Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(e) ISSN 0273-0979(p)

     

Not all free arrangements are $ K(\pi,1)$

Author(s): Paul H. Edelman; Victor Reiner
Journal: Bull. Amer. Math. Soc. 32 (1995), 61-65.
MathSciNet review: 1273396
Retrieve article in: PDF

Abstract | References | Additional information

Abstract: We produce a one-parameter family of hyperplane arrangements that are counterexamples to the conjecture of Saito that the complexified complement of a free arrangement is $ K(\pi,1)$. These arrangements are the restriction of a one-parameter family of arrangements that arose in the study of tilings of certain centrally symmetric octagons. This other family is discussed as well.

References:

Bibliography

[Ar]
V. I. Arnold, The cohomology ring of the colored braid group, Mat. Zametki 5 (1969), 227-231; Math. Notes, vol. 5, Princeton Univ. Press, Princeton, NJ, 1969, pp. 138-140. MR 0242196 (39:3529)

[Br]
E. Brieskorn, Sur les groupe de tresses, Séminaire Bourbaki 1971/72, Lecture Notes in Math., vol. 317, Springer, New York, 1973, pp. 21-44. MR 0422674 (54:10660)

[De]
P. Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273-302. MR 0422673 (54:10659)

[ER]
P. H. Edelman and V. Reiner, Free arrangements and tilings, preprint, 1994.

[FaN]
E. Fadell and L. Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111-118. MR 0141126 (25:4537)

[FR]
M. Falk and R. Randell, The lower central series of a fiber-type arrangement, Invent. Math. 82 (1985), 77-88. MR 808110 (87c:32015b)

[FoN]
R. H. Fox and L. Neuwirth, The braid groups, Math. Scand. 10 (1962), 119-126. MR 0150755 (27:742)

[Ha]
A. Hattori, Topology of $ {\mathbb{C}^n}$ minus a finite number of affine hyperplanes in general position, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), 205-219. MR 0379883 (52:788)

[Hu]
J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge, 1990. MR 1066460 (92h:20002)

[OS]
P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167-189. MR 558866 (81e:32015)

[OT]
P. Orlik and H. Terao, Arrangements of hyperplanes, Springer-Verlag, Berlin, 1992. MR 1217488 (94e:52014)

[Ra]
R. Randell, Lattice-isotopic arrangements are topologically isomorphic, Proc. Amer. Math. Soc. 107 (1989), 555-559. MR 984812 (90a:57032)

[Sa]
K. Saito, On the uniformization of complements of discriminant loci, Conference Notes, Summer Institute, Amer. Math. Soc., Williamstown, 1975.

[Te]
H. Terao, Modular elements of lattices and topological fibrations, Adv. in Math. 62 (1986), 135-154. MR 865835 (88b:32032)

[Yu]
S. Yuzvinsky, Free and locally free arrangements with a given intersection lattice, Proc. Amer. Math. Soc. 118 (1993), 745-752. MR 1160307 (93i:52022)

[Zi]
G. Ziegler, Combinatorial constructions of logarithmic differential forms, Adv. in Math. 76 (1989), 116-154. MR 1004488 (90j:32016)

Additional Information:

DOI: 10.1090/S0273-0979-1995-00557-4
PII: S 0273-0979(1995)00557-4
Copyright of article: Copyright 1995, American Mathematical Society




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia