Homotopy types of line arrangements

“…The first should be compared with Theorems 1.3 and 2.5. The original examples of non-isomorphic matroids with isomorphic O S algebras, which appeared in [10,11,22], are truncations of G and G , where the factors G 0 and G 1 both have rank two. In an NSF-sponsored REU undergraduate research project directed by the author, C. Pendergrass showed that truncation of matroids always preserves isomorphisms of the associated O S algebra [24].…”

Section: Proof This Is Immediate From the Identitiesmentioning

confidence: 99%

Combinatorial and Algebraic Structure in Orlik–Solomon Algebras

Falk

2001

European Journal of Combinatorics

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The Orlik-Solomon algebra A(G) of a matroid G is the free exterior algebra on the points, modulo the ideal generated by the circuit boundaries. On one hand, this algebra is a homotopy invariant of the complement of any complex hyperplane arrangement realizing G. On the other hand, some features of the matroid G are reflected in the algebraic structure of A(G).In this mostly expository article, we describe recent developments in the construction of algebraic invariants of A(G). We develop a categorical framework for the statement and proof of recently discovered isomorphism theorems which suggests a possible setting for classification theorems. Several specific open problems are formulated.

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“…Building on previous work by Hattori [19], Falk [11] and Cohen-Suciu [9], around 2000 Dimca-Papadima [10] and Randell [23] independently showed that the complement of any complex hyperplane arrangement is a minimal space. The idea is to establish a Lefschetz-type hyperplane theorem for the complement of the arrangement by first establishing a Lefschetz hyperplane theorem for the Milnor fiber of the arrangement, building on earlier work of Hamm and Lê [17] and Hamm [16].…”

Section: Introductionmentioning

confidence: 99%

Combinatorial stratifications and minimality of 2-arrangements

Adiprasito

2014

Journal of Topology

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We prove that the complement of any affine 2-arrangement in R d is minimal, that is, it is homotopy equivalent to a cell complex with as many i-cells as its i-th rational Betti number. For the proof, we provide a Lefschetz-type hyperplane theorem for complements of 2-arrangements, and introduce Alexander duality for combinatorial Morse functions. Our results greatly generalize previous work by Falk, Dimca-Papadima, Hattori, Randell, and Salvetti-Settepanella and others, and they demonstrate that in contrast to previous investigations, a purely combinatorial approach suffices to show minimality and the Lefschetz Hyperplane Theorem for complements of complex hyperplane arrangements.

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Homotopy types of line arrangements

Cited by 48 publications

References 10 publications

Arrangements of Hyperplanes

Arrangements of Hyperplanes

Combinatorial and Algebraic Structure in Orlik–Solomon Algebras

Combinatorial stratifications and minimality of 2-arrangements

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