Algebraic & Geometric Topology

The topology of arrangements of ideal type

Nils Amend and Gerhard Röhrle

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Abstract

In 1962, Fadell and Neuwirth showed that the configuration space of the braid arrangement is aspherical. Having generalized this to many real reflection groups, Brieskorn conjectured this for all finite Coxeter groups. This in turn follows from Deligne’s seminal work from 1972, where he showed that the complexification of every real simplicial arrangement is a K ( π , 1 ) –arrangement.

We study the K ( π , 1 ) –property for a certain class of subarrangements of Weyl arrangements, the so-called arrangements of ideal type A . These stem from ideals  in the set of positive roots of a reduced root system. We show that the K ( π , 1 ) –property holds for all arrangements A if the underlying Weyl group is classical and that it extends to most of the A if the underlying Weyl group is of exceptional type. Conjecturally this holds for all A . In general, the A are neither simplicial nor is their complexification of fiber type.

Article information

Source
Algebr. Geom. Topol., Volume 19, Number 3 (2019), 1341-1358.

Dates
Received: 30 January 2018
Revised: 21 August 2018
Accepted: 24 October 2018
First available in Project Euclid: 29 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.agt/1559095428

Digital Object Identifier
doi:10.2140/agt.2019.19.1341

Mathematical Reviews number (MathSciNet)
MR3954284

Zentralblatt MATH identifier
07078606

Subjects
Primary: 14N20: Configurations and arrangements of linear subspaces 20F55: Reflection and Coxeter groups [See also 22E40, 51F15] 52C35: Arrangements of points, flats, hyperplanes [See also 32S22]
Secondary: 13N15: Derivations

Keywords
Weyl arrangement arrangement of ideal type $K(\pi,1)$ arrangement

Citation

Amend, Nils; Röhrle, Gerhard. The topology of arrangements of ideal type. Algebr. Geom. Topol. 19 (2019), no. 3, 1341--1358. doi:10.2140/agt.2019.19.1341. https://projecteuclid.org/euclid.agt/1559095428


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References

  • T Abe, M Barakat, M Cuntz, T Hoge, H Terao, The freeness of ideal subarrangements of Weyl arrangements, J. Eur. Math. Soc. 18 (2016) 1339–1348
  • M Barakat, M Cuntz, Coxeter and crystallographic arrangements are inductively free, Adv. Math. 229 (2012) 691–709
  • N Bourbaki, Groupes et algèbres de Lie, Chapitre I: Algèbres de Lie, 2nd edition, Actualités Scientifiques et Industrielles 1285, Hermann, Paris (1971)
  • E Brieskorn, Sur les groupes de tresses (d'après V I Arnold), from “Séminaire Bourbaki, 1971/1972”, Lecture Notes in Math. 317, Springer (1973) Exposé 401, 21–44
  • D C Cohen, Monodromy of fiber-type arrangements and orbit configuration spaces, Forum Math. 13 (2001) 505–530
  • M Cuntz, I Heckenberger, Finite Weyl groupoids, J. Reine Angew. Math. 702 (2015) 77–108
  • M Cuntz, G Röhrle, A Schauenburg, Arrangements of ideal type are inductively free, Internat. J. Algebra Comput. (online publication February 2019)
  • P Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972) 273–302
  • E Fadell, L Neuwirth, Configuration spaces, Math. Scand. 10 (1962) 111–118
  • M J Falk, N J Proudfoot, Parallel connections and bundles of arrangements, Topology Appl. 118 (2002) 65–83
  • M Falk, R Randell, The lower central series of a fiber-type arrangement, Invent. Math. 82 (1985) 77–88
  • M Falk, R Randell, On the homotopy theory of arrangements, from “Complex analytic singularities” (T Suwa, P Wagreich, editors), Adv. Stud. Pure Math. 8, North-Holland, Amsterdam (1987) 101–124
  • A Hultman, Supersolvability and the Koszul property of root ideal arrangements, Proc. Amer. Math. Soc. 144 (2016) 1401–1413
  • A Leibman, D Markushevich, The monodromy of the Brieskorn bundle, from “Geometric topology” (C Gordon, Y Moriah, B Wajnryb, editors), Contemp. Math. 164, Amer. Math. Soc., Providence, RI (1994) 91–117
  • J N Mather, Stratifications and mappings, from “Dynamical systems” (M M Peixoto, editor), Academic, New York (1973) 195–232
  • P Orlik, H Terao, Arrangements of hyperplanes, Grundl. Math. Wissen. 300, Springer (1992)
  • L Paris, The Deligne complex of a real arrangement of hyperplanes, Nagoya Math. J. 131 (1993) 39–65
  • G Röhrle, Arrangements of ideal type, J. Algebra 484 (2017) 126–167
  • E Sommers, J Tymoczko, Exponents for $B$–stable ideals, Trans. Amer. Math. Soc. 358 (2006) 3493–3509
  • R P Stanley, Supersolvable lattices, Algebra Universalis 2 (1972) 197–217
  • H Terao, Modular elements of lattices and topological fibration, Adv. in Math. 62 (1986) 135–154
  • R Thom, Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc. 75 (1969) 240–284