Geometric relationship between cohomology of the complement of real and complexified arrangements
type: Item
instance of: scholarly article
title: Geometric relationship between cohomology of the complement of real and complexified arrangements
zbMATH Open document ID: 1010.52017
DOI: 10.1016/S0166-8641(01)00045-1
author: Ken Jewell
author: Peter Orlik
published in: Topology and its Applications
publication date: 2002-04-07T00:00:00Z
reviewed by: Günter M. Ziegler
Mathematics Subject Classification ID: 52C35
Mathematics Subject Classification ID: 55T99
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 1724927
zbMATH Keywords: real and complex hyperplane arrangements
zbMATH Keywords: cohomology
zbMATH Keywords: spectral sequences
MaRDI profile type: Publication
cites work: On the homology of geometric lattices
cites work: Q4012033
cites work: Q4871782
cites work: The Broken-Circuit Complex
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cites work: Q5522948
cites work: Q4039743
cites work: Complements of sphere and subspace arrangements
cites work: On the complements of affine subspace arrangements
cites work: Combinatorics and topology of complements of hyperplanes
cites work: Q4003206
cites work: On the foundations of combinatorial theory I. Theory of M�bius Functions
cites work: On geometric semilattices
cites work: Matroid shellability, \(\beta\)-systems, and affine hyperplane arrangements
Wikidata QID: Q126401759
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review text: This paper addresses the problem to find a relation between (bases for) the cohomology of an affine real hyperplane arrangement and the cohomology of its complexification. The authors propose as an answer to this problem that a Mayer-Vietoris spectral sequence establishes (rather indirect) such a relationship. NEWLINENEWLINENEWLINEUnfortunately, they choose to ignore much more direct correspondences obtained earlier, by saying ``Some constructions in this section have analogs in the literature [13, 14]. We decided to include them here to make this paper selfcontained.'' NEWLINENEWLINENEWLINEIn particular, they thus miss, resp. recreate only partially, substantial structural insights, such as the structure of the intersection lattice of an affine hyperplane arrangement as a geometric semilattice in the sense of \textit{M. L. Wachs} and \textit{J. W. Walker} [Order 2, 367-385 (1986; Zbl 0589.06005)], the geometric cycles in the reduced broken-circuit complex as constructed by \textit{A. Björner} [Encycl. Math. Appl. 40, 226-283 (1992; Zbl 0772.05027)], the explicit cycles in real and complex hyperplane arrangements, indexed by the broken circuit complex, as described by \textit{A. Björner} and the reviewer [J. Am. Math. Soc. 5, No. 1, 105-149; Zbl 0754.52003)], the combinatorics and geometry of cycles in affine hyperplane arrangements as described by the reviewer [J. Algebr. Comb. 1, No. 3, 283-300 (1992; Zbl 0782.05022)], etc.
description: scientific article; zbMATH DE number 1724927