Finite complex reflection arrangements are K(pi,1)

Abstract

Let $V$ be a finite dimensional complex vector space and $W\subseteq \GL(V)$ 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 $K(\pi ,1)$ 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 Orlik-Solomon, only six exceptional cases remained open. In addition to solving this 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 $π_1(W\cq V^{\reg})$, 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.

71 pages. v5 minor fixes over v4; v4 contains an entirely rewritten Section 11, a new appendix on Garside theory, and many more improvements (most notably to Section 7). arXiv admin note: text overlap with arXiv:math/0411645

References in corpus (2)

• Zariski theorems and diagrams for braid groups
• Garside categories, periodic loops and cyclic sets

Cited by in corpus (7)

• Counting factorizations of Coxeter elements into products of reflections
• Reflection factorizations of Singer cycles
• The freeness conjecture for Hecke algebras of complex reflection groups, and the case of the Hessian group G26
• Monoids in the fundamental groups of the complement of logarithmic free divisors in C^3
• Knizhnik-Zamolodchikov bundles are topologically trivial
• Abelianization of Subgroups of Reflection Group and their Braid Group; an Application to Cohomology
• q,t-Fuss-Catalan numbers for finite reflection groups