to several related problems on convex polytopes and line arrangements. A geometric construction and the isotopy property are obtained for a large class of neighborly polytopes. We improve a result of M. Las Vergnas by constructing a smallest counterexample to a conjecture of G Ringel, and, finally, we discuss the solution to a problem of R. Cordovil and P. Duchet on the realizability of cyclic matroid polytopes. 1.
Homotopy groups of the combinatorial Grassmannian
"... We prove that the homotopy groups of the oriented matroid Grassmannian MacP(k; n) are stable as n ! 1, that ß 1 (\Delta MacP(k; n)) ¸ = ß 1 (G(k; R n )), and that there is a surjection ß 2 (G(k; R n )) ! ß 2 (\Delta MacP(k; n)). The theory of oriented matroids gives rise to a combinatorial a ..."
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We prove that the homotopy groups of the oriented matroid Grassmannian MacP(k; n) are stable as n ! 1, that ß 1 (\Delta MacP(k; n)) ¸ = ß 1 (G(k; R n )), and that there is a surjection ß 2 (G(k; R n )) ! ß 2 (\Delta MacP(k; n)). The theory of oriented matroids gives rise to a combinatorial analog to the Grassmannian G(k; R n ). By thinking of an oriented matroid as a "combinatorial vector space", one is led to define the combinatorial Grassmannian (or MacPhersonian MacP(k; n)) as a partially ordered set, whose order complex is hoped to have topology similar that of G(k; R n ). While there are some immediate and natural correspondences between these two topological spaces, the topology of the MacPhersonian is on the whole a mystery. The topology of MacP(k; n) and the direct limit MacP(k; 1) is of interest from several perspectives. From the topological perspective, the MacPhersonian MacP(k; 1) is the classifying space for the combinatorial vector bundles (or matroid bundles...
Testing Extendability for Partial Chirotopes is NPComplete
 PROCEEDINGS OF THE 13TH CANADIAN CONFERENCE ON COMPUTATIONAL GEOMETRY, U. OF
, 2001
"... This paper shows that it is NPcomplete to algorithmically decide whether there exists a chirotope which extends a given partial chirotope. While Crippen and Havel ([3]) gave an algorithm to compute all possible extensions of a given partial chirotope running in exponential time, Gärtner et al. ([6] ..."
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This paper shows that it is NPcomplete to algorithmically decide whether there exists a chirotope which extends a given partial chirotope. While Crippen and Havel ([3]) gave an algorithm to compute all possible extensions of a given partial chirotope running in exponential time, Gärtner et al. ([6]) discussed a special class of partial chirotopes which turned out to be always extendable.