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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 Aℐ. These stem from ideals ℐ in the set of positive roots of a reduced root system. We show that the K(π,1)–property holds for all arrangements Aℐ if the underlying Weyl group is classical and that it extends to most of the Aℐ if the underlying Weyl group is of exceptional type. Conjecturally this holds for all Aℐ. In general, the Aℐ are neither simplicial nor is their complexification of fiber type.
Keywords: Weyl arrangement, arrangement of ideal type, $K(\pi,1)$ arrangement
Amend, Nils 1 ; Röhrle, Gerhard 2
@article{10_2140_agt_2019_19_1341,
author = {Amend, Nils and R\"ohrle, Gerhard},
title = {The topology of arrangements of ideal type},
journal = {Algebraic and Geometric Topology},
pages = {1341--1358},
publisher = {mathdoc},
volume = {19},
number = {3},
year = {2019},
doi = {10.2140/agt.2019.19.1341},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.1341/}
}
TY - JOUR AU - Amend, Nils AU - Röhrle, Gerhard TI - The topology of arrangements of ideal type JO - Algebraic and Geometric Topology PY - 2019 SP - 1341 EP - 1358 VL - 19 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.1341/ DO - 10.2140/agt.2019.19.1341 ID - 10_2140_agt_2019_19_1341 ER -
Amend, Nils; Röhrle, Gerhard. The topology of arrangements of ideal type. Algebraic and Geometric Topology, Tome 19 (2019) no. 3, pp. 1341-1358. doi: 10.2140/agt.2019.19.1341
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