Oriented matroids

Folkman, Jon; Lawrence, Jim

doi:10.1016/0095-8956(78)90039-4

Cited by 260 publications

(236 citation statements)

References 7 publications

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“…The inspiration for this paper came from Swartz's similar result ([Swa03]), which in turn was inspired by Folkman and Lawrence's Topological Representation Theorem ( [FL78]). Folkman and Lawrence's result concerns oriented matroids, which are combinatorial analogs to arrangements of oriented hyperplanes in R n (or F n , where F is any field of characteristic 0).…”

Section: Oriented Matroidsmentioning

confidence: 99%

“…A fundamental result in oriented matroid theory is the Topological Representation Theorem ( [FL78]), which says that every rank r oriented matroid can be represented by an arrangement of oriented pseudospheres in S r−1 . In [Swa03] Swartz made the startling discovery that any rank r matroid can be represented by an arrangement of homotopy spheres in a (r − 1)-dimensional CW complex homotopic to S r−1 .…”

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confidence: 99%

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Homotopy Sphere Representations for Matroids

Anderson

2012

Ann. Comb.

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For any rank r oriented matroid M , a construction is given of a "topological representation" of M by an arrangement of homotopy spheres in a simplicial complex which is homotopy equivalent to S r−1 . The construction is completely explicit and depends only on a choice of maximal flag in M . If M is orientable, then all Folkman-Lawrence representations of all orientations of M embed in this representation in a homotopically nice way.A fundamental result in oriented matroid theory is the Topological Representation Theorem ([FL78]), which says that every rank r oriented matroid can be represented by an arrangement of oriented pseudospheres in S r−1 . In [Swa03] Swartz made the startling discovery that any rank r matroid can be represented by an arrangement of homotopy spheres in a (r − 1)-dimensional CW complex homotopic to S r−1 . The representation is far from canonical: it depends on, among other things, a choice of tree for each rank 2 contraction and choices of cells glued in to kill off homotopy groups.The present paper, inspired by Swartz's work, gives a topological representation of any rank r matroid by an arrangement of homotopy spheres in a simplicial complex which is homotopy equivalent to S r−1 . The construction is completely explicit and depends only on a choice of maximal flag. For oriented matroids, there is a nice homotopy relationship between this representation and the representation given by the Topological Representation Theorem of Folkman and Lawrence.

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Section: Oriented Matroidsmentioning

confidence: 99%

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confidence: 99%

Homotopy Sphere Representations for Matroids

Anderson

2012

Ann. Comb.

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“…Recall [8] [2,3,7]. It is then natural to ask for the infimum, over all s' '" S, of the spread (1(S'); we call this the intrinsic spread fJ of S. It is not hard to see, by compactness, that this infimum is actually a minimum; see §4 below.…”

Section: Where (P(o) •• P(d)} Denotes the Simplex Spanned By The mentioning

confidence: 99%

The intrinsic spread of a configuration in 𝑅^{𝑑}

Goodman¹,

Pollack²,

Sturmfels³

1990

J. Amer. Math. Soc.

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“…One of the foundations of oriented matroid theory is the topological representation theorem of Folkman and Lawrence [8]. It says that an oriented (simple) matroid can be realized uniquely as an arrangement of pseudospheres.…”

Section: Introductionmentioning

confidence: 99%

Topological representations of matroids

Swartz

2002

J. Amer. Math. Soc.

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There is a one-to-one 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. Bounded subcomplexes of an arrangement of homotopy spheres correspond to minimal cellular resolutions of the dual matroid Steiner ideal. 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.

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Oriented matroids

Cited by 260 publications

References 7 publications

Homotopy Sphere Representations for Matroids

Homotopy Sphere Representations for Matroids

The intrinsic spread of a configuration in 𝑅^{𝑑}

Topological representations of matroids

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