Finite complex reflection arrangements are $K(\pi,1)$

By David Bessis. 2006

145 citations recorded by the source corpus.

Let V be a finite dimensional complex vector space and W ⊆ GL(V ) be a finite complex reflection group. Let V reg be the complement in V of the reflecting hyperplanes. We prove that V reg is a K(π, 1) space. This was predicted by a classical conjecture, originally stated by Brieskorn for complexified real reflection groups. The complexified real case follows from a theorem of Deligne and, after contributions by Nakamura and OrlikSolomon, only six exceptional cases remained open. In addition to solving these six cases, our approach is applicable to most previously known cases, including complexified real groups for which we obtain a new proof, based on new geometric objects. We also address a number of questions about π1(W\V ), the braid group of W . This includes a description of periodic elements in terms of a braid analog of Springer’s theory of regular elements.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3c0a3f182f6298b36b0dbdde89b477250b38d152

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In Discrete Mathematics and Combinatorics

1. Juan González-Meneses
2. J. Birman
3. Volker Gebhardt

In Geometry and Topology

1. Juan González-Meneses
2. Volker Gebhardt
3. J. Birman
4. O. Plamenevskaya

In Mathematical Physics

1. Juan González-Meneses
2. Volker Gebhardt
3. J. Birman
4. O. Plamenevskaya

Citations in the corpus

• Conjugacy in Garside groups II: structure of the ultra summit set (2006, methodology): “15 of [1], show the above conjecture in its general setting.”