Algebraic & Geometric Topology

The topology of arrangements of ideal type

Nils Amend and Gerhard Röhrle

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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