Finite complex reflection arrangements are K(pi,1)

2014/12/22 by David Bessis · 8 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory

paper · pdf · doi:10.4007/annals.2015.181.3.1

Abstract

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(, 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 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 1(W \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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