Finite complex reflection arrangements are K(pi,1)

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Let V be a finite dimensional complex vector space and W (V) be a finite complex reflection group. Let V be the complement in V of the reflecting hyperplanes. We prove that V 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 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 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.