Topology of plane arrangements and their complements.

*(English. Russian original)*Zbl 1002.55015
Russ. Math. Surv. 56, No. 2, 365-401 (2001); translation from Usp. Mat. Nauk 56, No. 2, 167-203 (2001).

From the introduction: “Finite sets of affine planes in \(\mathbb{R}^N\) or in \(\mathbb{C}^N\) (referred to as affine plane arrangements) form a remarkable class of algebraic varieties. Indeed,

(1) topology, combinatorics, linear algebra, representation theory, algebraic geometry, complexity theory, mathematical physics, and differential equations are closely interwoven in the study of affine plane arrangements;

(2) they are an excellent proving ground for various methods and motivations having very far-reaching generalizations in these fields;

(3) they provide a good elementary visualization of abstract algebraic and combinatorial notions and constructions.

Formulae, constructions, and theorems once arisen in this theory then appear over and over again in very remote fields and problems. One of the main problems in the theory is the extent to which the topological properties of a union of planes (and of its complement) are determined by formal data, that is, by the dimensions of all subcollections of planes. We use this problem to demonstrate notions and methods such as braid groups, configuration spaces, order complexes, stratified Morse theory, simplicial resolutions, complex of graphs, Orlik-Solomon rings, matroids, Spanier-Whitehead duality, twisted homology groups, monodromy theory, hypergeometric functions, and so on.

There are several very good expositions of the theory of arrangements and its various aspects [for example, I. M. Gel’fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants (1994; Zbl 0827.14036); P. Cartier, Les arrangements d’hyperplans: un chapitre de géométrie combinatoire, Lect. Notes Math. 901, 1-22 (1981; Zbl 0483.51011); P. Orlik and H. Terao, Arrangements of hyperplanes, Grundlehren Math. Wiss. 300 (1992; Zbl 0757.55001); G. M. Ziegler, Combinatorial models for subspace arrangements, Habiliations-Schrift, Techn. Univ. Berlin, Berlin (1992; Zbl 0779.52015); A. N. Varchenko, Multidimensional hypergeometric functions in conformal field theory, algebraic \(K\)-theory, algebraic geometry, Proc. Int. Congr. Math., Kyoto Japan 1990, Vol. I, 281-300 (1991; Zbl 0747.33002); and especially A. Björner, Subspace arrangements, Prog. Math. 119, 321-370 (1994; Zbl 0844.52008)]. A comprehensive survey of algebraic aspects of the theory of complex hyperplane arrangements is given in S. Yuzvinsky’s article [Orlik-Solomon algebras in algebra and topology, Russ. Math. Surv. 56, No. 2, 293-364 (2001); translation from Usp. Mat. Nauk 56, No. 2, 87-166 (2001; Zbl 1033.52019)].

In this short article, I have tried to give an elementary introduction to the theory, putting emphasis on (a) the most geometric aspects and motivations of the theory, (b) recent results not yet reflected in introductory texts, (c) topics traditionally treated too formally and abstractly, and (d) results having important applications and generalizations in fields familiar to me, like differential topology, singularity theory, integral geometry, complexity theory,…”

Contents: §1. Introduction, §2. Main definitions, notion, and examples, §3. The basic example: cohomology rings and pure braid groups, §4. The Orlik-Solomon ring and the cohomology of complements of complex hyperplane arrangements, §5. The extent to which the topology of the complement is determined by the dimension data: a summary, §6. The order complex of a partially order set. The Goresky-McPherson formula, §7. Simplicial resolutions and the inclusion-exclusion formula. The Mayer-Vietoris spectral sequence and its modifications, §8. The homotopy type of an affine plane arrangement and the stable homotopy type of its complement, §9. Examples: resolutions of some important arrangements. Complexes of connected graphs and hypergraphs, §10. A combinatorial realization of cohomology classes of complements of plane arrangements, §11. Multiplication in cohomology, §12. The Salvetti complex of a complexified real hyperplane arrangement, §13. Twisted homology of complements of plane arrangements. Resonances, §14. Matroids and configuration spaces, §15. Applications in integral geometry: general hypergeometric functions, §16. What if the collection of planes is infinite?, §17. Applications and analogies in differential topology, Bibliography.

