×

Not all free arrangements are \(K(\pi,1)\). (English) Zbl 0815.52012

The complexification of a centred arrangement \(A\) of subspaces of codimension one is the arrangement of hyperplances in \(\mathbb{C}^ n\) defined by \(A_{\mathbb{C}} = \{H \otimes_ R \mathbb{C} \mid H\in A\}\). Let \(M(A)\) be the complement of \(A_{\mathbb{C}}\) in \(\mathbb{C}^ d\). Then \(A\) is said to be \(K(\pi,1)\) if the universal covering space of \(M(A)\) is contractible and the fundamental group \(\pi_ 1(M(A)) = \pi\).
The authors produce a one-parameter family of hyperplane arrangements which are counterexamples to the conjecture of K. Saito that the complexified complement of a free arrangement is \(K(\pi,1)\). These arrangements are restrictions of arrangement families that arose in the study of tilings of certain centrally symmetric octagons.

MSC:

52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
55P20 Eilenberg-Mac Lane spaces
20G10 Cohomology theory for linear algebraic groups
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] V. I. Arnol\(^{\prime}\)d, The cohomology ring of the group of dyed braids, Mat. Zametki 5 (1969), 227 – 231 (Russian).
[2] Egbert Brieskorn, Sur les groupes de tresses [d’après V. I. Arnol\(^{\prime}\)d], Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, Springer, Berlin, 1973, pp. 21 – 44. Lecture Notes in Math., Vol. 317 (French).
[3] Pierre Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273 – 302 (French). · Zbl 0238.20034 · doi:10.1007/BF01406236
[4] P. H. Edelman and V. Reiner, Free arrangements and tilings, preprint, 1994. · Zbl 0853.52013
[5] Edward Fadell and Lee Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111-118. · Zbl 0136.44104 · doi:10.7146/math.scand.a-10517
[6] Toshitake Kohno, Série de Poincaré-Koszul associée aux groupes de tresses pures, Invent. Math. 82 (1985), no. 1, 57 – 75 (French). , https://doi.org/10.1007/BF01394779 Michael Falk and Richard Randell, The lower central series of a fiber-type arrangement, Invent. Math. 82 (1985), no. 1, 77 – 88. · Zbl 0574.55010 · doi:10.1007/BF01394780
[7] R. Fox and L. Neuwirth, The braid groups, Math. Scand. 10 (1962), 119 – 126. · Zbl 0117.41101 · doi:10.7146/math.scand.a-10518
[8] Akio Hattori, Topology of \?\(^{n}\) minus a finite number of affine hyperplanes in general position, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 2, 205 – 219. · Zbl 0306.55011
[9] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. · Zbl 0725.20028
[10] Peter Orlik and Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), no. 2, 167 – 189. · Zbl 0432.14016 · doi:10.1007/BF01392549
[11] Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. · Zbl 0757.55001
[12] Richard Randell, Lattice-isotopic arrangements are topologically isomorphic, Proc. Amer. Math. Soc. 107 (1989), no. 2, 555 – 559. · Zbl 0681.57016
[13] K. Saito, On the uniformization of complements of discriminant loci, Conference Notes, Summer Institute, Amer. Math. Soc., Williamstown, 1975.
[14] Hiroaki Terao, Modular elements of lattices and topological fibration, Adv. in Math. 62 (1986), no. 2, 135 – 154. · Zbl 0612.05019 · doi:10.1016/0001-8708(86)90097-6
[15] Sergey Yuzvinsky, Free and locally free arrangements with a given intersection lattice, Proc. Amer. Math. Soc. 118 (1993), no. 3, 745 – 752. · Zbl 0797.52009
[16] Günter M. Ziegler, Combinatorial construction of logarithmic differential forms, Adv. Math. 76 (1989), no. 1, 116 – 154. · Zbl 0725.05032 · doi:10.1016/0001-8708(89)90045-5
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.