×

zbMATH — the first resource for mathematics

Diffeomorphic types of the complements of arrangements of hyperplanes. (English) Zbl 0828.57018
A fundamental open problem in the theory of complex hyperplane arrangements has been the conjecture that the homotopy or topological type of the complement of a finite collection of hyperplanes is a function only of the underlying matroid or intersection lattice [P. Orlik and H. Terao, Arrangements of hyperplanes, Grundlehren Math. Wiss. 300 (1992; Zbl 0757.55001)]. This problem has now been resolved in the negative in unpublished work of Grigory Rybnikov [On the fundamental group of a complex hyperplane arrangement (preprint, 1993)]. In the paper under review, the authors provide a sufficient condition for the conjecture to hold. Given an arrangement \({\mathcal A}\) of lines in \(CP^2\), construct the graph \(G({\mathcal A})\) whose vertices are the points of multiplicity greater than two, with two vertices adjacent when there is a line of \({\mathcal A}\) containing them. The star of a vertex is the set of lines containing it (which may contain several collinear edges). An arrangement is called “nice” if there is a set of pairwise disjoint stars in \(G({\mathcal A})\) whose complement is a forest. In this case it is shown that for any arrangement \({\mathcal A}'\) with the same intersection lattice as \({\mathcal A}\), the arrangements along the segment from \({\mathcal A}\) to \({\mathcal A}'\) have constant intersection lattice. In other words, the realization space of the matroid of \({\mathcal A}\) is convex. Then the lattice isotopy theorem of R. Randell [Proc. Am. Math. Soc. 107, 555-559 (1989; Zbl 0681.57016)] implies that \({\mathcal A}\) and \({\mathcal A}'\) have diffeomorphic complements.
Reviewer: M.J.Falk (Madison)