(1) topology, combinatorics, linear algebra, representation theory, algebraic geometry, complexity theory, mathematical physics, and differential equations are closely interwoven in the study of affine plane arrangements;

(2) they are an excellent proving ground for various methods and motivations having very far-reaching generalizations in these fields;

(3) they provide a good elementary visualization of abstract algebraic and combinatorial notions and constructions.

Formulae, constructions, and theorems once arisen in this theory then appear over and over again in very remote fields and problems. One of the main problems in the theory is the extent to which the topological properties of a union of planes (and of its complement) are determined by formal data, that is, by the dimensions of all subcollections of planes. We use this problem to demonstrate notions and methods such as braid groups, configuration spaces, order complexes, stratified Morse theory, simplicial resolutions, complex of graphs, Orlik-Solomon rings, matroids, Spanier-Whitehead duality, twisted homology groups, monodromy theory, hypergeometric functions, and so on.

There are several very good expositions of the theory of arrangements and its various aspects [for example, I. M. Gel’fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants (1994; Zbl 0827.14036); P. Cartier, Les arrangements d’hyperplans: un chapitre de géométrie combinatoire, Lect. Notes Math. 901, 1-22 (1981; Zbl 0483.51011); P. Orlik and H. Terao, Arrangements of hyperplanes, Grundlehren Math. Wiss. 300 (1992; Zbl 0757.55001); G. M. Ziegler, Combinatorial models for subspace arrangements, Habiliations-Schrift, Techn. Univ. Berlin, Berlin (1992; Zbl 0779.52015); A. N. Varchenko, Multidimensional hypergeometric functions in conformal field theory, algebraic \(K\)-theory, algebraic geometry, Proc. Int. Congr. Math., Kyoto Japan 1990, Vol. I, 281-300 (1991; Zbl 0747.33002); and especially A. Björner, Subspace arrangements, Prog. Math. 119, 321-370 (1994; Zbl 0844.52008)]. A comprehensive survey of algebraic aspects of the theory of complex hyperplane arrangements is given in S. Yuzvinsky’s article [Orlik-Solomon algebras in algebra and topology, Russ. Math. Surv. 56, No. 2, 293-364 (2001); translation from Usp. Mat. Nauk 56, No. 2, 87-166 (2001; Zbl 1033.52019)].

In this short article, I have tried to give an elementary introduction to the theory, putting emphasis on (a) the most geometric aspects and motivations of the theory, (b) recent results not yet reflected in introductory texts, (c) topics traditionally treated too formally and abstractly, and (d) results having important applications and generalizations in fields familiar to me, like differential topology, singularity theory, integral geometry, complexity theory,…”

Contents: §1. Introduction, §2. Main definitions, notion, and examples, §3. The basic example: cohomology rings and pure braid groups, §4. The Orlik-Solomon ring and the cohomology of complements of complex hyperplane arrangements, §5. The extent to which the topology of the complement is determined by the dimension data: a summary, §6. The order complex of a partially order set. The Goresky-McPherson formula, §7. Simplicial resolutions and the inclusion-exclusion formula. The Mayer-Vietoris spectral sequence and its modifications, §8. The homotopy type of an affine plane arrangement and the stable homotopy type of its complement, §9. Examples: resolutions of some important arrangements. Complexes of connected graphs and hypergraphs, §10. A combinatorial realization of cohomology classes of complements of plane arrangements, §11. Multiplication in cohomology, §12. The Salvetti complex of a complexified real hyperplane arrangement, §13. Twisted homology of complements of plane arrangements. Resonances, §14. Matroids and configuration spaces, §15. Applications in integral geometry: general hypergeometric functions, §16. What if the collection of planes is infinite?, §17. Applications and analogies in differential topology, Bibliography.

Reviewer: Ioan Pop (Iaşi)

##### MSC:

55R80 | Discriminantal varieties and configuration spaces in algebraic topology |

52C35 | Arrangements of points, flats, hyperplanes (aspects of discrete geometry) |

57N65 | Algebraic topology of manifolds |

20F36 | Braid groups; Artin groups |

06A99 | Ordered sets |

05C99 | Graph theory |