The topology of arrangements of ideal type

Amend, Nils; Röhrle, Gerhard

doi:10.2140/agt.2019.19.1341

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The topology of arrangements of ideal type

Nils Amend and Gerhard Röhrle

Algebraic & Geometric Topology 19 (2019) 1341–1358

DOI: 10.2140/agt.2019.19.1341

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 $A_{ℐ}$ . 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 $A_{ℐ}$ if the underlying Weyl group is classical and that it extends to most of the $A_{ℐ}$ if the underlying Weyl group is of exceptional type. Conjecturally this holds for all $A_{ℐ}$ . In general, the $A_{ℐ}$ are neither simplicial nor is their complexification of fiber type.

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Keywords

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

Mathematical Subject Classification 2010

Primary: 14N20, 20F55, 52C35

Secondary: 13N15

References

Publication

Received: 30 January 2018

Revised: 21 August 2018

Accepted: 24 October 2018

Published: 21 May 2019

Authors

Nils Amend
Institut für Algebra, Zahlentheorie und Diskrete Mathematik Fakultät für Mathematik und Physik Gottfried Wilhelm Leibniz Universität Hannover Hannover Germany

Gerhard Röhrle
Fakultät für Mathematik Ruhr-Universität Bochum Bochum Germany

Volume 19, issue 3 (2019)