MSC:
57R19 Algebraic topology on manifolds and differential topology
57M05 Fundamental group, presentations, free differential calculus
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] V.I. Arnold : The cohomology ring of the colored braid group . Mat. Sametki 5 (1969) 227-231; Math. Notes 5 (1969) 138-140. · Zbl 0277.55002 · doi:10.1007/BF01098313
[2] W. Arvola : The fundamental group of the complement of an arrangement of complex hyperplanes (preprint). · Zbl 0772.57001 · doi:10.1016/0040-9383(92)90006-4
[3] W. Arvola : Complexified real arrangements of hyperplanes (preprint). · Zbl 0731.57011 · doi:10.1007/BF02568407 · eudml:155617
[4] A. Björner and G. Ziegler : Combinatorial stratification of complex arrangements . J. Amer. Math. Soc., 1992. · Zbl 0754.52003 · doi:10.2307/2152753
[5] E. Brieskorn : Sur les groups de tresses , in ” Séminaire Bourbaki 1971/72”, Lecture Notes in Math. 317. Springer-Verlag: Berlin, Heidelberg, New York (1973) 21-44. · Zbl 0277.55003 · numdam:SB_1971-1972__14__21_0 · eudml:109814
[6] P. Cartier : Les arrangements d’hyperplans: un chapitre de géometrie combinatoire , in ”Séminaire Bourbaki 1980/81,” Lecture Notes in Math. 901. Springer-Verlag, Berlin, Heidelberg, New York (1981) 1-22. · Zbl 0483.51011 · numdam:SB_1980-1981__23__1_0 · eudml:109972
[7] H.S.M. Coxeter : Discrete groups generated by reflections . Annals of Math. 35 (1934) 588-621. · Zbl 0010.01101 · doi:10.2307/1968753
[8] P. Deligne : Les immeubles des groups de tresses génèralisés . Invent. Math. 17 (1972) 273-302. · Zbl 0238.20034 · doi:10.1007/BF01406236 · eudml:142173
[9] P. Deligne and G.D. Mostow : Monodromy of hypergeometric functions and non-lattice integral monodromy . Publ. Math. IHES 63 (1986) 5-89. · Zbl 0615.22008 · doi:10.1007/BF02831622 · numdam:PMIHES_1986__63__5_0 · eudml:104012
[10] M. Falk : On the algebra associated with a geometric lattice . Advances in Mathematics, vol. 80(2) ((1990) 152-163. · Zbl 0733.05026 · doi:10.1016/0001-8708(90)90024-H
[11] I.M. Gelfand : General theory of hypergeometric functions . Soviet Math. Doklady 33 (1986) 573-577. · Zbl 0645.33010
[12] M. Goresky and R. Macpherson : Stratified Morse Theory, Ergebnisse der Mathematik und ihre Grenzgebiete, 3 Folge, Band 14. Springer-Verlag, Berlin (1988). · Zbl 0639.14012
[13] F. Hirzzebriech : Arrangements of lines and algebraic surfaces, in ”Arithmetic and Geometry” , vol. II. Progress in Math. 36. Birkhauser, Boston (1983) 113-140. · Zbl 0527.14033
[14] T. Jiang and S.S.-T. Yau : Topological and differentiable structures of the complement of an arrangement of hyperplanes (preprint). · Zbl 0795.57012
[15] J. Mather : Notes on topological stability . Harvard University, July 1970. · Zbl 1260.57049 · doi:10.1090/S0273-0979-2012-01383-6
[16] B. Moishezon : Simply connected algebraic surfaces of general type . Invent. Math. 89 (1987) 601-643. · Zbl 0627.14019 · doi:10.1007/BF01388987 · eudml:143496
[17] P. Orlik : Introduction to Arrangements , C.B.M.S. Regional Conference Series in Mathematics, No. 72, A.M.S. · Zbl 0722.51003
[18] P. Orlik : Complements of subspace arrangements (preprint). · Zbl 0795.52003
[19] P. Orlik and L. Solomon : Combinations and topology of complements of hyperplanes . Invent. Math. 56 (1980) 167-189. · Zbl 0432.14016 · doi:10.1007/BF01392549 · eudml:142694
[20] P. Orlik and L. Solomon : Unitary reflection groups and cohomology . Invent. Math. 59 (1980) 77-94. · Zbl 0452.20050 · doi:10.1007/BF01390316 · eudml:142731
[21] P. Orlik and L. Solomon : Discriminants in the invariant theory of reflection groups . Bagoya Math. J.. 109 (1988) 23-45. · Zbl 0614.20032 · doi:10.1017/S0027763000002749
[22] R. Randall : The fundamental group of the complement of a union of complex hyperplanes . Invent. Math. 80 (1985) 467-468. A correction of this paper by the same author is in Invent. Math. 80 (1985) 467-468. · Zbl 0596.14014 · doi:10.1007/BF01388726 · eudml:142944
[23] R. Randall : Lattice-isotopic arrangements are topologically isomorphic . · Zbl 0681.57016 · doi:10.2307/2047847
[24] M. Salvetti : Topology of the complement of real hyperplanes in CN . Invent. Math. 88 (1987) 603-618. · Zbl 0594.57009 · doi:10.1007/BF01391833 · eudml:143468
[25] M. Salvetti : Arrangements of lines and monodromy of plane curves . Compositio Math. 68 (1988) 103-122. · Zbl 0661.14038 · numdam:CM_1988__68_1_103_0 · eudml:89924
[26] M. Salvetti : On the homotopy theory of complexes associated to metrical-hemisphere complexes . Discrete Mathematics 113 (1993) 155-177. · Zbl 0774.52007 · doi:10.1016/0012-365X(93)90514-T
[27] G.C. Shepherd and J.A. Todd : Finite unitary reflection groups . Canad. J. Math. 6 (1954) 274-304. · Zbl 0055.14305 · doi:10.4153/CJM-1954-028-3
[28] B. Teissier : Cycles évanescents, sections planes et conditions de Whitney . Asterisque (Société Mathématique de France) No. 7-8, 1973. · Zbl 0295.14003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.