Complexified real arrangements of hyperplanes

Arvola, William A.

doi:10.1007/bf02568407

Cited by 9 publications

(6 citation statements)

References 2 publications

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“…In Section 5.2 we construct a finite simplicial complex M of the homotopy type of M. The construction uses an embedding in V of the order complex of the face poset of a real arrangement.In the special case of a complexified real arrangement, Salvetti[203] constructed a smaller complex W of the homotopy type of M and Arvola[15] constructed a simplicial map M ----t W, which is a homotopy equivalence. The subject of this chapter is the topology of the complement of a complex arrangement, M(A).…”

mentioning

confidence: 99%

Arrangements of Hyperplanes

Orlik¹,

Terao²

1992

Grundlehren Der Mathematischen Wissenschaften

1,172

1,597

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mentioning

confidence: 99%

Arrangements of Hyperplanes

Orlik¹,

Terao²

1992

Grundlehren Der Mathematischen Wissenschaften

1,172

1,597

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“…Salvetti's construction has been intensively studied and several alternative descriptions and interpretations are known. We use the description in terms of matroid product developed in [GR89], [Arv91], and [BZ92]. The properties of the filtration on the space of little cubes are summarized in §3.…”

Section: Statement Of Resultsmentioning

confidence: 99%

The Salvetti complex and the little cubes

Tamaki¹

2012

J. Eur. Math. Soc.

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For a real central arrangement A, Salvetti introduced a construction of a finite complex Sal(A) which is homotopy equivalent to the complement of the complexified arrangement in [Sal87]. For the braid arrangement A k−1 , the Salvetti complex Sal(A k−1 ) serves as a good combinatorial model for the homotopy type of the configuration space F (C, k) of k points in C, which is homotopy equivalent to the space C2(k) of k little 2-cubes. Motivated by the importance of little cubes in homotopy theory, especially in the study of iterated loop spaces, we study how the combinatorial structure of the Salvetti complexes of the braid arrangements are related to homotopy theoretic properties of iterated loop spaces.As a consequence, we prove the skeletal filtrations on the Salvetti complexes of the braid arrangements give rise to the cobar-type Eilenberg-Moore spectral sequence converging to the homology of Ω 2 Σ 2 X. We also construct a new spectral sequence that computes the homology of Ω ℓ Σ ℓ X for ℓ > 2 by using a higher order analogue of the Salvetti complex. The E 1 -term of the spectral sequence is described in terms of the homology of X. The spectral sequence is different from known spectral sequences that compute the homology of iterated loop spaces, such as the Eilenberg-Moore spectral sequence and the spectral sequence studied by Ahearn and Kuhn in [AK02].

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“…The tangency can be computed directly from Example 1.54 as σ 2 and the tacnode, whose local equation is y 2 = x 8 , that is two smooth branches with multiplicity of intersection 4, can be obtained from Example 1.53 as σ 8 1 . However, the remaining braids depend on global monodromy for two different reasons:…”

Section: Braid Monodromy Of Curves: Local Versus Globalmentioning

confidence: 99%

“…Very extensive literature has been written on this topic (see [64,66,8,9,69,68,31,21,22,40,30,20] among others).…”

Section: Line Arrangementsmentioning

confidence: 99%