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The polyhedral product functor: a method of computation for momentangle complexes, arrangements and related spaces, arXiv:0711.4689v1 [math.AT
"... Abstract. This article gives a natural decomposition of the suspension of generalized momentangle complexes or partial product spaces which arise as polyhedral product functors described below. In the special case of the complements of certain subspace arrangements, the geometrical decomposition im ..."
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Abstract. This article gives a natural decomposition of the suspension of generalized momentangle complexes or partial product spaces which arise as polyhedral product functors described below. In the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the homological decomposition in GoreskyMacPherson [20], Hochster[22], Baskakov [3], Panov [36], and BuchstaberPanov [7]. Since the splitting is geometric, an analogous homological decomposition for a generalized momentangle complex applies for any homology theory. This decomposition gives an additive decomposition for the StanleyReisner ring of a finite simplicial complex and generalizations of certain homotopy theoretic results of Porter [39] and Ganea [19]. The spirit of the work here follows that of DenhamSuciu in [16]. 1.
Enumeration in convex geometries and associated polytopal subdivisions of spheres, arXiv:math.CO/0505576
"... Abstract. We construct CW spheres from the lattices that arise as the closed sets of a convex closure, the meetdistributive lattices. These spheres are nearly polytopal, in the sense that their barycentric subdivisions are simplicial polytopes. The complete information on the numbers of faces and c ..."
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Abstract. We construct CW spheres from the lattices that arise as the closed sets of a convex closure, the meetdistributive lattices. These spheres are nearly polytopal, in the sense that their barycentric subdivisions are simplicial polytopes. The complete information on the numbers of faces and chains of faces in these spheres can be obtained from the defining lattices in a manner analogous to the relation between arrangements of hyperplanes and their underlying geometric intersection lattices. 1.
Lectures on matroids and oriented matroids
"... Let’s begin with a little “pep talk”, some (very) brief history, and some of the motivating examples of matroids. 1.1. Motivation. Why learn about or study matroids/oriented matroids in geometric, topological, algebraic combinatorics? Here are a few of my personal reasons. • They are general, so res ..."
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Let’s begin with a little “pep talk”, some (very) brief history, and some of the motivating examples of matroids. 1.1. Motivation. Why learn about or study matroids/oriented matroids in geometric, topological, algebraic combinatorics? Here are a few of my personal reasons. • They are general, so results about them are widely applicable. • They have relatively few axioms and standard constructions/techniques, so they focus one’s approach to solving a problem. • They give examples of wellbehaved objects: polytopes, cell/simplicial complexes, rings. • They provide “duals ” for nonplanar graphs! 1.2. Brief early history. (in no way comprehensive...) 1.2.1. Matroids. • H. Whitney (1932, 1935) graphs, duality, and matroids as abstract linear independence.
The Affine Representation Theorem for Abstract Convex Geometries
"... A convex geometry is a combinatorial abstract model introduced by Edelman and Jamison which captures a combinatorial essence of "convexity" shared by some structures including finite point sets, partially ordered sets, trees, rooted graphs. In this paper, we introduce a generalized convex shelling, ..."
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Cited by 1 (1 self)
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A convex geometry is a combinatorial abstract model introduced by Edelman and Jamison which captures a combinatorial essence of "convexity" shared by some structures including finite point sets, partially ordered sets, trees, rooted graphs. In this paper, we introduce a generalized convex shelling, and we show that any convex geometry can be represented as a generalized convex shelling. This is "the representation theorem for convex geometries" similar to "the representation theorem for oriented matroids" by Folkman and Lawrence. An important feature is that our representation theorem is affinegeometric while that for oriented matroids is topological. So our representation theorem indicates the intrinsic simplicity of convex geometries.
Affine Representations of Abstract Convex Geometries
, 2003
"... A convex geometry is a combinatorial abstract model introduced by Edelman and Jamison which captures a combinatorial essence of "convexity " shared by some structures including finite point sets, partially ordered sets, trees, rooted graphs. In this paper, we introduce a generalized convex shelling, ..."
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A convex geometry is a combinatorial abstract model introduced by Edelman and Jamison which captures a combinatorial essence of "convexity " shared by some structures including finite point sets, partially ordered sets, trees, rooted graphs. In this paper, we introduce a generalized convex shelling, and we show that any convex geometry can be represented as a generalized convex shelling. This is "the representation theorem for convex geometries" similar to "the representation theorem for oriented matroids" by Folkman and Lawrence. An important feature is that our representation theorem is affinegeometric while that for oriented matroids is topological. Namely our representation theorem indicates the intrinsic simplicity of convex geometries.
Arrangements of symmetric products of spaces
, 2003
"... We study the combinatorics and topology of general arrangements of subspaces of the form D + SP n−d (X) in symmetric products SP n (X) where D ∈ SP d (X). Symmetric products SP m (X): = X m /Sm, also known as the spaces of effective “divisors ” of order m, together with their companion spaces of div ..."
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We study the combinatorics and topology of general arrangements of subspaces of the form D + SP n−d (X) in symmetric products SP n (X) where D ∈ SP d (X). Symmetric products SP m (X): = X m /Sm, also known as the spaces of effective “divisors ” of order m, together with their companion spaces of divisors/particles, have been studied from many points of view in numerous papers, see [7] and [21] for the references. In this paper we approach them from the point of view of geometric combinatorics. Using the topological technique of diagrams of spaces along the lines of [34] and [37], we calculate the homology of the union and the complement of these arrangements. As an application we include a computation of the homology of the homotopy end space of the open manifold SP n (Mg,k), where Mg,k is a Riemann surface of genus g punctured at k points, a problem which was originally motivated by the study of commutative (m + k, m)groups [32]. 1 Arrangements of symmetric products