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Geometric relationship between cohomology of the complement of real and complexified arrangements. (English) Zbl 1010.52017
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.
Unfortunately, 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.”
In 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 M. L. Wachs and J. W. Walker [Order 2, 367-385 (1986; Zbl 0589.06005)], the geometric cycles in the reduced broken-circuit complex as constructed by 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 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.

MSC:
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
55T99 Spectral sequences in algebraic topology
05B35 Combinatorial aspects of matroids and geometric lattices
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[2] Björner, A., Homology and shellability of matroids and geometric lattices, (), 228-283 · Zbl 0772.05027
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[12] Rota, G.C., On the foundations of combinatorial theory I. theory of Möbius functions, Z. wahrscheinlichkeitsrechnung, 2, 340-368, (1964) · Zbl 0121.02406
[13] Wachs, M.; Walker, J., On geometric semilattices, Order, 2, 367-385, (1986) · Zbl 0589.06005
[14] Zieger, G.M., Matroid shellability, β-systems, and affine hyperplane arrangements, J. algebraic combin., 1, 283-300, (1992) · Zbl 0782.05022
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