Geometric relationship between cohomology of the complement of real and complexified arrangements.

*(English)*Zbl 1010.52017This 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.

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.

Reviewer: Günter M.Ziegler (Berlin)

##### 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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\textit{K. Jewell} and \textit{P. Orlik}, Topology Appl. 118, No. 1--2, 113--129 (2002; Zbl 1010.52017)

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##### References:

[1] | Björner, A., On the homology of geometric lattices, Algebra universalis, 14, 107-128, (1982) |

[2] | Björner, A., Homology and shellability of matroids and geometric lattices, (), 228-283 · Zbl 0772.05027 |

[3] | Björner, A., Topological methods, (), 1819-1872 · Zbl 0851.52016 |

[4] | Brylawski, T., The broken-circuit complex, Trans. amer. math. soc., 234, 417-433, (1977) · Zbl 0368.05022 |

[5] | Bott, R.; Tu, L., Differential forms in algebraic topology, (1982), Springer-Verlag Berlin · Zbl 0496.55001 |

[6] | Folkman, J., The homology groups of a lattice, J. math. and mech., 15, 631-636, (1966) · Zbl 0146.01602 |

[7] | Goresky, M.; MacPherson, R., Stratified Morse theory, (1988), Springer-Verlag Berlin |

[8] | Jewell, K., Complements of sphere and subspace arrangements, Topology appl., 6, 199-214, (1994) · Zbl 0797.55015 |

[9] | Jewell, K.; Orlik, P.; Shapiro, B., On the complements of affine and subspace arrangements, Topology appl., 56, 215-233, (1994) · Zbl 0797.55016 |

[10] | Orlik, P.; Solomon, L., Combinatorics and topology of complements of hyperplanes, Invent. math., 56, 167-189, (1980) · Zbl 0432.14016 |

[11] | Orlik, P.; Terao, H., Arrangements of hyperplanes, (1992), Springer-Verlag Berlin · Zbl 0757.55001 |

[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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