Finite complex reflection arrangements are K(pi,1)

Details Finite complex reflection arrangements are K(pi,1) Finite complex reflection arrangements are K(pi,1)

2006-10-26

arxiv.org/abs/math/0610777v3

Mathematics

Geometric Topology

54 pages. Improves and replaces my earlier arXiv preprint "Topology of complex reflection arrangements". v3: many details adde

Scientific paper

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 $\pi_1(W\cq V^{\reg})$, 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.

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