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The generalized Baues problem
, 1998
"... Abstract. We survey the generalized Baues problem of Billera and Sturmfels. The problem is one of discrete geometry and topology, and asks about the topology of the set of subdivisions of a certain kind of a convex polytope. Along with a discussion of most of the known results, we survey the motivat ..."
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Abstract. We survey the generalized Baues problem of Billera and Sturmfels. The problem is one of discrete geometry and topology, and asks about the topology of the set of subdivisions of a certain kind of a convex polytope. Along with a discussion of most of the known results, we survey the motivation for the problem and its relation to triangulations, zonotopal tilings, monotone paths in linear programming, oriented matroid Grassmannians, singularities, and homotopy theory. Included are several open questions and problems. 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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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
Matroid bundles
 IN NEW PERSPECTIVES IN ALGEBRAIC COMBINATORICS, MSRI BOOK SERIES
, 1999
"... Combinatorial vector bundles, or matroid bundles, are a combinatorial analog to real vector bundles. Combinatorial objects called oriented matroids play the role of real vector spaces. This combinatorial analogy is remarkably strong, and has led to combinatorial results in topology and bundletheore ..."
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Combinatorial vector bundles, or matroid bundles, are a combinatorial analog to real vector bundles. Combinatorial objects called oriented matroids play the role of real vector spaces. This combinatorial analogy is remarkably strong, and has led to combinatorial results in topology and bundletheoretic proofs in combinatorics. This paper surveys recent results on matroid bundles, and describes a canonical functor from real vector bundles to matroid bundles.
On The Baues Conjecture in corank 3.
, 2000
"... A special case of the Generalized Baues Conjecture states that the order complex of the Baues poset of an acyclic vector configuration A (the Baues complex of A) is homotopy equivalent to a sphere of dimension equal to the corank of A minus one. The Baues poset of A is the set of proper polyhedr ..."
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A special case of the Generalized Baues Conjecture states that the order complex of the Baues poset of an acyclic vector configuration A (the Baues complex of A) is homotopy equivalent to a sphere of dimension equal to the corank of A minus one. The Baues poset of A is the set of proper polyhedral subdivisions of A ordered by refinement. Recently, Santos has found a counterexample in corank 317 to the Baues conjecture. Here, we study the case of corank 3. The techniques we use also allow us to show that if a corank 3 vector configuration is not acyclic, then its Baues complex is contractible. Introduction The Baues problem concerns the study of the space of the polyhedral subdivisions of a vector configuration [15]. A vector configuration A in R d+1 is a finite spanning set of labelled vectors (we allow repetitions) in the linear space R d+1 . A cell of A is any spanning subset of A. The number d + 1 is called the rank of A, while the difference between the cardinality of...
CIRCUIT ADMISSIBLE TRIANGULATIONS OF ORIENTED MATROIDS
, 2000
"... All triangulations of euclidean oriented matroids are of the same PLhomeomorphism type by a result of Anderson. That means all triangulations of euclidean acyclic oriented matroids are PLhomeomorphic to PLballs and that all triangulations of totally cyclic oriented matroids are PLhomeomorphic to ..."
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All triangulations of euclidean oriented matroids are of the same PLhomeomorphism type by a result of Anderson. That means all triangulations of euclidean acyclic oriented matroids are PLhomeomorphic to PLballs and that all triangulations of totally cyclic oriented matroids are PLhomeomorphic to PLspheres. For noneuclidean oriented matroids this question is wide open. One key point in the proof of Anderson is the following fact: for every triangulation of a euclidean oriented matroid the adjacency graph of the set of all simplices “intersecting” a segment ����� � is a path. We call this graph the ������adjacency graph of the triangulation. While we cannot solve the problem of the topological type of triangulations of general oriented matroids we show in this note that for every circuit admissible triangulation of an arbitrary oriented matroid the ������adjacency graph is path.
CIRCUIT ADMISSIBLE TRIANGULATIONS OF ORIENTED MATROIDS
"... ABSTRACT. All triangulations of euclidean oriented matroids are of the same PLhomeomorphism type by a result of Anderson. That means all triangulations of euclidean acyclic oriented matroids are PLhomeomorphic to PLballs and that all triangulations of totally cyclic oriented matroids are PLhomeom ..."
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ABSTRACT. All triangulations of euclidean oriented matroids are of the same PLhomeomorphism type by a result of Anderson. That means all triangulations of euclidean acyclic oriented matroids are PLhomeomorphic to PLballs and that all triangulations of totally cyclic oriented matroids are PLhomeomorphic to PLspheres. For noneuclidean oriented matroids this question is wide open. One key point in the proof of Anderson is the following fact: for every triangulation of a euclidean oriented matroid the adjacency graph of the set of all simplices “intersecting ” a segment [p_p+] is a path. We call this graph the [p_p+]adjacency graph of the triangulation. While we cannot solve the problem of the topological type of triangulations of general oriented matroids we show in this note that for every circuit admissible triangulation of an arbitrary oriented matroid the [p_p+]adjacency graph is path. Triangulations of oriented matroids appeared in the literature as natural combinatorial models for triangulations of point configurations [2]. However, since not all oriented matroids model point configurations the notion of a triangulation of