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Finite complex reflection arrangements are \(K(\pi,1)\). (English) Zbl 1372.20036
Summary: Let \(V\) be a finite dimensional complex vector space and \(W\subseteq \mathrm{GL}(V)\) be a finite complex reflection group. Let \(V^{\mathrm{reg}}\) be the complement in \(V\) of the reflecting hyperplanes. We prove that \(V^{\mathrm{reg}}\) is a \(K(\pi,1)\) space. This was predicted by a classical conjecture, originally stated by E. Brieskorn [Invent. Math. 12, 57–61 (1971; Zbl 0204.56502)] for complexified real reflection groups. The complexified real case follows from a theorem of P. Deligne [ibid. 17, 273–302 (1972; Zbl 0238.20034)] and, after contributions by T. Nakamura [Sci. Pap. Coll. Arts Sci., Univ. Tokyo 33, 1–6 (1983; Zbl 0524.20027)] and P. Orlik and L. Solomon [Nagoya Math. J. 109, 23–45 (1988; Zbl 0614.20032)], 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 \(\pi_1(W\backslash V^{\mathrm{reg}})\), the braid group of \(W\). This includes a description of periodic elements in terms of a braid analog of T. A. Springer’s theory of regular elements [Invent. Math. 25, 159–198 (1974; Zbl 0287.20043)].

MSC:
20F55 Reflection and Coxeter groups (group-theoretic aspects)
32S22 Relations with arrangements of hyperplanes
20F36 Braid groups; Artin groups
51F15 Reflection groups, reflection geometries
05E15 Combinatorial aspects of groups and algebras (MSC2010)
05A18 Partitions of sets
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